Lemonade from LemonsInformation Design and Adverse Selectionlowast

Navin Kartikdagger Weijie ZhongDagger

December 2 2019

Abstract

A seller posts a price for a single object The sellerrsquos and buyerrsquos values may be

interdependent We characterize the set of payoff vectors across all information struc-

tures Simple feasibility and individual-rationality constraints identify the payoff set

The buyer can obtain the entire surplus often other mechanisms cannot enlarge the

payoff set We also study payoffs when the buyer is more informed than the seller and

when the buyer is fully informed All three payoff sets coincide (only) in notable special

casesmdashin particular when there is complete breakdown in a ldquolemons marketrdquo with an

uninformed seller and fully informed buyer

Incomplete draft Please do not circulate without permission

lowastWe thank Laura Doval and Pietro Ortoleva for useful conversations and various audiences for helpfulcomments

daggerDepartment of Economics Columbia University E-mail nkartikgmailcomDaggerCowles Foundation Yale University E-mail wz2269columbiaedu

1 IntroductionMotivation Asymmetric information affects market outcomes both in terms of efficiencyand distribution For example adverse selection can generate dramatic market failure (Ak-erlof 1970) or skew wages in labor markets (Greenwald 1986) while consumers can secureinformation rents from a monopolist (Mussa and Rosen 1978) Much existing work takesthe market participantsrsquo information as given and studies properties of a particular marketstructure or mechanism or tackles these properties across various mechanisms

This paper instead asks what is the scope for different market outcomes as the partici-pantsrsquo information varies We are motivated by the fact that in the digital age the nature ofinformation that sellers (eg Amazon) have about consumers is ever changing Consumersand regulators do have some control over this information of course In some cases it isplausible that a sellerrsquos information is a subset of the consumerrsquos But in other cases theseller may well know more about the consumerrsquo value for a product or at least have someinformation the consumer herself does not This is especially relevant for products theconsumer is not already familiar with Indeed numerous firms make tailored recommen-dations to consumers about products they carry With social media and other sources ofinformation diffusion the possible correlation in information across two sides of a marketseems truly limitless

Our paper fixes a simple canonical market mechanism and studies the possible marketoutcomes across a variety of information structures including all of them We model twoparties Buyer and Seller who can trade a single object Buyerrsquos value for the object is arandom v isin [v v] sub R Sellerrsquos cost of providing the object or equivalently her valuefrom not trading is c(v) le v Thus values may be interdependent but trade is alwaysefficient1 The environment ie the function c(v) and the distribution of v is commonlyknown Seller posts a price p isin R and Buyer decides whether to buy

This stylized setting subsumes a variety of possibilities depending on the shape of thecost function c(middot) and the partiesrsquo information about the value v With an informed Buyerand an uninformed Seller there is adverse selection when c(middot) is increasing while there isfavorable or advantageous selection when c(middot) is decreasing2 If on the other hand Seller isbetter informed than Buyer signaling becomes relevant the price can serve as credible sig-nal if the two partiesrsquo information is suitably correlated (eg Bagwell and Riordan 1991)

1 We describe here our baseline model presented in Section 2 Section 4 discusses extensions includingcases when Buyerrsquos value does not pin down Sellerrsquos cost and when trade is not always efficient

2 Jovanovic (1982) uses the term lsquofavorable selectionrsquo Einav and Finkelstein (2011) use lsquoadvantageousrsquoand discuss both adverse and advantageous selection in the context of insurance markets with empiricalreferences to empirical evidence on both

1

A constant c(middot) captures an environment in which there no uncertainty about Sellerrsquos costthis is the canonical monopoly pricing problem when Seller is uninformed about v andthird-degree price discrimination when Seller has some partial information while Buyer isbetter informed

Summary of results For any given environment (ie Sellerrsquos cost function and the dis-tribution of Buyerrsquos values) we seek to identify the possible market outcomes Specificallywe are interested in the ex-ante expected payoffs that obtain given sequentially rational be-havior in an equilibrium under some information structure3 We provide three results eachof which covers a different class of information structures Our main theorems are Theo-rem 1Theorem 1lowast which impose no restrictions on information and Theorem 2 whichapplies when Buyer is better informed than Seller in the sense of Blackwell (1953) in factTheorem 2 applies more broadly as elaborated later Theorem 3 concerns a fully-informedBuyer (ie who knows his value v) We view each of these three cases as intellectuallysalient and economically relevant Plainly these payoff sets must be ordered by set inclu-sion Theorem 1rsquos is the largest Theorem 2rsquos is intermediate and Theorem 3rsquos the smallestFigure 1 below summarizes

Akerlof

Full or No Information

Fully-Informed BuyerABC

Uninformed Seller(More Informed Buyer)

ADE

All Info StructuresAFG

Expected SurplusFrontier

C

E

G

0πb

Maxv-E[c(v)]0 F

A

B

D

πs

Figure 1 Outcome under different restrictions on information structures

In the figurersquos axes πb and πs represent respectively Buyerrsquos and Sellerrsquos ex-ante ex-

3 As detailed in Section 2 an information structure specifies a joint distribution of private signals foreach party conditional on the value v This induces an extensive-form game of incomplete informationOur primary solution concept is weak Perfect Bayesian equilibrium we also address refinements for ourconstructive arguments

2

pected utilities or payoffs (for readability we often drop the ldquoexpectedrdquo qualifier) Theno-trade payoffs are normalized to zero The three triangles AFG ADE and ABC depictTheorems 1ndash3 respectively That payoffs must lie within AFG is straightforward Buyercan guarantee himself a payoff of zero by not purchasing Seller can guarantee herself notonly a payoff of zero (by posting any price p gt v which will not be accepted) but alsov minus E[c(v)] (by pricing at or just below v which will be accepted) and the sum of pay-offs cannot exceed the trading surplus E[v minus c(v)] We refer to the first two constraints asindividual rationality and the third as feasibility

Theorem 1 says that every feasible and individually-rational payoff pair can be imple-mented ie obtains in an equilibrium under some information structure It is immediatethat point A obtains whens both parties learn v (full information) or neither party has anyinformation (no information) More interestingly at the point G trade occurs with probabil-ity one and Buyer obtains the entire surplus despite Seller posting the price While perhapssurprising this outcome obtains with sparse information structures For simplicitly sup-pose E[c(v)|v gt v] le maxvE[c(v)] Then Buyer can be uninformed while Seller learnswhether v = v or v gt v In equilibrium Seller prices at p = maxvE[c(v)] regardless ofher signal and Buyer purchases If Seller were to deviate to a higher price Buyer wouldreject because he believes v = v Subsection 31 explains how a single information struc-ture in fact implements every point in the triangle AFG Theorem 1lowast there discusses how aricher information structure using imperfectly-correlated signals ensures implementationin Kreps and Wilsonrsquos (1982) sequential equilibrium in discretized versions of the model

Turning to Figure 1rsquos triangle ADE Theorem 2 establishes that the payoff pair in anyequilibrium when Buyer is better informed than Seller arises in an equilibrium of an(other)information structure in which Seller is uninformed4 In other words there is no loss ofgenerality in studying an uninformed Seller so long as Buyer is better informed Whenc(middot) is increasing such information generates a game with adverse selection when c(middot)is decreasing there is favorable selection Sellerrsquos payoff along the line segment DE isthe lowest payoff she can get in any information structure in which she is uninformed5

Theorem 2 further establishes that any point within the ADE triangle can be implementedwith some such information structure by suitably varying Buyerrsquos information In fact weshow that higher slices of the triangle (ie those corresponding to larger Sellerrsquos payoff)

4 We stipulate that a better-informed Buyer does not update his value from Sellerrsquos price (even off theequilibrium path) in line with the ldquono signaling what you donrsquot knowrdquo requirement (Fudenberg and Tirole1991) that is standard in versions of Perfect Bayesian Equilibrium and implied by sequential equilibrium

5 It is because Seller cannot commit to the price as a function of her signal that she can be harmed (iereceive a payoff lower than that on the DE segment) with more information However Theorem 2 assuresthat Seller is not harmed so long as Buyer is better informed

3

can always be implemented by reducing Buyerrsquos information in the sense of Blackwell(1953) We also explain in Subsection 32 why the triangle ADE actually characterizes allpayoffs that can obtain when Buyer does not update from Sellerrsquos price even if Buyer isnot better informed than Seller

Finally the ABC triangle in Figure 1 depicts Theorem 3 which characterizes all payoffpairs when Buyer is fully informed ie learns v We use the term ldquoAkerlofrdquo to describe afully-informed Buyer and an uninformed Seller as these information structures are stan-dard in the adverse-selection literature the corresponding payoff pair is marked as suchin the figure Depending on the environmentrsquos primitives the Akerlof point can be any-where on the segment BC including at the extreme points Any feasible payoff pair thatsatisfies Buyerrsquos individual rationality and gives Seller at least her Akerlof payoff can beimplemented with a fully-informed Buyer by suitably varying Sellerrsquos information

An implication of Theorems 1ndash3 is that it is without loss in terms of ex-ante equilibriumpayoffs to focus on information structures in which Buyer is fully informed if and onlyif Sellerrsquos Akerlof payoff coincides with her individual rationality constraint This coinci-dence occurs if and only if the Akerlof market can have full trade (Seller prices at p = v

and gets payoff v minus E[c(v)] ge 0) or no trade (the price is p ge v and both partiesrsquo payoffsare 0) As detailed in Remark 4 of Subsection 33 in all other cases the point B in Fig-ure 1 is distinct from the point D (and hence also F ) which means that Buyer can obtaina higher payoff with less-than-full information while keeping Seller uninformed Further-more under a reasonable condition if Sellerrsquos individual rationality constraint is zero (iev le E[c(v)]) then point D is also distinct from F see Remark 3 in Subsection 32 WhenD and F are distinct maximizing Buyerrsquos payoff ie achieving point G requires Seller tohave some information Buyer does not and an equilibrium with price-dependent beliefsafter conditioning on his signal Buyer must update about v from the price either on or offthe equilibrium path

Related literature Our perspective and results are most closely related to BergemannBrooks and Morris (2015) Roesler and Szentes (2017) and Makris and Renou (2018 Sec-tion 4) These papersmdashonly the relevant section of the third papermdashstudy the monopolypricing problem in which there is uncertainty only about Buyerrsquos valuation This is thespecial case of our interdependent-values model with a constant function c(v) le v6

6 The online appendix of Roesler and Szentes (2017) relaxes the assumption that there are always gainsfrom trade but maintains no uncertainty about Sellerrsquos cost Related to Roesler and Szentes (2017) are alsoDu (2018) and Libgober and Mu (2019) who consider worst-case profit guarantees for Seller in static and dy-namic environments respectively Terstiege and Wasser (2019) qualify Roesler and Szentes (2017) by allowing

4

Bergemann et al (2015) assume Buyer is fully informed and hence can only vary Sellerrsquosinformation Our Theorem 3 corresponding to triangle ABC in Figure 1 is a generaliza-tion of their main result to our environment the key step in our proof methodology is toconstruct ldquoincentive compatible distributionsrdquo which reduces to their ldquoextreme marketsrdquoin the monopoly-pricing environment Another contribution of our result is to ensure ap-proximately unique implementation in a sense explained in Subsection 33

Roesler and Szentes (2017) assume Seller is uninformed and only vary Buyerrsquos infor-mation For the monopoly-pricing environment they derive one part of our Theorem 2viz they identify the triangle ADE in Figure 1 as the implementable set given Seller isuninformed Even for this result our methodology is quite different from theirs becausewe do not stipulate a linear c(middot) function our methodology delivers new insights includ-ing that noted in Remark 2 of Subsection 32 When we specialize to a linear c(middot) we canobtain a sharper characterization of the point E which extends Roesler and Szentesrsquo char-acterization of Buyer-optimal information to an interdependent-values environment seeProposition 2 in Subsection 43

While our main interest is in interdependent values our results provide new insightseven for monopoly pricing Theorem 2 implies that the Roesler and Szentes (2017) boundsare without loss so long as Buyer is better informed than Seller or more generally in equi-libria in which Buyerrsquos belief is price independent (after conditioning on his own signal)On the other hand Theorem 1 establishes that any feasible and individually rational pay-off pair can be implemented absent these restrictions in particular Buyer may even get allthe surplus This latter point has a parallel with Makris and Renou (2018) As an applica-tion of their general results on ldquorevelation principlesrdquo for information design in multi-stagegames Makris and Renoursquos (2018) Proposition 1 deduces an analog of our Theorem 1 forthe (independent values) monopoly pricing problem We share with Makris and Renou anemphasis on sequential rationality7 we differ in showing that a single information struc-ture implements all payoffs in the relevant triangle and also in establishing in Theorem 1lowast

off-the-equilibrium-path belief consistency in the sense of sequential equilibrium (Krepsand Wilson 1982) in discretizations

Other authors have studied different aspects of more specific changes of information inadverse-selection settings maintaining that one side of the market is better informed than

Seller to supply Buyer with additional information although Seller cannot have any private information ofher own

7 Makris and Renou use an apparatus of ldquosequential Bayes correlated equilibriumrdquo which we do notThey draw a contrast with Bergemann and Morrisrsquos (2016) Bayes correlated equilibrium In our approachnote that Sellerrsquos individual rationality constraint described earlier hinges when E[c(v)] lt v on Buyerrsquosbehavior being sequentially rational even off the equilibrium path

5

the other Levin (2001) identifies conditions under which the volume of trade decreaseswhen one party is kept uninformed and the otherrsquos information become more effective inthe sense of Lehmann (1988) see also Kessler (2001) Assuming a linear payoff structureBar-Isaac Jewitt and Leaver (2018) consider how certain changes in Gaussian informationaffect the volume of trade surplus and a certain quantification of adverse selection

Finally Garcia Teper and Tsur (2018) solve for socially optimal information provisionmdashrather than characterizing the range of market outcomesmdashin an insurance setting with ad-verse selection owing to a cross-subsidization motive full information disclosure is typi-cally not optimal

The rest of the paper proceeds as follows We introduce our model equilibrium con-cept(s) and certain classes of information structures in Section 2 Section 3 presents themain results implementable payoffs when the information structure is arbitrary or varieswithin canonical classes Section 4 contains discussion and extensions Proofs and someadditional material are contained in the Appendices

2 Model

21 Primitives

Seller may sell an indivisible good to Buyer Buyerrsquos value for the good is v isin V sub Rwhere V is a compact (finite or infinite) set with v equiv minV lt maxV equiv v The valuev is drawn from a probability measure micro with support V Sellerrsquos cost of production isgiven by a function c(v)8 We assume c V rarr R is continuous v minus c(v) ge 0 for all vand E[v minus c(v)] gt 0 So trading surplus is nonnegative for all v and positive for a positivemeasure of v (Throughout expectations are with respect to the prior measure micro unlessindicated otherwise lsquopositiversquo means lsquostrictly positiversquo and similarly elsewhere) Note thatfunction c(v) need not be monotonic We call Γ = (c micro) an environment We refer to anenvironment with a constant c(middot) function as that of monopoly pricing

An information structure consists of signal spaces for each party and a joint signal dis-tribution (We abuse notation and sometimes write lsquodistributionrsquo even though lsquomeasurersquowould be more precise) Formally there is a probability space (ΩF P ) complete and sep-arable metric spaces Ts and Tb (with measurable structure given by their Borel sigma alge-bras) and an integrable function X Ω rarr TstimesTbtimesV We hereafter suppress the probabilityspace and define with an abuse of notation P (D) = P (Xminus1(D)) for any D isin Ts times Tb times V

measurable Each realization of random variable X is a triplet (tb ts v) where tb isin Tb

8 Subsection 41 discusses Sellerrsquos cost being stochastic even conditional on v

6

is Buyerrsquos signal and ts isin Ts is Sellerrsquos signal For i isin s b v let Pi denote the corre-sponding marginal distribution of P on dimension of Ti with the convention Tv equiv V Werequire Pv = micro this is the iterated expectation or Bayes plausibility requirement Denotean information structure by τ

The environment Γ and information structure τ define the following game

1 The random variables (tb ts v) are realized Signal tb is privately observed by Buyerand signal ts privately observed by Seller Neither party observes v

2 Seller posts a price p isin R

3 Buyer accepts or rejects the price If Buyer accepts his von-Neumann Morgensternpayoff is v minus p and Sellerrsquos is p minus c(v) If Buyer rejects both partiesrsquo payoffs arenormalized to 0

Note that because the signal spaces Tb and Ts are abstract and the two partiesrsquo signalscan be arbitrarily correlated conditional on v there is no loss of generality in assuming thateach party privately observes their own signal For example public information can becaptured by perfectly correlating (components of) tb and ts

22 Strategies and Equilibria

In the game defined by (Γ τ) denote Sellerrsquos strategy by σ and Buyerrsquos by α FollowingMilgrom and Weber (1985) we define σ as a distributional strategy σ is a joint distributionon R times Ts whose marginal distribution on Ts must be the Sellerrsquos signal distribution Soσ(middot|ts) is Sellerrsquos price distribution given her signal ts9 Buyerrsquos strategy α R times Tb rarr[0 1] maps each price-signal pair (p tb) into a trading probability A strategy profile (σα)

induces expected utilities for Buyer and Seller (πb πs) in the natural way

πb =

983133(v minus p)α(tb p)σ(dp|ts)P (dts dtb dv)

πs =

983133(pminus c(v))α(tb p)σ(dp|ts)P (dts dtb dv)

Our baseline equilibrium concept is weak Perfect Bayesian equilibrium Since Sellerrsquosaction is not preceded by Buyerrsquos we can dispense with specifying beliefs for Seller ForBuyer it suffices to focus on his belief about the value v given his signal and the price wedenote this distribution by ν(v|p tb)

9 Here σ(middot|ts) is the regular conditional distribution which exists and is unique almost everywhere becauseTs is a standard Borel space (Durrett 1995 pp 229ndash230) Similarly for subsequent such notation we dropldquoalmost everywhererdquo qualifiers unless essential

7

Definition 1 A strategy profile (σα) and a belief ν(v|p tb) is a weak perfect Bayesian equilib-rium (wPBE) of game (Γ τ) if

1 Buyer plays optimally at every information set given his belief

α(p tb) =

983099983103

9831011 if Eν(v|ptb)[v] gt p

0 if Eν(v|ptb)[v] lt p

2 Seller plays optimally

σ isin argmaxσ

983133(pminus c(v))α(p tb)σ(dp|ts)P (dts dtb dv)

3 Beliefs satisfy Bayes rule on path for every measurable D sube Rtimes Ts times Tb times V

983133

D

ν(dv|p tb)σ(dp|ts)P (dts dtb V ) =

983133

D

σ(dp|ts)P (dts dtb dv)

Notice that we have formulated Sellerrsquos optimality requirement ex ante but Buyerrsquos ateach information set The latter is needed to capture sequential rationality The formeris for (notational) convenience this choice is inconsequential because Seller moves beforeBuyer

Hereafter ldquoequilibriumrdquo without qualification refers to a wPBE As is well understoodwPBE permits significant latitude in beliefs off the equilibrium path We will discuss re-finements when appropriate

23 Implementable Payoffs and Canonical Information Structures

We now define the set of implementable equilibrium outcomesmdashthat is the payoffs thatobtain in some equilibrium under some information structuremdashand some canonical classesof information structures

For a game (Γ τ) let the equilibrium payoff set be

Π(Γ τ) equiv (πb πs) exist wPBE of (Γ τ) with payoffs (πb πs)

Denote the class of all information structures by T and define

Π(Γ) equiv983134

τisinT

Π(Γ τ)

8

That is for environment Γ Π(Γ) is the set of all equilibrium payoff pairs that obtain undersome information structure

Uninformed Seller An information structure has uninformed Seller if Ts is a singletonSellerrsquos own signal contains no information about Buyerrsquos value v and hence neither aboutSellerrsquos cost c(v) When discussing such information structures we write the associateddistribution as just P (tb v) and Sellerrsquos strategy as just σ(p) omitting the argument ts inboth cases The class of all uninformed-Seller information structure is denoted Tus

Fully-informed Buyer An information structure has fully-informed Buyer if Buyerrsquos signalfully reveals his value v Formally this holds if Tb = V and the conditional distribution onV P (middot|tb) satisfies P (tb|tb) = 1 We denote the class of fully-informed-Buyer informationstructures Tfb

More-informed Buyer An information structure has more-informed Buyer if Buyer hasmore information than Seller Formally this holds when v and ts are independent condi-tional on tb ie for any measurable Ds sub Ts and Dv sub V P (DscapDv|tb) = P (Ds|tb)P (Dv|tb)10

Another way to interpret this requirement is that random variable tb must be statisticallysufficient for ts with respect to v ie tb is more informative than ts about v in the sense ofBlackwell (1953) We denote the class of more-informed-Buyer information structures byTmb Naturally information structures with uninformed Seller or fully-informed Buyer arecases of more-informed Buyer both Tus and Tfb are subclasses of Tmb

No updating from price For more-informed-Buyer information structures it is desirableto impose further requirements on Buyerrsquos equilibrium belief Since Sellerrsquos price can onlydepend on her own signal and this signal contains no additional information about v givenBuyerrsquos signal the price is statistically uninformative about v given Buyerrsquos signal Con-sequently Buyerrsquos posterior belief should be price independent once his signal has beenconditioned upon Formally regardless of the price p the equilibrium belief ν(middot|p tb) mustsatisfy 983133

ν(dv|p tb)P (dtb Ts V ) =

983133P (dtb Ts dv)

We refer to this condition as price-independent belief Note that the condition is meaningfulregardless of whether Buyer is more informed than Seller In a more-informed-Buyer in-formation structure price-independent belief would be implied by the ldquono signaling what

10 We write sub to mean ldquoweak subsetrdquo

9

you donrsquot knowrdquo requirement (Fudenberg and Tirole 1991) frequently imposed in versionsof perfect Bayesian equilibrium and the concept of sequential equilibrium (Kreps and Wil-son 1982) in finite versions of our setting11

Some notation will be helpful Define

Πlowast(Γ τ) equiv (πb πs) exist wPBE of (Γ τ) with price-independent belief and payoffs(πb πs)

Πlowast(Γ) equiv983134

τisinT

Πlowast(Γ τ)

Πlowasti (Γ) equiv

983134

τisinTi

Πlowast(Γ τ) for i = us fbmb

So Πlowast and Πlowast are analogous to the implementable payoff sets Π and Π defined earlier butrestricted to equilibria with price-independent beliefs Πlowast

us Πlowastfb and Πlowast

mb are the imple-mentable payoff sets when further restricted to uninformed-Seller fully-informed-Buyerand more-informed-Buyer information structures Plainly for any environment Γ

Πlowastus(Γ) cupΠlowast

fb(Γ) sub Πlowastmb(Γ) sub Πlowast(Γ) sub Π(Γ)

3 Main Results

Our goal is to characterize equilibrium payoff pairs across information structures in anarbitrary environment Γ In particular we seek to characterize the five sets Π(Γ) Πlowast(Γ)Πlowast

mb(Γ) Πlowastus(Γ) and Πlowast

fb(Γ) Let

S(Γ) equiv E[v minus c(v)]

be the (expected) surplus from trade in environment Γ This quantity will play an impor-tant role

31 All Information Structures

Define Sellerrsquos payoff guarantee as

πs(Γ) equiv max v minus E [c(v)] 0

11 An example clarifying our terminology may be helpful If both Seller and Buyer are fully informed of vthen the natural equilibriummdashthe unique sequential equilibrium in a finite version of the gamemdashhas Sellerpricing at p = v and Buyerrsquos belief being degenerate on v regardless of Sellerrsquos price This equilibrium hasprice-independent belief even though ex-ante Sellerrsquos price are Buyerrsquos belief are perfectly correlated Thepoint is that Buyerrsquos belief does not depend on price conditional on his signal

10

To interpret this observe that it is optimal for Buyer to accept the price v no matter hisbelief Therefore Seller can guarantee herself the (expected) profit v minus E[c(v)] no matterwhat the information structure is More precisely she can guarantee v minus E[c(v)] minus ε forany ε gt 0 since sequential rationality requires Buyer to accept any price v minus ε SimilarlySeller can also guarantee zero profit offering a price p gt v Hence Sellerrsquos payoff in anyequilibrium of any information structure must be at least the payoff guarantee πs(Γ) above

On the other hand Buyer can guarantee himself the payoff πb = 0 by rejecting all pricesIt follows that the implementable set Π(Γ) must satisfy three simple constraints 1) Sellerrsquosldquoindividual rationalityrdquo constraint πs ge πs(Γ) 2) Buyerrsquos ldquoindividual rationalityrdquo con-straint πb ge 0 and 3) the feasibility constraint πb + πs le S(Γ)

Our first result is that these individual rationality and feasibility constraints are alsosufficient for a payoff pair to be implementable

Theorem 1 Consider all information structures and equilibria

Π(Γ) =

983099983105983103

983105983101

πb ge 0

(πb πs) πs ge πs(Γ)

πb + πs le S(Γ)

983100983105983104

983105983102

Theorem 1 says that the set Π(Γ) corresponds to the triangle AFG in Figure 1 In par-ticular Buyer can receive the entire surplus beyond Sellerrsquos payoff guarantee This is sur-prising as Seller has substantial bargaining power Note that when v le E[c(v)] an entirelyreasonable condition Sellerrsquos payoff guarantee is zero12

The proof of Theorem 1 is in fact straightforward Suppose for expositional simplicityv ge E[c(v)] Fix the trivial information structure in which neither player receives any infor-mation and consider the following family of strategy profiles Seller randomizes betweentwo prices some pl isin [vE[v]] and ph = E[v] Buyer accepts pl with probability one andaccepts ph with probability α(ph) where α(ph) isin [0 1] is specified to make Seller indifferentbetween the two prices That is α(ph)(ph minus E[c(v)]) = pl minus E[c(v)] The expected payoffsfrom this strategy profile are

πb = σ(pl)(E[v]minus pl) and πs = pl minus E[c(v)]

As pl traverses the interval [vE[v]] Sellerrsquos payoff πs traverses [πs(Γ) S(Γ)] Given any

12 In monopoly pricing with c(middot) = v Sellerrsquos payoff guarantee of zero is lower than the revenue guaranteeidentified in Du (2018 Section 5) which is typically positive Dursquos notion is different from ours

11

pl Buyerrsquos payoff πb traverses [0 S(Γ)minusπs] as σ(pl) traverses [0 1] Therefore the proposedstrategy profiles induce all the payoff pairs stated in Theorem 1

We are left to specify beliefs ν for Buyer After prices pl and ph Buyer holds the priorbelief micro After any other (necessarily off-path) price Buyerrsquos belief is that v = v and soBuyer rejects all prices p isin [vinfin)pl ph It is straightforward to confirm that the specified(σα ν) constitute a wPBE

To get more insight into this construction consider its implication for the monopolypricing with c(middot) = v The equilibrium with pl = v (hence α(ph) = 0) and σ(pl) = 1

corresponds to the monopolist deterministically pricing at v and Buyer purchasing Giventhat both sides of the market receive no information why doesnrsquot the monopolist deviate toany price in (vE[v]) The reason is that in this equilibrium the consumer will then not buybecause he updates to v = v Such belief is compatible with wPBE because the equilibriumconcept places no restrictions on off-path beliefs This may seem like a game-theoreticmisdirection Buyerrsquos belief is not consistent with ldquono signaling what you donrsquot knowrdquoPut differently since we have a (weakly) more-informed Buyer information structure weought to impose the price-independent belief condition described in Subsection 23 thatwould imply Buyer must purchase at any price p lt E[v]

But the message of Theorem 1 does not rely on the permissiveness of wPBE To illus-trate continue with the above monopoly-pricing environment and suppose v has positiveprior probability Consider Buyer remaining uninformed but Seller learning whether v = v

or v gt v Now Buyerrsquos off-path belief that v = v is consistent with ldquono signaling whatyou donrsquot knowrdquo More generally using richer information structures we can prove thatany payoff pair identified in Theorem 1 can be approximately implemented as a sequentialequilibrium (Kreps and Wilson 1982) in a suitably discretized game

Theorem 1lowast Fix any ε gt 0 There is a finite information structure and a finite price grid inducinga game with a set of sequential equilibrium payoffs that is an ε-net of Π(Γ)13

In fact the proof of Theorem 1lowast establishes even more the sequential equilibria in thediscretized games satisfy a natural version of the D1 refinement (Cho and Kreps 1987) Werelegate the logic to the Appendix but mention here that we use imperfectly-correlatedsignals for Buyer and Seller14

13 An information structure is finite if the signal spaces Tb and Ts are finite Sequential equilibrium isdefined in the obvious way for the ldquoinducedrdquo finite game where Nature directly draws (tb ts) rather thanfirst drawing v and playersrsquo payoffs are defined directly using (tb ts) For X sub R2 and ε gt 0 the set A sub Xis an ε-net of X if for each x isin X there is a isin A such that 983042xminus a983042 lt ε where 983042middot983042 is the Euclidean distance

14 The idea behind D1 is to ask for any off-path price whether one type of Seller would deviate for any

12

Even when our environment is specialized to monopoly pricing it is worth highlightingtwo contrasts between Theorem 11lowast and results of Bergemann et al (2015) and Roeslerand Szentes (2017) First we find that by not restricting the monopolist to be uninformedthe implementable payoff set typically expands rather dramatically trade can be efficientwith the monopolist securing none of the surplus beyond her payoff guarantee (πs) whichmay be zero (Roesler and Szentes establish implicitly that the implementable set withan uninformed monopolist is a superset of Bergemann et alrsquos which has a fully-informedconsumer) We will see in Subsection 32 that what is crucial to this expansion is price-dependent beliefs In particular the proof of Theorem 1lowast uses an information structure inwhich Buyer is not better informed than Seller mdash if he were then sequential equilibriumwould imply price-independent beliefs Second Theorem 1lowast establishes that for a givenε gt 0 a single information structure (and price grid) can be used to approximate the entirepayoff set Π(Γ) analogously to the construction described after Theorem 1 that used asingle information structure15 Bergemann et al (2015) and Roesler and Szentes (2017) onthe other hand vary information structures to span their payoff sets

32 More-informed Buyer and Price-independent Beliefs

In some economic settings it is plausible that Buyer is more informed than Seller Howdoes a restriction to such information structures ie τ isin Tmb affect the implementablepayoff set It turns out that what is in fact crucial is price-independent beliefs We have ex-plained earlier why it is desirable to impose this condition when Buyer is more informedbut that the condition is well defined even otherwise If Buyer is not more informed thanSeller price-independent beliefs ought to be viewed as an equilibrium restriction in gen-eral To reduce repetition readers should be bear in mind for the rest of this subsectionthe qualifier ldquowith price-independent beliefsrdquo applies unless stated explicitly otherwise

It is useful to define

πuss (Γ) equiv inf πs exist(πb πs) isin Πlowast

us(Γ) (1)

as the lowest payoff that an uninformed Seller can obtain no matter Buyerrsquos information(among equilibria with price-independent beliefs) Plainly πus

s (Γ) ge πs(Γ) In monopoly

Buyer mixed response that another type would Our construction has multiple Buyer types that are imper-fectly correlated with Seller types So different types of Seller have different beliefs about Buyer types Thisblunts the power of dominance considerations to the point where D1 does not exclude any Seller type fromthe support of Buyerrsquos off-path belief

15 In fact if one lets the price grid vary with ε then a single information structure implements exactlyrather than approximately in sequential equilibrium all payoffs ε-away from the boundary of Π(Γ) See in the Appendix for a formal statement

13

pricing with V = [v v] and c(middot) = v Roesler and Szentesrsquos (2017) characterization of theconsumer-optimal information structure identifies πus

s establishing that πuss gt πs If there

is no trade due to adverse selection when Seller is uniformed and Buyer has some informa-tion then πus

s = πs = 0 We do not have a general explicit formula for πuss Subsection 43

provides it for linear c(middot) Nonetheless we establish next that (i) the only additional restric-tion on equilibrium payoffs imposed by price-independent beliefs is a lower bound of πus

s

for Seller and (ii) uninformed-Seller information structures implement all such payoffs

Theorem 2 Consider equilibria with price-independent beliefs

1 Πlowast(Γ) = Πlowastmb(Γ) = Πlowast

us(Γ)

2 Πlowastus(Γ) = (πb πs) isin Π(Γ) πs ge πus

s (Γ)

3 For any (πb πs) isin Πlowastus(Γ) with πs gt πus

s (Γ) there is τ isin Tus with Π(Γ τ) = (πb πs)

Remark 1 We believe the substance of Theorem 2 would hold using discretizations and se-quential equilibria analogous to Theorem 1lowast As previously noted sequential equilibriumimplies price-independent beliefs when Buyer is more informed than Seller

To digest Theorem 2 note that Πlowast(Γ) sup Πlowastmb(Γ) sup Πlowast

us(Γ) is trivial So part 1 of thetheorem amounts to establishing the reverse inclusions The intuition for thismdashgiven part2rsquos characterization of Πlowast

usmdashis fairly straightforward with price-independent beliefs ad-ditional information cannot harm Seller even though it could alter the set of equilibria SoSellerrsquos lowest payoff obtains when she is uninformed

The characterization in part 2 of payoffs with an uninformed Seller corresponds to thetriangle ADE in Figure 1 Part 3 of the theorem assures ldquounique implementationrdquo of allimplementable payoffs satisfying πs gt πus

s (Γ) That is for any such payoff pair there is anuninformed-Seller information structure such that all equilibria (with price-independentbeliefs) induce exactly that payoff pair Unique implementation is appealing for multiplereasons one of which is that it obviates concerns about which among multiple payoff-distinct equilibria is more reasonable For example Ravid Roesler and Szentes (2019)bring to the fore the issue of equilibrium multiplicity in Roesler and Szentesrsquos (2017) infor-mation structure

Let us describe how we obtain the characterization of Πlowastus(Γ) and unique implementa-

tion There are two steps The first ensures that there is some information structure callit τ lowast isin Tus that implements Sellerrsquos payoff πus

s (Γ) That is we ensure that the infimum

14

in (1) is in fact a minimum16 While this argument is technical knowing τ lowast exists is usefulin what follows The second and economically insightful step is to construct informa-tion structures that implement every point in the triangle Πlowast

us(Γ) by suitably garbling theinformation structure τ lowast The construction is illustrated in Figure 2 Consider the distri-bution of Buyerrsquos posterior mean of his valuation v in information structure τ lowast (Givenprice-independent beliefs Buyerrsquos posterior mean is sufficient for his decision) For sim-plicity suppose this posterior-mean distribution has a density as depicted by the red curvein Figure 2 Fix any (πb πs) isin Πlowast

us(Γ)

First there is some number zlowast such that πb + πs is the total surplus from trading onlywhen Buyerrsquos posterior mean is greater than zlowast Next there is some price plowast ge zlowast suchthat Sellerrsquos payoff is πs if all these trades were to occur at price plowast17 Note that plowast must beno larger than the expected Buyer posterior mean conditional on that being above zlowast forotherwise πb lt 0 We claim that the information structure τ lowast can be garbled so that plowast isan equilibrium price and trade occurs only when Buyerrsquos posterior mean is greater thanzlowast The garbling is illustrated in Figure 2 by the blue distribution There is one signal that

pzv

ν(v)

Figure 2 Construction of information structure in Theorem 2

Buyer receives when the original posterior mean is between zlowast and plowast and also receiveswith some probability when the original posterior mean is above plowast The probability ischosen to make the posterior mean from this signal exactly plowast Apart from this one newsignal Buyer receives the original signal in τ lowast Plainly this is a garbling of τ lowast and hence isfeasible

16 The difficulty is in establishing suitable continuity Uninformed-Seller information structures can beviewed as probability measures over Buyerrsquos beliefs with convergence in the sense of the weak topologyThis topology ensures continuity with respect to probability measures of expectations of continuous orat least Lipschitz (and bounded) functions However Sellerrsquos expected payoff is not the expectation of aLipschitz function as Sellerrsquos profit is truncated at the price she charges

17 That plowast ge zlowast follows from πs ge πuss (Γ) as πs(Γ) itself is weakly larger than Sellerrsquos payoff from posting

price zlowast (and thus trading with the same set of Buyer posterior means) under information structure τlowast

15

Figure 2 makes clear why the new information structure has an equilibrium with priceplowast and Buyer breaking indifference in favor of trading (i) Sellerrsquos profit from posting anyprice below zlowast is the same as under τ lowast and hence no larger than πus

s (Γ) (ii) similarly Sellerrsquosprofit from posting any price above plowast is no higher than some fraction of πus

s (Γ) and (iii)any price between zlowast and plowast is worse that price plowast Moreover since Sellerrsquos profit fromoffering any price other than plowast is no more than πus

s (Γ) it follows that when πs gt πuss (Γ)

Buyer must break indifference as specified for Seller to have an optimal price and theequilibrium payoffs are unique

Remark 2 The above logic establishes that given any τ isin Tus that implements some (πb πs)τ can be garbled to uniquely implement any (πprime

b πprimes) isin Π(Γ) such that πprime

s gt πs That isan uninformed Sellerrsquos payoff can always be strictly raised and Buyerrsquos payoff reduced(strictly so long as it was not already zero) by garbling Buyerrsquos information

Remark 3 According to Theorem 11lowast and Theorem 2 uninformed-Seller information struc-tures cannot implement all implementable payoff pairs in an environment Γ if and only ifπuss (Γ) gt πs(Γ) This inequality fails in environment Γ if πus

s (Γ) = 0 since that impliesπuss (Γ) = πs(Γ) = 0 An example is when there is no trade due to adverse selection when

Seller is uniformed and Buyer has some information On the other hand it is sufficient forthe inequality that

v le E[c(v)] and forallv isin V c(v) lt v (2)

To see why (2) implies πuss (Γ) gt πs(Γ) notice that in any uninformed-Seller information

structure Seller can price at slightly less than Buyerrsquos highest posterior mean valuationand guarantee trade with only (a neighborhood of) that Buyer type If c(v) lt v for all v thisgives Seller a positive expected payoff and hence πus

s (Γ) gt 018 But the first inequality in (2)is equivalent to πs(Γ) = 0 We observe that Condition (2) is compatible with severe adverseselection resulting in very little trade when Seller is uninformed and Buyer (partially orfully) informed

33 Fully-Informed Buyer

We now turn to the third canonical class of information structures Buyer is fully in-formed of his value v As this is a special case of more-informed Buyer we maintain price-independent beliefs throughout this subsection

18 More precisely as V is compact c(v) lt v for all v implies there exists ε gt 0 such that v minus c(v) gt εGiven any uninformed-Seller information structure let mv be the highest posterior mean valuation in thesupport of the posterior means induced by Buyerrsquos signals So there is positive probability of Buyer signalswith posterior mean valuations at least mv minus ε2 By pricing at mv minus ε2 Sellerrsquos expected cost conditionalon trade is bounded above by mv minus ε and hence Sellerrsquos profit conditional on trade is at least ε2 gt 0 Itfollows that πus

s gt 0

16

Faced with a fully informed Buyer and any sequentially rational Buyer strategy an un-informed Seller can guarantee a profit level

πfbs (Γ) equiv sup

p

983133 v

p

(pminus c(v))micro(dv)

regardless of her information Plainly πfbs (Γ) ge πus

s (Γ) In monopoly pricing with c(middot) = vRoesler and Szentes (2017) have shown that πfb

s gt πuss if there is no trade due to adverse

selection when Buyer is fully informed and Seller is uniformed then πfbs = πus

s = 0 Belowwe establish that when Buyer is fully informed πfb

s is the only additional constraint onequilibrium payoffs Let Bε(πb πs) denote an ε-ball around (πb πs)

Theorem 3 Consider a fully informed Buyer and price-independent beliefs

1 Πlowastfb(Γ) = (πb πs) isin Π(Γ) πs ge πfb

s (Γ)

2 For any (πb πs) isin Πlowastfb(Γ) and any ε gt 0 there is τ isin Tfb with Π(Γ τ) sub Bε(πb πs)

The payoff set characterized in Theorem 3 corresponds to the triangle ABE in Figure 1Part 2 of the theorem establishes ldquoapproximately unique implementationrdquo for any point inthe triangle there is an information structure under which all equilibria yield payoff pairsin a neighborhood of that point

Here is the idea behind part 1 of Theorem 3 When Buyer is fully informed an informa-tion structure can be viewed as dividing vrsquos prior distribution micro into a set of microi that aver-age to micro with Seller informed of which microi she faces Theorem 3 is proven by establishingthat we can divide micro suitably so that against each microi Seller is indifferent between pricingat all prices in the support of microi including the price corresponding to πfb

s in that environ-ment We call such a microi a (Seller-price) incentive compatible distribution or ICD (Bergemannet al (2015) call their analogous construct an extreme market) The Appendix provides aldquogreedyrdquo algorithm to compute ICDs19

We can sketch how the algorithm works and construct a set of ICDs that average to theprior Suppose V = v1 v2 vK with vi lt vi+1 for i isin 1 K minus 1 and c(v) lt v forall v Given any small-enough mass of vK there is a unique mass of type vKminus1 that makesSeller indifferent between charging price vK and vKminus1 (If the mass is too low Seller prefersvK if it is too high she prefers vKminus1) Iterating down to keep Seller indifferent betweenall prices pins down an ICD Choose the maximum mass of type vK for which this worksRemove that ICD and repeat

19 The algorithm is defined for finite V and we take limits to handle the infinite case

17

Crucially whenever an ICD is removed the price corresponding to πfbs (Γ) remains op-

timal in the remaining ldquomarketrdquo this follows from the ICDrsquos defining property of Sellerindifference and an accounting identity Therefore Sellerrsquos profit in this segmentation ofICDs remains πfb

s (Γ) Moreover it is also optimal for Seller to always (ie for each microi) priceso that there is full trade or no trade Hence Buyerrsquos expected payoff can be either 0 or theentire surplus less πfb

s (Γ) It follows that the fully-informed Buyer information structuredefined by this set of ICDs implements point B and C in Figure 1 The entire triangle ABC

can then be implemented by convexification randomizing over this information structure(and two equilibria) and full information (where Seller obtains all the surplus)

The above logic of incentive compatible distributions reduces to that of Bergemann et al(2015) in the context of monopoly pricing Even in the monopoly pricing case Theorem 3part 2 strengthens the conclusions of Bergemann et al (2015) by ensuring approximatelyunique implementation

Remark 4 Theorems 1ndash3 imply that fully-informed-Buyer information structures imple-ment all implementable payoff pairs if and only if πfb

s (Γ) = πs(Γ) In that case trianglesAFG and ABC coincide in Figure 1 It follows that πfb

s (Γ) = πs(Γ) only when a fully-informed Buyer and uninformed Seller can result in full trade (v ge E[c(v)] and Sellerprices at v) or no trade (v le E[c(v)] and Seller prices at some p ge v) Interestinglywhen πfb

s (Γ) gt πs(Γ) fully-informed-Buyer information structures cannot even implementall payoff pairs implementable by uninformed-Seller information structures ie trianglesABC and ADE in Figure 1 are distinct That is when πfb

s (Γ) gt πs(Γ) there is an uninformed-Seller information structure that implements some πs lt πfb

s (Γ)20 Theorems 2ndash3 further im-ply in this case that Buyer can obtain a strictly higher payoff when he is not fully informed(and Seller is uninformed)

4 Discussion

41 Multidimensionality

Suppose Buyer and Sellerrsquos cost and valuation pair (c v) is a two-dimensional randomvariable distributed according to joint distribution micro with a compact support The ex-tension of our maintained assumption of commonly known gains from trade is for all

20 Pick any pprime gt v such that pprimeminusE[c(v)] lt πfbs (Γ) Following the construction described after Theorem 2 we

can mix all valuations v le pprime with a fraction λ gt 0 of valuations v gt pprime so that the mixture has posterior meanexactly pprime The remaining fraction 1 minus λ of valuations above pprime are revealed to Buyer With this uninformed-Seller information structure consider any equilibrium in which Buyer purchases when indifferent (Suchan equilibrium with price-independent belief exists) Sellerrsquos profit is at most (1 minus λ)πfb

s (Γ) from any pricep gt pprime and pprime minus E[c(v)] from price p = pprime Hence Sellerrsquos profit is strictly less than πfb

s (Γ)

18

(c v) isin Supp(micro) v ge c and E[v minus c] gt 0 An information structure is now a joint distribu-tion P (tb ts v c) whose marginal distribution on (v c) is micro

We claim that the content of our key results Theorem 1Theorem 1lowast Theorem 2 andTheorem 3 still hold To see why let v be the lowest valuation in the support of micro Sellerrsquosindividual rationality constraint is now maxv minus E[c] 0 as she can guarantee this by set-ting either a sufficiently high price or a price (arbitrarily close to) v regardless of her signalWe can define a cost function c(vprime) equiv Emicro[c|v = vprime] ge vprime This results in an environment sat-isfying all the maintained assumptions of our baseline model except that c(middot) may not becontinuous Recall that such continuity is not needed for proving Theorem 1 Theorem 1lowastBoth Theorem 2 and Theorem 3 use continuity of c(middot) to guarantee that Eν [c(v)] is a contin-uous function of ν isin ∆(V ) for a key convergence result However in the two-dimensionaltype environment Eν [c] is still a continuous function of ν isin ∆(V times C) Note that Theo-rem 3 uses upper semi-continuity of Sellerrsquos profit in price boundedness of c and c le v issufficient for such upper semi-continuity

42 Negative Trading Surplus

Returning to our baseline model we next discuss cases in which trade sometimes gen-erates negative surplus That is we drop the assumption that c(v) le v we do not requireE[v minus c(v)] gt 0 either Define Sλ(Γ) for λ isin [1infin) as

Sλ(Γ) equiv983133 v

v

[v minus c(v) + λ(v minus v)]+ micro(dv)

where [middot]+ equiv max middot 0 It is readily verified that S1(Γ) = E [[v minus c(v)]+] and limλrarr0

Sλ(Γ)λ =

E[v] minus v Allowing negative trading surplus does not affect our definition of wPBE Sothe notation Π(Γ) and πs(Γ) still have the same meanings as before The next propositionshows that Π(Γ) is now characterized by three constraints as before the two individualrationality constraints πs ge πs(Γ) and πb ge 0 and different now a Pareto frontier definedby all Sλ(Γ)

Proposition 1 Consider all information structures and equilibria when trade can generate negativesurplus

Π(Γ) =

983099983105983103

983105983101

πb ge 0

(πb πs) πs ge πs(Γ)

λπb + πs le Sλ(Γ) forallλ ge 1

983100983105983104

983105983102

Proposition 1 shows that when the trade can be inefficient it may be impossible to lower

19

Sellerrsquos profit without losing total surplus (ie πb + πs) This result is intuitive at the ex-tremes Suppose for simplicity v gt E[c(v)] If πs = πs(Γ) then since πs(Γ) = v minus E[c(v)]Seller must find it optimal to price at v and trade with probability one including whentrade generates negative surplus So Buyerrsquos payoff is E[v]minus v = limλrarrinfin Sλ(Γ)λ and totalsurplus is E[v minus c(v)] which includes the maximum amount of inefficient trade On theother hand an information structure that publicly reveals (only) whether trade is efficient(ie whether v ge c(v)) can implement efficient trade with all the surplus accruing to SellerIn this case Seller obtains S1(Γ) while Buyer obtains zero The proposition establishes thatin between S1(Γ) and πs(Γ) information structures and equilibria that lower Sellerrsquos profitmust reduce total surplus as the Pareto frontier Sλ(Γ) is a concave function with maximalslope minus1

The function capturing the Pareto frontier Sλ(Γ) is the maximal weighted sum of Buyerand Seller payoff assuming that Seller will trade at the low price p = v whenever it createspositive weighted total payoff Since we place higher weight on buyer (λ ge 1) S1(Γ) isthe upper bound for the weighted total payoff In the proof of Proposition 1 we showthat these bounds are tight mdash any payoff pair on the frontier can be implemented by someinformation structure The key observation is that whenever v minus c(v) + λ(v minus v) lt 0 itfollows that vminus c(v) lt (1minusλ)(vminus v) le 0 That is it is incentive compatible for Seller to notsell when it is common knowledge that v creates negative weighted total payoff Thereforewe can simply release a public signal indicating the sign of weighted total payoff and ifpositive induce Seller to sell at v which implies all bounds Sλ(Γ)

43 Linear Cost

When Sellerrsquos cost function c(v) is linear we can explicitly characterize an informationstructure that implements Sellerrsquos minimum implementable payoff πus

s (Γ) when Seller isuninformed (or per Theorem 2 when Buyer is better informed) subject to price-independentbeliefs We note that given the discussion in Subsection 41 a linear c(v) subsumes richerenvironments in which conditional expectations are linear including Gaussian environ-ments (cf Bar-Isaac et al 2018)

Condition 1 c(v) = λv + γ for some λ isin R

Let F (v) be the cumulative distribution function (CDF) corresponding to the prior mea-sure micro Let D(micro) be the set of all distributions whose CDF G satisfies

983133

V

vdG(v) =

983133

V

vdF (v) and983133 v

v

G(s)ds le983133 v

v

F (s)ds forallv isin V

20

That is D(micro) contains all distributions that are mean-preserving-contractions (MPC) of microIt is well known that D(micro) characterizes the set of distributions of Buyer posterior meansthat can be generated by any (uninformed-Seller) information structure

Definition 2 G is an incentive compatible distribution (ICD) if

983133

sgep

(pminus c(s))G(ds) = constant ge 0 forallp isin Supp(G)

That is when Buyerrsquos (mean) valuation is distributed according to an ICD G Seller isindifferent among all prices in the support of G assuming Buyer buys when indifferent Wefocus on a special family of ICDs whose supports are intervals [vlowast v

lowast] Given Condition 1such ICDs have analytical an expression

G(v) =

983099983105983105983103

983105983105983101

1minus983061

(1minus λ)v minus γ

(1minus λ)vlowast minus γ

983062 1λminus1

if λ ∕= 1

1minus evminusvlowast

γ if λ = 1

(3)

Given any vlowast and the above G(v) an upper bound vlowast is pinned down by the conditionEG[v] = EF [v] So a family of ICDs is parametrized by a single parameter vlowast and hence anICD is denoted by CDF 983144Gvlowast(v) with density 983144gvlowast(v) It is not difficult to verify that 983144Gvlowast(v) ismonotonic in vlowast for v in the support Consequently all ICDs are ordered according to theMPC order For vlowast isin [vE[v]] when vlowast increases the ICD increases in the MPC order

Proposition 2 Suppose Condition 1 is satisfied and let plowast equiv min983153vlowast983055983055 983144Gvlowast isin D(micro)

983154 It holds that

πuss (Γ) = plowast minus Emicro[c(v)]

Proposition 2 states that an uninformed Sellerrsquos minimum payoff (with price-independentbeliefs) is characterized by the ICDs that are MPCs of the prior distribution micro By defini-tion of ICDs Sellerrsquos profit when facing ICD 983144Gvlowast is vlowast minus E[v] Sellerrsquos minimum payoff isimplemented by the most dispersed ICD that is a MPC of the prior distribution

In proving Proposition 2 the key step is to show that given the prior micro garbling Buyerrsquosinformation such that the posterior mean distribution is an ICD makes Seller weakly worseoff It is then without loss to only consider ICDs to implement πus

s (Γ) We can thus workwith a tractable one-dimensional problem Consider the set D(micro) Suppose we find G(v)

such that (i) G isin D(micro) (ii) G is an ICD and (iii) existp such that983125 p

vG(s)ds =

983125 p

vF (s)ds Such

a G is the most-dispersed ICD that is a mean preserving contraction of F Consider the

21

following two manipulations

λ

983133 v

p

F (s)ds =

983133 v

p

(pminus c(s))dF (s)minus (1minus F (v)) (pminus c(v)) + λ(v minus p) (4)

λ

983133 v

p

G(s)ds =

983133 v

p

(pminus c(s))dG(s)minus (1minusG(v)) (pminus c(v)) + λ(v minus p) (5)

First by property (iii) above the LHS of Equation 4 equals the LHS of Equation 5 Secondsince

983125 v

p(F (s) minus G(s))ds is minimized at v this implies F (p) le G(p) Therefore the first

term on RHS of Equation 4 must be larger than that of Equation 5 Notice that the firstterm on RHS is Sellerrsquos profit when offering price p This implies that the profit fromoffering p given Buyer mean-valuation distribution G(v) is lower than that given F (v)which is lower than Sellerrsquos maximum profit given F (v) On the other hand by the ICDproperty p is already Sellerrsquos optimal price given G(v) The optimal profit given valuationdistribution F must be no lower than that of G Therefore we have proved that to minimizeSellerrsquos profit it is without loss to consider only ICDs within the set D(micro) which is a onedimensional subspace The detailed proof that such ICDs exist is relegated to the appendix

The monopoly-pricing environment with c(middot) = v is covered by Condition 1 with λ = 0

and γ = v The distribution G in (3) then reduces to that identified by Roesler and Szentes(2017)

If the prior micro has binary support then the cost function c(v) is linear In this special casewe can solve for πus

s (Γ) explicitly

Corollary 1 Suppose micro has binary support V = v1 v2 Let λ = c(v2)minusc(v1)v2minusv1

and let p be theunique solution to

(pminus c(p))1

λminus1 (pminus E[c(v)]) = (v2 minus c(v2))λ

λminus1 (6)

Then πuss (Γ) = max p v2minus E[c(v)]

References

AKERLOF G A (1970) ldquoThe Market for rdquoLemonsrdquo Quality Uncertainty and the MarketMechanismrdquo Quarterly Journal of Economics 84 488ndash500

BAGWELL K AND M H RIORDAN (1991) ldquoHigh and Declining Prices Signal ProductQualityrdquo American Economic Review 81 224ndash239

BAR-ISAAC H I JEWITT AND C LEAVER (2018) ldquoAdverse Selection Efficiency and theStructure of Informationrdquo Unpublished

22

BERGEMANN D B BROOKS AND S MORRIS (2015) ldquoThe Limits of Price Discrimina-tionrdquo American Economic Review 105 921ndash57

BERGEMANN D AND S MORRIS (2016) ldquoBayes Correlated Equilibrium and the Compar-ison of Information Structures in Gamesrdquo Theoretical Economics 11 487ndash522

BLACKWELL D (1953) ldquoEquivalent Comparisons of Experimentsrdquo Annals of MathematicalStatistics 24 265ndash272

CHO I-K AND D KREPS (1987) ldquoSignaling Games and Stable Equilibriardquo Quarterly Jour-nal of Economics 102 179ndash221

DU S (2018) ldquoRobust Mechanisms Under Common Valuationrdquo Econometrica 86 1569ndash1588

DURRETT R (1995) Probability Theory and Examples Duxbury Press 2nd ed

EINAV L AND A FINKELSTEIN (2011) ldquoSelection in Insurance Markets Theory and Em-pirics in Picturesrdquo Journal of Economic Perspectives 25 115ndash38

FUDENBERG D AND J TIROLE (1991) ldquoPerfect Bayesian Equilibrium and Sequential Equi-libriumrdquo Journal of Economic Theory 53 236ndash260

GARCIA D R TEPER AND M TSUR (2018) ldquoInformation Design in Insurance MarketsSelling Peaches in a Market for Lemonsrdquo Unpublished

GREENWALD B C (1986) ldquoAdverse Selection in the Labour Marketrdquo Review of EconomicStudies 53 325ndash347

JOVANOVIC B (1982) ldquoFavorable Selection with Asymmetric Informationrdquo QuarterlyJournal of Economics 97 535ndash539

KESSLER A S (2001) ldquoRevisiting the Lemons Marketrdquo International Economic Review 4225ndash41

KREPS D AND R WILSON (1982) ldquoSequential Equilibriardquo Econometrica 50 863ndash894

LEHMANN E L (1988) ldquoComparing Location Experimentsrdquo The Annals of Statistics 16521ndash533

LEVIN J (2001) ldquoInformation and the Market for Lemonsrdquo RAND Journal of Economics32 657ndash666

23

LIBGOBER J AND X MU (2019) ldquoInformational Robustness in Intertemporal PricingrdquoWorking paper

MAKRIS M AND L RENOU (2018) ldquoInformation Design in Multi-Stage Gamesrdquo Workingpaper

MILGROM P R AND R J WEBER (1985) ldquoDistributional Strategies for Games with In-complete Informationrdquo Mathematics of Operations Research 10 619ndash632

MUSSA M AND S ROSEN (1978) ldquoMonopoly and Product Qualityrdquo Journal of EconomicTheory 18 301ndash317

RAVID D A-K ROESLER AND B SZENTES (2019) ldquoLearning Before Trading On theInefficiency of Ignoring Free Informationrdquo Working paper

ROESLER A-K AND B SZENTES (2017) ldquoBuyer-Optimal Learning and Monopoly Pric-ingrdquo American Economic Review 107 2072ndash80

TERSTIEGE S AND C WASSER (2019) ldquoBuyer-optimal Extensionproof InformationrdquoWorking paper

Appendices omitted from this version but are available on request

24

1 IntroductionMotivation Asymmetric information affects market outcomes both in terms of efficiencyand distribution For example adverse selection can generate dramatic market failure (Ak-erlof 1970) or skew wages in labor markets (Greenwald 1986) while consumers can secureinformation rents from a monopolist (Mussa and Rosen 1978) Much existing work takesthe market participantsrsquo information as given and studies properties of a particular marketstructure or mechanism or tackles these properties across various mechanisms

This paper instead asks what is the scope for different market outcomes as the partici-pantsrsquo information varies We are motivated by the fact that in the digital age the nature ofinformation that sellers (eg Amazon) have about consumers is ever changing Consumersand regulators do have some control over this information of course In some cases it isplausible that a sellerrsquos information is a subset of the consumerrsquos But in other cases theseller may well know more about the consumerrsquo value for a product or at least have someinformation the consumer herself does not This is especially relevant for products theconsumer is not already familiar with Indeed numerous firms make tailored recommen-dations to consumers about products they carry With social media and other sources ofinformation diffusion the possible correlation in information across two sides of a marketseems truly limitless

Our paper fixes a simple canonical market mechanism and studies the possible marketoutcomes across a variety of information structures including all of them We model twoparties Buyer and Seller who can trade a single object Buyerrsquos value for the object is arandom v isin [v v] sub R Sellerrsquos cost of providing the object or equivalently her valuefrom not trading is c(v) le v Thus values may be interdependent but trade is alwaysefficient1 The environment ie the function c(v) and the distribution of v is commonlyknown Seller posts a price p isin R and Buyer decides whether to buy

This stylized setting subsumes a variety of possibilities depending on the shape of thecost function c(middot) and the partiesrsquo information about the value v With an informed Buyerand an uninformed Seller there is adverse selection when c(middot) is increasing while there isfavorable or advantageous selection when c(middot) is decreasing2 If on the other hand Seller isbetter informed than Buyer signaling becomes relevant the price can serve as credible sig-nal if the two partiesrsquo information is suitably correlated (eg Bagwell and Riordan 1991)

1 We describe here our baseline model presented in Section 2 Section 4 discusses extensions includingcases when Buyerrsquos value does not pin down Sellerrsquos cost and when trade is not always efficient

2 Jovanovic (1982) uses the term lsquofavorable selectionrsquo Einav and Finkelstein (2011) use lsquoadvantageousrsquoand discuss both adverse and advantageous selection in the context of insurance markets with empiricalreferences to empirical evidence on both

1

A constant c(middot) captures an environment in which there no uncertainty about Sellerrsquos costthis is the canonical monopoly pricing problem when Seller is uninformed about v andthird-degree price discrimination when Seller has some partial information while Buyer isbetter informed

Summary of results For any given environment (ie Sellerrsquos cost function and the dis-tribution of Buyerrsquos values) we seek to identify the possible market outcomes Specificallywe are interested in the ex-ante expected payoffs that obtain given sequentially rational be-havior in an equilibrium under some information structure3 We provide three results eachof which covers a different class of information structures Our main theorems are Theo-rem 1Theorem 1lowast which impose no restrictions on information and Theorem 2 whichapplies when Buyer is better informed than Seller in the sense of Blackwell (1953) in factTheorem 2 applies more broadly as elaborated later Theorem 3 concerns a fully-informedBuyer (ie who knows his value v) We view each of these three cases as intellectuallysalient and economically relevant Plainly these payoff sets must be ordered by set inclu-sion Theorem 1rsquos is the largest Theorem 2rsquos is intermediate and Theorem 3rsquos the smallestFigure 1 below summarizes

Akerlof

Full or No Information

Fully-Informed BuyerABC

Uninformed Seller(More Informed Buyer)

ADE

All Info StructuresAFG

Expected SurplusFrontier

C

E

G

0πb

Maxv-E[c(v)]0 F

A

B

D

πs

Figure 1 Outcome under different restrictions on information structures

In the figurersquos axes πb and πs represent respectively Buyerrsquos and Sellerrsquos ex-ante ex-

3 As detailed in Section 2 an information structure specifies a joint distribution of private signals foreach party conditional on the value v This induces an extensive-form game of incomplete informationOur primary solution concept is weak Perfect Bayesian equilibrium we also address refinements for ourconstructive arguments

2

pected utilities or payoffs (for readability we often drop the ldquoexpectedrdquo qualifier) Theno-trade payoffs are normalized to zero The three triangles AFG ADE and ABC depictTheorems 1ndash3 respectively That payoffs must lie within AFG is straightforward Buyercan guarantee himself a payoff of zero by not purchasing Seller can guarantee herself notonly a payoff of zero (by posting any price p gt v which will not be accepted) but alsov minus E[c(v)] (by pricing at or just below v which will be accepted) and the sum of pay-offs cannot exceed the trading surplus E[v minus c(v)] We refer to the first two constraints asindividual rationality and the third as feasibility

Theorem 1 says that every feasible and individually-rational payoff pair can be imple-mented ie obtains in an equilibrium under some information structure It is immediatethat point A obtains whens both parties learn v (full information) or neither party has anyinformation (no information) More interestingly at the point G trade occurs with probabil-ity one and Buyer obtains the entire surplus despite Seller posting the price While perhapssurprising this outcome obtains with sparse information structures For simplicitly sup-pose E[c(v)|v gt v] le maxvE[c(v)] Then Buyer can be uninformed while Seller learnswhether v = v or v gt v In equilibrium Seller prices at p = maxvE[c(v)] regardless ofher signal and Buyer purchases If Seller were to deviate to a higher price Buyer wouldreject because he believes v = v Subsection 31 explains how a single information struc-ture in fact implements every point in the triangle AFG Theorem 1lowast there discusses how aricher information structure using imperfectly-correlated signals ensures implementationin Kreps and Wilsonrsquos (1982) sequential equilibrium in discretized versions of the model

Turning to Figure 1rsquos triangle ADE Theorem 2 establishes that the payoff pair in anyequilibrium when Buyer is better informed than Seller arises in an equilibrium of an(other)information structure in which Seller is uninformed4 In other words there is no loss ofgenerality in studying an uninformed Seller so long as Buyer is better informed Whenc(middot) is increasing such information generates a game with adverse selection when c(middot)is decreasing there is favorable selection Sellerrsquos payoff along the line segment DE isthe lowest payoff she can get in any information structure in which she is uninformed5

Theorem 2 further establishes that any point within the ADE triangle can be implementedwith some such information structure by suitably varying Buyerrsquos information In fact weshow that higher slices of the triangle (ie those corresponding to larger Sellerrsquos payoff)

4 We stipulate that a better-informed Buyer does not update his value from Sellerrsquos price (even off theequilibrium path) in line with the ldquono signaling what you donrsquot knowrdquo requirement (Fudenberg and Tirole1991) that is standard in versions of Perfect Bayesian Equilibrium and implied by sequential equilibrium

5 It is because Seller cannot commit to the price as a function of her signal that she can be harmed (iereceive a payoff lower than that on the DE segment) with more information However Theorem 2 assuresthat Seller is not harmed so long as Buyer is better informed

3

can always be implemented by reducing Buyerrsquos information in the sense of Blackwell(1953) We also explain in Subsection 32 why the triangle ADE actually characterizes allpayoffs that can obtain when Buyer does not update from Sellerrsquos price even if Buyer isnot better informed than Seller

Finally the ABC triangle in Figure 1 depicts Theorem 3 which characterizes all payoffpairs when Buyer is fully informed ie learns v We use the term ldquoAkerlofrdquo to describe afully-informed Buyer and an uninformed Seller as these information structures are stan-dard in the adverse-selection literature the corresponding payoff pair is marked as suchin the figure Depending on the environmentrsquos primitives the Akerlof point can be any-where on the segment BC including at the extreme points Any feasible payoff pair thatsatisfies Buyerrsquos individual rationality and gives Seller at least her Akerlof payoff can beimplemented with a fully-informed Buyer by suitably varying Sellerrsquos information

An implication of Theorems 1ndash3 is that it is without loss in terms of ex-ante equilibriumpayoffs to focus on information structures in which Buyer is fully informed if and onlyif Sellerrsquos Akerlof payoff coincides with her individual rationality constraint This coinci-dence occurs if and only if the Akerlof market can have full trade (Seller prices at p = v

and gets payoff v minus E[c(v)] ge 0) or no trade (the price is p ge v and both partiesrsquo payoffsare 0) As detailed in Remark 4 of Subsection 33 in all other cases the point B in Fig-ure 1 is distinct from the point D (and hence also F ) which means that Buyer can obtaina higher payoff with less-than-full information while keeping Seller uninformed Further-more under a reasonable condition if Sellerrsquos individual rationality constraint is zero (iev le E[c(v)]) then point D is also distinct from F see Remark 3 in Subsection 32 WhenD and F are distinct maximizing Buyerrsquos payoff ie achieving point G requires Seller tohave some information Buyer does not and an equilibrium with price-dependent beliefsafter conditioning on his signal Buyer must update about v from the price either on or offthe equilibrium path

Related literature Our perspective and results are most closely related to BergemannBrooks and Morris (2015) Roesler and Szentes (2017) and Makris and Renou (2018 Sec-tion 4) These papersmdashonly the relevant section of the third papermdashstudy the monopolypricing problem in which there is uncertainty only about Buyerrsquos valuation This is thespecial case of our interdependent-values model with a constant function c(v) le v6

6 The online appendix of Roesler and Szentes (2017) relaxes the assumption that there are always gainsfrom trade but maintains no uncertainty about Sellerrsquos cost Related to Roesler and Szentes (2017) are alsoDu (2018) and Libgober and Mu (2019) who consider worst-case profit guarantees for Seller in static and dy-namic environments respectively Terstiege and Wasser (2019) qualify Roesler and Szentes (2017) by allowing

4

Bergemann et al (2015) assume Buyer is fully informed and hence can only vary Sellerrsquosinformation Our Theorem 3 corresponding to triangle ABC in Figure 1 is a generaliza-tion of their main result to our environment the key step in our proof methodology is toconstruct ldquoincentive compatible distributionsrdquo which reduces to their ldquoextreme marketsrdquoin the monopoly-pricing environment Another contribution of our result is to ensure ap-proximately unique implementation in a sense explained in Subsection 33

Roesler and Szentes (2017) assume Seller is uninformed and only vary Buyerrsquos infor-mation For the monopoly-pricing environment they derive one part of our Theorem 2viz they identify the triangle ADE in Figure 1 as the implementable set given Seller isuninformed Even for this result our methodology is quite different from theirs becausewe do not stipulate a linear c(middot) function our methodology delivers new insights includ-ing that noted in Remark 2 of Subsection 32 When we specialize to a linear c(middot) we canobtain a sharper characterization of the point E which extends Roesler and Szentesrsquo char-acterization of Buyer-optimal information to an interdependent-values environment seeProposition 2 in Subsection 43

While our main interest is in interdependent values our results provide new insightseven for monopoly pricing Theorem 2 implies that the Roesler and Szentes (2017) boundsare without loss so long as Buyer is better informed than Seller or more generally in equi-libria in which Buyerrsquos belief is price independent (after conditioning on his own signal)On the other hand Theorem 1 establishes that any feasible and individually rational pay-off pair can be implemented absent these restrictions in particular Buyer may even get allthe surplus This latter point has a parallel with Makris and Renou (2018) As an applica-tion of their general results on ldquorevelation principlesrdquo for information design in multi-stagegames Makris and Renoursquos (2018) Proposition 1 deduces an analog of our Theorem 1 forthe (independent values) monopoly pricing problem We share with Makris and Renou anemphasis on sequential rationality7 we differ in showing that a single information struc-ture implements all payoffs in the relevant triangle and also in establishing in Theorem 1lowast

off-the-equilibrium-path belief consistency in the sense of sequential equilibrium (Krepsand Wilson 1982) in discretizations

Other authors have studied different aspects of more specific changes of information inadverse-selection settings maintaining that one side of the market is better informed than

Seller to supply Buyer with additional information although Seller cannot have any private information ofher own

7 Makris and Renou use an apparatus of ldquosequential Bayes correlated equilibriumrdquo which we do notThey draw a contrast with Bergemann and Morrisrsquos (2016) Bayes correlated equilibrium In our approachnote that Sellerrsquos individual rationality constraint described earlier hinges when E[c(v)] lt v on Buyerrsquosbehavior being sequentially rational even off the equilibrium path

5

the other Levin (2001) identifies conditions under which the volume of trade decreaseswhen one party is kept uninformed and the otherrsquos information become more effective inthe sense of Lehmann (1988) see also Kessler (2001) Assuming a linear payoff structureBar-Isaac Jewitt and Leaver (2018) consider how certain changes in Gaussian informationaffect the volume of trade surplus and a certain quantification of adverse selection

Finally Garcia Teper and Tsur (2018) solve for socially optimal information provisionmdashrather than characterizing the range of market outcomesmdashin an insurance setting with ad-verse selection owing to a cross-subsidization motive full information disclosure is typi-cally not optimal

The rest of the paper proceeds as follows We introduce our model equilibrium con-cept(s) and certain classes of information structures in Section 2 Section 3 presents themain results implementable payoffs when the information structure is arbitrary or varieswithin canonical classes Section 4 contains discussion and extensions Proofs and someadditional material are contained in the Appendices

2 Model

21 Primitives

Seller may sell an indivisible good to Buyer Buyerrsquos value for the good is v isin V sub Rwhere V is a compact (finite or infinite) set with v equiv minV lt maxV equiv v The valuev is drawn from a probability measure micro with support V Sellerrsquos cost of production isgiven by a function c(v)8 We assume c V rarr R is continuous v minus c(v) ge 0 for all vand E[v minus c(v)] gt 0 So trading surplus is nonnegative for all v and positive for a positivemeasure of v (Throughout expectations are with respect to the prior measure micro unlessindicated otherwise lsquopositiversquo means lsquostrictly positiversquo and similarly elsewhere) Note thatfunction c(v) need not be monotonic We call Γ = (c micro) an environment We refer to anenvironment with a constant c(middot) function as that of monopoly pricing

An information structure consists of signal spaces for each party and a joint signal dis-tribution (We abuse notation and sometimes write lsquodistributionrsquo even though lsquomeasurersquowould be more precise) Formally there is a probability space (ΩF P ) complete and sep-arable metric spaces Ts and Tb (with measurable structure given by their Borel sigma alge-bras) and an integrable function X Ω rarr TstimesTbtimesV We hereafter suppress the probabilityspace and define with an abuse of notation P (D) = P (Xminus1(D)) for any D isin Ts times Tb times V

measurable Each realization of random variable X is a triplet (tb ts v) where tb isin Tb

8 Subsection 41 discusses Sellerrsquos cost being stochastic even conditional on v

6

is Buyerrsquos signal and ts isin Ts is Sellerrsquos signal For i isin s b v let Pi denote the corre-sponding marginal distribution of P on dimension of Ti with the convention Tv equiv V Werequire Pv = micro this is the iterated expectation or Bayes plausibility requirement Denotean information structure by τ

The environment Γ and information structure τ define the following game

1 The random variables (tb ts v) are realized Signal tb is privately observed by Buyerand signal ts privately observed by Seller Neither party observes v

2 Seller posts a price p isin R

3 Buyer accepts or rejects the price If Buyer accepts his von-Neumann Morgensternpayoff is v minus p and Sellerrsquos is p minus c(v) If Buyer rejects both partiesrsquo payoffs arenormalized to 0

Note that because the signal spaces Tb and Ts are abstract and the two partiesrsquo signalscan be arbitrarily correlated conditional on v there is no loss of generality in assuming thateach party privately observes their own signal For example public information can becaptured by perfectly correlating (components of) tb and ts

22 Strategies and Equilibria

In the game defined by (Γ τ) denote Sellerrsquos strategy by σ and Buyerrsquos by α FollowingMilgrom and Weber (1985) we define σ as a distributional strategy σ is a joint distributionon R times Ts whose marginal distribution on Ts must be the Sellerrsquos signal distribution Soσ(middot|ts) is Sellerrsquos price distribution given her signal ts9 Buyerrsquos strategy α R times Tb rarr[0 1] maps each price-signal pair (p tb) into a trading probability A strategy profile (σα)

induces expected utilities for Buyer and Seller (πb πs) in the natural way

πb =

983133(v minus p)α(tb p)σ(dp|ts)P (dts dtb dv)

πs =

983133(pminus c(v))α(tb p)σ(dp|ts)P (dts dtb dv)

Our baseline equilibrium concept is weak Perfect Bayesian equilibrium Since Sellerrsquosaction is not preceded by Buyerrsquos we can dispense with specifying beliefs for Seller ForBuyer it suffices to focus on his belief about the value v given his signal and the price wedenote this distribution by ν(v|p tb)

9 Here σ(middot|ts) is the regular conditional distribution which exists and is unique almost everywhere becauseTs is a standard Borel space (Durrett 1995 pp 229ndash230) Similarly for subsequent such notation we dropldquoalmost everywhererdquo qualifiers unless essential

7

Definition 1 A strategy profile (σα) and a belief ν(v|p tb) is a weak perfect Bayesian equilib-rium (wPBE) of game (Γ τ) if

1 Buyer plays optimally at every information set given his belief

α(p tb) =

983099983103

9831011 if Eν(v|ptb)[v] gt p

0 if Eν(v|ptb)[v] lt p

2 Seller plays optimally

σ isin argmaxσ

983133(pminus c(v))α(p tb)σ(dp|ts)P (dts dtb dv)

3 Beliefs satisfy Bayes rule on path for every measurable D sube Rtimes Ts times Tb times V

983133

D

ν(dv|p tb)σ(dp|ts)P (dts dtb V ) =

983133

D

σ(dp|ts)P (dts dtb dv)

Notice that we have formulated Sellerrsquos optimality requirement ex ante but Buyerrsquos ateach information set The latter is needed to capture sequential rationality The formeris for (notational) convenience this choice is inconsequential because Seller moves beforeBuyer

Hereafter ldquoequilibriumrdquo without qualification refers to a wPBE As is well understoodwPBE permits significant latitude in beliefs off the equilibrium path We will discuss re-finements when appropriate

23 Implementable Payoffs and Canonical Information Structures

We now define the set of implementable equilibrium outcomesmdashthat is the payoffs thatobtain in some equilibrium under some information structuremdashand some canonical classesof information structures

For a game (Γ τ) let the equilibrium payoff set be

Π(Γ τ) equiv (πb πs) exist wPBE of (Γ τ) with payoffs (πb πs)

Denote the class of all information structures by T and define

Π(Γ) equiv983134

τisinT

Π(Γ τ)

8

That is for environment Γ Π(Γ) is the set of all equilibrium payoff pairs that obtain undersome information structure

Uninformed Seller An information structure has uninformed Seller if Ts is a singletonSellerrsquos own signal contains no information about Buyerrsquos value v and hence neither aboutSellerrsquos cost c(v) When discussing such information structures we write the associateddistribution as just P (tb v) and Sellerrsquos strategy as just σ(p) omitting the argument ts inboth cases The class of all uninformed-Seller information structure is denoted Tus

Fully-informed Buyer An information structure has fully-informed Buyer if Buyerrsquos signalfully reveals his value v Formally this holds if Tb = V and the conditional distribution onV P (middot|tb) satisfies P (tb|tb) = 1 We denote the class of fully-informed-Buyer informationstructures Tfb

More-informed Buyer An information structure has more-informed Buyer if Buyer hasmore information than Seller Formally this holds when v and ts are independent condi-tional on tb ie for any measurable Ds sub Ts and Dv sub V P (DscapDv|tb) = P (Ds|tb)P (Dv|tb)10

Another way to interpret this requirement is that random variable tb must be statisticallysufficient for ts with respect to v ie tb is more informative than ts about v in the sense ofBlackwell (1953) We denote the class of more-informed-Buyer information structures byTmb Naturally information structures with uninformed Seller or fully-informed Buyer arecases of more-informed Buyer both Tus and Tfb are subclasses of Tmb

No updating from price For more-informed-Buyer information structures it is desirableto impose further requirements on Buyerrsquos equilibrium belief Since Sellerrsquos price can onlydepend on her own signal and this signal contains no additional information about v givenBuyerrsquos signal the price is statistically uninformative about v given Buyerrsquos signal Con-sequently Buyerrsquos posterior belief should be price independent once his signal has beenconditioned upon Formally regardless of the price p the equilibrium belief ν(middot|p tb) mustsatisfy 983133

ν(dv|p tb)P (dtb Ts V ) =

983133P (dtb Ts dv)

We refer to this condition as price-independent belief Note that the condition is meaningfulregardless of whether Buyer is more informed than Seller In a more-informed-Buyer in-formation structure price-independent belief would be implied by the ldquono signaling what

10 We write sub to mean ldquoweak subsetrdquo

9

you donrsquot knowrdquo requirement (Fudenberg and Tirole 1991) frequently imposed in versionsof perfect Bayesian equilibrium and the concept of sequential equilibrium (Kreps and Wil-son 1982) in finite versions of our setting11

Some notation will be helpful Define

Πlowast(Γ τ) equiv (πb πs) exist wPBE of (Γ τ) with price-independent belief and payoffs(πb πs)

Πlowast(Γ) equiv983134

τisinT

Πlowast(Γ τ)

Πlowasti (Γ) equiv

983134

τisinTi

Πlowast(Γ τ) for i = us fbmb

So Πlowast and Πlowast are analogous to the implementable payoff sets Π and Π defined earlier butrestricted to equilibria with price-independent beliefs Πlowast

us Πlowastfb and Πlowast

mb are the imple-mentable payoff sets when further restricted to uninformed-Seller fully-informed-Buyerand more-informed-Buyer information structures Plainly for any environment Γ

Πlowastus(Γ) cupΠlowast

fb(Γ) sub Πlowastmb(Γ) sub Πlowast(Γ) sub Π(Γ)

3 Main Results

Our goal is to characterize equilibrium payoff pairs across information structures in anarbitrary environment Γ In particular we seek to characterize the five sets Π(Γ) Πlowast(Γ)Πlowast

mb(Γ) Πlowastus(Γ) and Πlowast

fb(Γ) Let

S(Γ) equiv E[v minus c(v)]

be the (expected) surplus from trade in environment Γ This quantity will play an impor-tant role

31 All Information Structures

Define Sellerrsquos payoff guarantee as

πs(Γ) equiv max v minus E [c(v)] 0

11 An example clarifying our terminology may be helpful If both Seller and Buyer are fully informed of vthen the natural equilibriummdashthe unique sequential equilibrium in a finite version of the gamemdashhas Sellerpricing at p = v and Buyerrsquos belief being degenerate on v regardless of Sellerrsquos price This equilibrium hasprice-independent belief even though ex-ante Sellerrsquos price are Buyerrsquos belief are perfectly correlated Thepoint is that Buyerrsquos belief does not depend on price conditional on his signal

10

To interpret this observe that it is optimal for Buyer to accept the price v no matter hisbelief Therefore Seller can guarantee herself the (expected) profit v minus E[c(v)] no matterwhat the information structure is More precisely she can guarantee v minus E[c(v)] minus ε forany ε gt 0 since sequential rationality requires Buyer to accept any price v minus ε SimilarlySeller can also guarantee zero profit offering a price p gt v Hence Sellerrsquos payoff in anyequilibrium of any information structure must be at least the payoff guarantee πs(Γ) above

On the other hand Buyer can guarantee himself the payoff πb = 0 by rejecting all pricesIt follows that the implementable set Π(Γ) must satisfy three simple constraints 1) Sellerrsquosldquoindividual rationalityrdquo constraint πs ge πs(Γ) 2) Buyerrsquos ldquoindividual rationalityrdquo con-straint πb ge 0 and 3) the feasibility constraint πb + πs le S(Γ)

Our first result is that these individual rationality and feasibility constraints are alsosufficient for a payoff pair to be implementable

Theorem 1 Consider all information structures and equilibria

Π(Γ) =

983099983105983103

983105983101

πb ge 0

(πb πs) πs ge πs(Γ)

πb + πs le S(Γ)

983100983105983104

983105983102

Theorem 1 says that the set Π(Γ) corresponds to the triangle AFG in Figure 1 In par-ticular Buyer can receive the entire surplus beyond Sellerrsquos payoff guarantee This is sur-prising as Seller has substantial bargaining power Note that when v le E[c(v)] an entirelyreasonable condition Sellerrsquos payoff guarantee is zero12

The proof of Theorem 1 is in fact straightforward Suppose for expositional simplicityv ge E[c(v)] Fix the trivial information structure in which neither player receives any infor-mation and consider the following family of strategy profiles Seller randomizes betweentwo prices some pl isin [vE[v]] and ph = E[v] Buyer accepts pl with probability one andaccepts ph with probability α(ph) where α(ph) isin [0 1] is specified to make Seller indifferentbetween the two prices That is α(ph)(ph minus E[c(v)]) = pl minus E[c(v)] The expected payoffsfrom this strategy profile are

πb = σ(pl)(E[v]minus pl) and πs = pl minus E[c(v)]

As pl traverses the interval [vE[v]] Sellerrsquos payoff πs traverses [πs(Γ) S(Γ)] Given any

12 In monopoly pricing with c(middot) = v Sellerrsquos payoff guarantee of zero is lower than the revenue guaranteeidentified in Du (2018 Section 5) which is typically positive Dursquos notion is different from ours

11

pl Buyerrsquos payoff πb traverses [0 S(Γ)minusπs] as σ(pl) traverses [0 1] Therefore the proposedstrategy profiles induce all the payoff pairs stated in Theorem 1

We are left to specify beliefs ν for Buyer After prices pl and ph Buyer holds the priorbelief micro After any other (necessarily off-path) price Buyerrsquos belief is that v = v and soBuyer rejects all prices p isin [vinfin)pl ph It is straightforward to confirm that the specified(σα ν) constitute a wPBE

To get more insight into this construction consider its implication for the monopolypricing with c(middot) = v The equilibrium with pl = v (hence α(ph) = 0) and σ(pl) = 1

corresponds to the monopolist deterministically pricing at v and Buyer purchasing Giventhat both sides of the market receive no information why doesnrsquot the monopolist deviate toany price in (vE[v]) The reason is that in this equilibrium the consumer will then not buybecause he updates to v = v Such belief is compatible with wPBE because the equilibriumconcept places no restrictions on off-path beliefs This may seem like a game-theoreticmisdirection Buyerrsquos belief is not consistent with ldquono signaling what you donrsquot knowrdquoPut differently since we have a (weakly) more-informed Buyer information structure weought to impose the price-independent belief condition described in Subsection 23 thatwould imply Buyer must purchase at any price p lt E[v]

But the message of Theorem 1 does not rely on the permissiveness of wPBE To illus-trate continue with the above monopoly-pricing environment and suppose v has positiveprior probability Consider Buyer remaining uninformed but Seller learning whether v = v

or v gt v Now Buyerrsquos off-path belief that v = v is consistent with ldquono signaling whatyou donrsquot knowrdquo More generally using richer information structures we can prove thatany payoff pair identified in Theorem 1 can be approximately implemented as a sequentialequilibrium (Kreps and Wilson 1982) in a suitably discretized game

Theorem 1lowast Fix any ε gt 0 There is a finite information structure and a finite price grid inducinga game with a set of sequential equilibrium payoffs that is an ε-net of Π(Γ)13

In fact the proof of Theorem 1lowast establishes even more the sequential equilibria in thediscretized games satisfy a natural version of the D1 refinement (Cho and Kreps 1987) Werelegate the logic to the Appendix but mention here that we use imperfectly-correlatedsignals for Buyer and Seller14

13 An information structure is finite if the signal spaces Tb and Ts are finite Sequential equilibrium isdefined in the obvious way for the ldquoinducedrdquo finite game where Nature directly draws (tb ts) rather thanfirst drawing v and playersrsquo payoffs are defined directly using (tb ts) For X sub R2 and ε gt 0 the set A sub Xis an ε-net of X if for each x isin X there is a isin A such that 983042xminus a983042 lt ε where 983042middot983042 is the Euclidean distance

14 The idea behind D1 is to ask for any off-path price whether one type of Seller would deviate for any

12

Even when our environment is specialized to monopoly pricing it is worth highlightingtwo contrasts between Theorem 11lowast and results of Bergemann et al (2015) and Roeslerand Szentes (2017) First we find that by not restricting the monopolist to be uninformedthe implementable payoff set typically expands rather dramatically trade can be efficientwith the monopolist securing none of the surplus beyond her payoff guarantee (πs) whichmay be zero (Roesler and Szentes establish implicitly that the implementable set withan uninformed monopolist is a superset of Bergemann et alrsquos which has a fully-informedconsumer) We will see in Subsection 32 that what is crucial to this expansion is price-dependent beliefs In particular the proof of Theorem 1lowast uses an information structure inwhich Buyer is not better informed than Seller mdash if he were then sequential equilibriumwould imply price-independent beliefs Second Theorem 1lowast establishes that for a givenε gt 0 a single information structure (and price grid) can be used to approximate the entirepayoff set Π(Γ) analogously to the construction described after Theorem 1 that used asingle information structure15 Bergemann et al (2015) and Roesler and Szentes (2017) onthe other hand vary information structures to span their payoff sets

32 More-informed Buyer and Price-independent Beliefs

In some economic settings it is plausible that Buyer is more informed than Seller Howdoes a restriction to such information structures ie τ isin Tmb affect the implementablepayoff set It turns out that what is in fact crucial is price-independent beliefs We have ex-plained earlier why it is desirable to impose this condition when Buyer is more informedbut that the condition is well defined even otherwise If Buyer is not more informed thanSeller price-independent beliefs ought to be viewed as an equilibrium restriction in gen-eral To reduce repetition readers should be bear in mind for the rest of this subsectionthe qualifier ldquowith price-independent beliefsrdquo applies unless stated explicitly otherwise

It is useful to define

πuss (Γ) equiv inf πs exist(πb πs) isin Πlowast

us(Γ) (1)

as the lowest payoff that an uninformed Seller can obtain no matter Buyerrsquos information(among equilibria with price-independent beliefs) Plainly πus

s (Γ) ge πs(Γ) In monopoly

Buyer mixed response that another type would Our construction has multiple Buyer types that are imper-fectly correlated with Seller types So different types of Seller have different beliefs about Buyer types Thisblunts the power of dominance considerations to the point where D1 does not exclude any Seller type fromthe support of Buyerrsquos off-path belief

15 In fact if one lets the price grid vary with ε then a single information structure implements exactlyrather than approximately in sequential equilibrium all payoffs ε-away from the boundary of Π(Γ) See in the Appendix for a formal statement

13

pricing with V = [v v] and c(middot) = v Roesler and Szentesrsquos (2017) characterization of theconsumer-optimal information structure identifies πus

s establishing that πuss gt πs If there

is no trade due to adverse selection when Seller is uniformed and Buyer has some informa-tion then πus

s = πs = 0 We do not have a general explicit formula for πuss Subsection 43

provides it for linear c(middot) Nonetheless we establish next that (i) the only additional restric-tion on equilibrium payoffs imposed by price-independent beliefs is a lower bound of πus

s

for Seller and (ii) uninformed-Seller information structures implement all such payoffs

Theorem 2 Consider equilibria with price-independent beliefs

1 Πlowast(Γ) = Πlowastmb(Γ) = Πlowast

us(Γ)

2 Πlowastus(Γ) = (πb πs) isin Π(Γ) πs ge πus

s (Γ)

3 For any (πb πs) isin Πlowastus(Γ) with πs gt πus

s (Γ) there is τ isin Tus with Π(Γ τ) = (πb πs)

Remark 1 We believe the substance of Theorem 2 would hold using discretizations and se-quential equilibria analogous to Theorem 1lowast As previously noted sequential equilibriumimplies price-independent beliefs when Buyer is more informed than Seller

To digest Theorem 2 note that Πlowast(Γ) sup Πlowastmb(Γ) sup Πlowast

us(Γ) is trivial So part 1 of thetheorem amounts to establishing the reverse inclusions The intuition for thismdashgiven part2rsquos characterization of Πlowast

usmdashis fairly straightforward with price-independent beliefs ad-ditional information cannot harm Seller even though it could alter the set of equilibria SoSellerrsquos lowest payoff obtains when she is uninformed

The characterization in part 2 of payoffs with an uninformed Seller corresponds to thetriangle ADE in Figure 1 Part 3 of the theorem assures ldquounique implementationrdquo of allimplementable payoffs satisfying πs gt πus

s (Γ) That is for any such payoff pair there is anuninformed-Seller information structure such that all equilibria (with price-independentbeliefs) induce exactly that payoff pair Unique implementation is appealing for multiplereasons one of which is that it obviates concerns about which among multiple payoff-distinct equilibria is more reasonable For example Ravid Roesler and Szentes (2019)bring to the fore the issue of equilibrium multiplicity in Roesler and Szentesrsquos (2017) infor-mation structure

Let us describe how we obtain the characterization of Πlowastus(Γ) and unique implementa-

tion There are two steps The first ensures that there is some information structure callit τ lowast isin Tus that implements Sellerrsquos payoff πus

s (Γ) That is we ensure that the infimum

14

in (1) is in fact a minimum16 While this argument is technical knowing τ lowast exists is usefulin what follows The second and economically insightful step is to construct informa-tion structures that implement every point in the triangle Πlowast

us(Γ) by suitably garbling theinformation structure τ lowast The construction is illustrated in Figure 2 Consider the distri-bution of Buyerrsquos posterior mean of his valuation v in information structure τ lowast (Givenprice-independent beliefs Buyerrsquos posterior mean is sufficient for his decision) For sim-plicity suppose this posterior-mean distribution has a density as depicted by the red curvein Figure 2 Fix any (πb πs) isin Πlowast

us(Γ)

First there is some number zlowast such that πb + πs is the total surplus from trading onlywhen Buyerrsquos posterior mean is greater than zlowast Next there is some price plowast ge zlowast suchthat Sellerrsquos payoff is πs if all these trades were to occur at price plowast17 Note that plowast must beno larger than the expected Buyer posterior mean conditional on that being above zlowast forotherwise πb lt 0 We claim that the information structure τ lowast can be garbled so that plowast isan equilibrium price and trade occurs only when Buyerrsquos posterior mean is greater thanzlowast The garbling is illustrated in Figure 2 by the blue distribution There is one signal that

pzv

ν(v)

Figure 2 Construction of information structure in Theorem 2

Buyer receives when the original posterior mean is between zlowast and plowast and also receiveswith some probability when the original posterior mean is above plowast The probability ischosen to make the posterior mean from this signal exactly plowast Apart from this one newsignal Buyer receives the original signal in τ lowast Plainly this is a garbling of τ lowast and hence isfeasible

16 The difficulty is in establishing suitable continuity Uninformed-Seller information structures can beviewed as probability measures over Buyerrsquos beliefs with convergence in the sense of the weak topologyThis topology ensures continuity with respect to probability measures of expectations of continuous orat least Lipschitz (and bounded) functions However Sellerrsquos expected payoff is not the expectation of aLipschitz function as Sellerrsquos profit is truncated at the price she charges

17 That plowast ge zlowast follows from πs ge πuss (Γ) as πs(Γ) itself is weakly larger than Sellerrsquos payoff from posting

price zlowast (and thus trading with the same set of Buyer posterior means) under information structure τlowast

15

Figure 2 makes clear why the new information structure has an equilibrium with priceplowast and Buyer breaking indifference in favor of trading (i) Sellerrsquos profit from posting anyprice below zlowast is the same as under τ lowast and hence no larger than πus

s (Γ) (ii) similarly Sellerrsquosprofit from posting any price above plowast is no higher than some fraction of πus

s (Γ) and (iii)any price between zlowast and plowast is worse that price plowast Moreover since Sellerrsquos profit fromoffering any price other than plowast is no more than πus

s (Γ) it follows that when πs gt πuss (Γ)

Buyer must break indifference as specified for Seller to have an optimal price and theequilibrium payoffs are unique

Remark 2 The above logic establishes that given any τ isin Tus that implements some (πb πs)τ can be garbled to uniquely implement any (πprime

b πprimes) isin Π(Γ) such that πprime

s gt πs That isan uninformed Sellerrsquos payoff can always be strictly raised and Buyerrsquos payoff reduced(strictly so long as it was not already zero) by garbling Buyerrsquos information

Remark 3 According to Theorem 11lowast and Theorem 2 uninformed-Seller information struc-tures cannot implement all implementable payoff pairs in an environment Γ if and only ifπuss (Γ) gt πs(Γ) This inequality fails in environment Γ if πus

s (Γ) = 0 since that impliesπuss (Γ) = πs(Γ) = 0 An example is when there is no trade due to adverse selection when

Seller is uniformed and Buyer has some information On the other hand it is sufficient forthe inequality that

v le E[c(v)] and forallv isin V c(v) lt v (2)

To see why (2) implies πuss (Γ) gt πs(Γ) notice that in any uninformed-Seller information

structure Seller can price at slightly less than Buyerrsquos highest posterior mean valuationand guarantee trade with only (a neighborhood of) that Buyer type If c(v) lt v for all v thisgives Seller a positive expected payoff and hence πus

s (Γ) gt 018 But the first inequality in (2)is equivalent to πs(Γ) = 0 We observe that Condition (2) is compatible with severe adverseselection resulting in very little trade when Seller is uninformed and Buyer (partially orfully) informed

33 Fully-Informed Buyer

We now turn to the third canonical class of information structures Buyer is fully in-formed of his value v As this is a special case of more-informed Buyer we maintain price-independent beliefs throughout this subsection

18 More precisely as V is compact c(v) lt v for all v implies there exists ε gt 0 such that v minus c(v) gt εGiven any uninformed-Seller information structure let mv be the highest posterior mean valuation in thesupport of the posterior means induced by Buyerrsquos signals So there is positive probability of Buyer signalswith posterior mean valuations at least mv minus ε2 By pricing at mv minus ε2 Sellerrsquos expected cost conditionalon trade is bounded above by mv minus ε and hence Sellerrsquos profit conditional on trade is at least ε2 gt 0 Itfollows that πus

s gt 0

16

Faced with a fully informed Buyer and any sequentially rational Buyer strategy an un-informed Seller can guarantee a profit level

πfbs (Γ) equiv sup

p

983133 v

p

(pminus c(v))micro(dv)

regardless of her information Plainly πfbs (Γ) ge πus

s (Γ) In monopoly pricing with c(middot) = vRoesler and Szentes (2017) have shown that πfb

s gt πuss if there is no trade due to adverse

selection when Buyer is fully informed and Seller is uniformed then πfbs = πus

s = 0 Belowwe establish that when Buyer is fully informed πfb

s is the only additional constraint onequilibrium payoffs Let Bε(πb πs) denote an ε-ball around (πb πs)

Theorem 3 Consider a fully informed Buyer and price-independent beliefs

1 Πlowastfb(Γ) = (πb πs) isin Π(Γ) πs ge πfb

s (Γ)

2 For any (πb πs) isin Πlowastfb(Γ) and any ε gt 0 there is τ isin Tfb with Π(Γ τ) sub Bε(πb πs)

The payoff set characterized in Theorem 3 corresponds to the triangle ABE in Figure 1Part 2 of the theorem establishes ldquoapproximately unique implementationrdquo for any point inthe triangle there is an information structure under which all equilibria yield payoff pairsin a neighborhood of that point

Here is the idea behind part 1 of Theorem 3 When Buyer is fully informed an informa-tion structure can be viewed as dividing vrsquos prior distribution micro into a set of microi that aver-age to micro with Seller informed of which microi she faces Theorem 3 is proven by establishingthat we can divide micro suitably so that against each microi Seller is indifferent between pricingat all prices in the support of microi including the price corresponding to πfb

s in that environ-ment We call such a microi a (Seller-price) incentive compatible distribution or ICD (Bergemannet al (2015) call their analogous construct an extreme market) The Appendix provides aldquogreedyrdquo algorithm to compute ICDs19

We can sketch how the algorithm works and construct a set of ICDs that average to theprior Suppose V = v1 v2 vK with vi lt vi+1 for i isin 1 K minus 1 and c(v) lt v forall v Given any small-enough mass of vK there is a unique mass of type vKminus1 that makesSeller indifferent between charging price vK and vKminus1 (If the mass is too low Seller prefersvK if it is too high she prefers vKminus1) Iterating down to keep Seller indifferent betweenall prices pins down an ICD Choose the maximum mass of type vK for which this worksRemove that ICD and repeat

19 The algorithm is defined for finite V and we take limits to handle the infinite case

17

Crucially whenever an ICD is removed the price corresponding to πfbs (Γ) remains op-

timal in the remaining ldquomarketrdquo this follows from the ICDrsquos defining property of Sellerindifference and an accounting identity Therefore Sellerrsquos profit in this segmentation ofICDs remains πfb

s (Γ) Moreover it is also optimal for Seller to always (ie for each microi) priceso that there is full trade or no trade Hence Buyerrsquos expected payoff can be either 0 or theentire surplus less πfb

s (Γ) It follows that the fully-informed Buyer information structuredefined by this set of ICDs implements point B and C in Figure 1 The entire triangle ABC

can then be implemented by convexification randomizing over this information structure(and two equilibria) and full information (where Seller obtains all the surplus)

The above logic of incentive compatible distributions reduces to that of Bergemann et al(2015) in the context of monopoly pricing Even in the monopoly pricing case Theorem 3part 2 strengthens the conclusions of Bergemann et al (2015) by ensuring approximatelyunique implementation

Remark 4 Theorems 1ndash3 imply that fully-informed-Buyer information structures imple-ment all implementable payoff pairs if and only if πfb

s (Γ) = πs(Γ) In that case trianglesAFG and ABC coincide in Figure 1 It follows that πfb

s (Γ) = πs(Γ) only when a fully-informed Buyer and uninformed Seller can result in full trade (v ge E[c(v)] and Sellerprices at v) or no trade (v le E[c(v)] and Seller prices at some p ge v) Interestinglywhen πfb

s (Γ) gt πs(Γ) fully-informed-Buyer information structures cannot even implementall payoff pairs implementable by uninformed-Seller information structures ie trianglesABC and ADE in Figure 1 are distinct That is when πfb

s (Γ) gt πs(Γ) there is an uninformed-Seller information structure that implements some πs lt πfb

s (Γ)20 Theorems 2ndash3 further im-ply in this case that Buyer can obtain a strictly higher payoff when he is not fully informed(and Seller is uninformed)

4 Discussion

41 Multidimensionality

Suppose Buyer and Sellerrsquos cost and valuation pair (c v) is a two-dimensional randomvariable distributed according to joint distribution micro with a compact support The ex-tension of our maintained assumption of commonly known gains from trade is for all

20 Pick any pprime gt v such that pprimeminusE[c(v)] lt πfbs (Γ) Following the construction described after Theorem 2 we

can mix all valuations v le pprime with a fraction λ gt 0 of valuations v gt pprime so that the mixture has posterior meanexactly pprime The remaining fraction 1 minus λ of valuations above pprime are revealed to Buyer With this uninformed-Seller information structure consider any equilibrium in which Buyer purchases when indifferent (Suchan equilibrium with price-independent belief exists) Sellerrsquos profit is at most (1 minus λ)πfb

s (Γ) from any pricep gt pprime and pprime minus E[c(v)] from price p = pprime Hence Sellerrsquos profit is strictly less than πfb

s (Γ)

18

(c v) isin Supp(micro) v ge c and E[v minus c] gt 0 An information structure is now a joint distribu-tion P (tb ts v c) whose marginal distribution on (v c) is micro

We claim that the content of our key results Theorem 1Theorem 1lowast Theorem 2 andTheorem 3 still hold To see why let v be the lowest valuation in the support of micro Sellerrsquosindividual rationality constraint is now maxv minus E[c] 0 as she can guarantee this by set-ting either a sufficiently high price or a price (arbitrarily close to) v regardless of her signalWe can define a cost function c(vprime) equiv Emicro[c|v = vprime] ge vprime This results in an environment sat-isfying all the maintained assumptions of our baseline model except that c(middot) may not becontinuous Recall that such continuity is not needed for proving Theorem 1 Theorem 1lowastBoth Theorem 2 and Theorem 3 use continuity of c(middot) to guarantee that Eν [c(v)] is a contin-uous function of ν isin ∆(V ) for a key convergence result However in the two-dimensionaltype environment Eν [c] is still a continuous function of ν isin ∆(V times C) Note that Theo-rem 3 uses upper semi-continuity of Sellerrsquos profit in price boundedness of c and c le v issufficient for such upper semi-continuity

42 Negative Trading Surplus

Returning to our baseline model we next discuss cases in which trade sometimes gen-erates negative surplus That is we drop the assumption that c(v) le v we do not requireE[v minus c(v)] gt 0 either Define Sλ(Γ) for λ isin [1infin) as

Sλ(Γ) equiv983133 v

v

[v minus c(v) + λ(v minus v)]+ micro(dv)

where [middot]+ equiv max middot 0 It is readily verified that S1(Γ) = E [[v minus c(v)]+] and limλrarr0

Sλ(Γ)λ =

E[v] minus v Allowing negative trading surplus does not affect our definition of wPBE Sothe notation Π(Γ) and πs(Γ) still have the same meanings as before The next propositionshows that Π(Γ) is now characterized by three constraints as before the two individualrationality constraints πs ge πs(Γ) and πb ge 0 and different now a Pareto frontier definedby all Sλ(Γ)

Proposition 1 Consider all information structures and equilibria when trade can generate negativesurplus

Π(Γ) =

983099983105983103

983105983101

πb ge 0

(πb πs) πs ge πs(Γ)

λπb + πs le Sλ(Γ) forallλ ge 1

983100983105983104

983105983102

Proposition 1 shows that when the trade can be inefficient it may be impossible to lower

19

Sellerrsquos profit without losing total surplus (ie πb + πs) This result is intuitive at the ex-tremes Suppose for simplicity v gt E[c(v)] If πs = πs(Γ) then since πs(Γ) = v minus E[c(v)]Seller must find it optimal to price at v and trade with probability one including whentrade generates negative surplus So Buyerrsquos payoff is E[v]minus v = limλrarrinfin Sλ(Γ)λ and totalsurplus is E[v minus c(v)] which includes the maximum amount of inefficient trade On theother hand an information structure that publicly reveals (only) whether trade is efficient(ie whether v ge c(v)) can implement efficient trade with all the surplus accruing to SellerIn this case Seller obtains S1(Γ) while Buyer obtains zero The proposition establishes thatin between S1(Γ) and πs(Γ) information structures and equilibria that lower Sellerrsquos profitmust reduce total surplus as the Pareto frontier Sλ(Γ) is a concave function with maximalslope minus1

The function capturing the Pareto frontier Sλ(Γ) is the maximal weighted sum of Buyerand Seller payoff assuming that Seller will trade at the low price p = v whenever it createspositive weighted total payoff Since we place higher weight on buyer (λ ge 1) S1(Γ) isthe upper bound for the weighted total payoff In the proof of Proposition 1 we showthat these bounds are tight mdash any payoff pair on the frontier can be implemented by someinformation structure The key observation is that whenever v minus c(v) + λ(v minus v) lt 0 itfollows that vminus c(v) lt (1minusλ)(vminus v) le 0 That is it is incentive compatible for Seller to notsell when it is common knowledge that v creates negative weighted total payoff Thereforewe can simply release a public signal indicating the sign of weighted total payoff and ifpositive induce Seller to sell at v which implies all bounds Sλ(Γ)

43 Linear Cost

When Sellerrsquos cost function c(v) is linear we can explicitly characterize an informationstructure that implements Sellerrsquos minimum implementable payoff πus

s (Γ) when Seller isuninformed (or per Theorem 2 when Buyer is better informed) subject to price-independentbeliefs We note that given the discussion in Subsection 41 a linear c(v) subsumes richerenvironments in which conditional expectations are linear including Gaussian environ-ments (cf Bar-Isaac et al 2018)

Condition 1 c(v) = λv + γ for some λ isin R

Let F (v) be the cumulative distribution function (CDF) corresponding to the prior mea-sure micro Let D(micro) be the set of all distributions whose CDF G satisfies

983133

V

vdG(v) =

983133

V

vdF (v) and983133 v

v

G(s)ds le983133 v

v

F (s)ds forallv isin V

20

That is D(micro) contains all distributions that are mean-preserving-contractions (MPC) of microIt is well known that D(micro) characterizes the set of distributions of Buyer posterior meansthat can be generated by any (uninformed-Seller) information structure

Definition 2 G is an incentive compatible distribution (ICD) if

983133

sgep

(pminus c(s))G(ds) = constant ge 0 forallp isin Supp(G)

That is when Buyerrsquos (mean) valuation is distributed according to an ICD G Seller isindifferent among all prices in the support of G assuming Buyer buys when indifferent Wefocus on a special family of ICDs whose supports are intervals [vlowast v

lowast] Given Condition 1such ICDs have analytical an expression

G(v) =

983099983105983105983103

983105983105983101

1minus983061

(1minus λ)v minus γ

(1minus λ)vlowast minus γ

983062 1λminus1

if λ ∕= 1

1minus evminusvlowast

γ if λ = 1

(3)

Given any vlowast and the above G(v) an upper bound vlowast is pinned down by the conditionEG[v] = EF [v] So a family of ICDs is parametrized by a single parameter vlowast and hence anICD is denoted by CDF 983144Gvlowast(v) with density 983144gvlowast(v) It is not difficult to verify that 983144Gvlowast(v) ismonotonic in vlowast for v in the support Consequently all ICDs are ordered according to theMPC order For vlowast isin [vE[v]] when vlowast increases the ICD increases in the MPC order

Proposition 2 Suppose Condition 1 is satisfied and let plowast equiv min983153vlowast983055983055 983144Gvlowast isin D(micro)

983154 It holds that

πuss (Γ) = plowast minus Emicro[c(v)]

Proposition 2 states that an uninformed Sellerrsquos minimum payoff (with price-independentbeliefs) is characterized by the ICDs that are MPCs of the prior distribution micro By defini-tion of ICDs Sellerrsquos profit when facing ICD 983144Gvlowast is vlowast minus E[v] Sellerrsquos minimum payoff isimplemented by the most dispersed ICD that is a MPC of the prior distribution

In proving Proposition 2 the key step is to show that given the prior micro garbling Buyerrsquosinformation such that the posterior mean distribution is an ICD makes Seller weakly worseoff It is then without loss to only consider ICDs to implement πus

s (Γ) We can thus workwith a tractable one-dimensional problem Consider the set D(micro) Suppose we find G(v)

such that (i) G isin D(micro) (ii) G is an ICD and (iii) existp such that983125 p

vG(s)ds =

983125 p

vF (s)ds Such

a G is the most-dispersed ICD that is a mean preserving contraction of F Consider the

21

following two manipulations

λ

983133 v

p

F (s)ds =

983133 v

p

(pminus c(s))dF (s)minus (1minus F (v)) (pminus c(v)) + λ(v minus p) (4)

λ

983133 v

p

G(s)ds =

983133 v

p

(pminus c(s))dG(s)minus (1minusG(v)) (pminus c(v)) + λ(v minus p) (5)

First by property (iii) above the LHS of Equation 4 equals the LHS of Equation 5 Secondsince

983125 v

p(F (s) minus G(s))ds is minimized at v this implies F (p) le G(p) Therefore the first

term on RHS of Equation 4 must be larger than that of Equation 5 Notice that the firstterm on RHS is Sellerrsquos profit when offering price p This implies that the profit fromoffering p given Buyer mean-valuation distribution G(v) is lower than that given F (v)which is lower than Sellerrsquos maximum profit given F (v) On the other hand by the ICDproperty p is already Sellerrsquos optimal price given G(v) The optimal profit given valuationdistribution F must be no lower than that of G Therefore we have proved that to minimizeSellerrsquos profit it is without loss to consider only ICDs within the set D(micro) which is a onedimensional subspace The detailed proof that such ICDs exist is relegated to the appendix

The monopoly-pricing environment with c(middot) = v is covered by Condition 1 with λ = 0

and γ = v The distribution G in (3) then reduces to that identified by Roesler and Szentes(2017)

If the prior micro has binary support then the cost function c(v) is linear In this special casewe can solve for πus

s (Γ) explicitly

Corollary 1 Suppose micro has binary support V = v1 v2 Let λ = c(v2)minusc(v1)v2minusv1

and let p be theunique solution to

(pminus c(p))1

λminus1 (pminus E[c(v)]) = (v2 minus c(v2))λ

λminus1 (6)

Then πuss (Γ) = max p v2minus E[c(v)]

References

AKERLOF G A (1970) ldquoThe Market for rdquoLemonsrdquo Quality Uncertainty and the MarketMechanismrdquo Quarterly Journal of Economics 84 488ndash500

BAGWELL K AND M H RIORDAN (1991) ldquoHigh and Declining Prices Signal ProductQualityrdquo American Economic Review 81 224ndash239

BAR-ISAAC H I JEWITT AND C LEAVER (2018) ldquoAdverse Selection Efficiency and theStructure of Informationrdquo Unpublished

22

BERGEMANN D B BROOKS AND S MORRIS (2015) ldquoThe Limits of Price Discrimina-tionrdquo American Economic Review 105 921ndash57

BERGEMANN D AND S MORRIS (2016) ldquoBayes Correlated Equilibrium and the Compar-ison of Information Structures in Gamesrdquo Theoretical Economics 11 487ndash522

BLACKWELL D (1953) ldquoEquivalent Comparisons of Experimentsrdquo Annals of MathematicalStatistics 24 265ndash272

CHO I-K AND D KREPS (1987) ldquoSignaling Games and Stable Equilibriardquo Quarterly Jour-nal of Economics 102 179ndash221

DU S (2018) ldquoRobust Mechanisms Under Common Valuationrdquo Econometrica 86 1569ndash1588

DURRETT R (1995) Probability Theory and Examples Duxbury Press 2nd ed

EINAV L AND A FINKELSTEIN (2011) ldquoSelection in Insurance Markets Theory and Em-pirics in Picturesrdquo Journal of Economic Perspectives 25 115ndash38

FUDENBERG D AND J TIROLE (1991) ldquoPerfect Bayesian Equilibrium and Sequential Equi-libriumrdquo Journal of Economic Theory 53 236ndash260

GARCIA D R TEPER AND M TSUR (2018) ldquoInformation Design in Insurance MarketsSelling Peaches in a Market for Lemonsrdquo Unpublished

GREENWALD B C (1986) ldquoAdverse Selection in the Labour Marketrdquo Review of EconomicStudies 53 325ndash347

JOVANOVIC B (1982) ldquoFavorable Selection with Asymmetric Informationrdquo QuarterlyJournal of Economics 97 535ndash539

KESSLER A S (2001) ldquoRevisiting the Lemons Marketrdquo International Economic Review 4225ndash41

KREPS D AND R WILSON (1982) ldquoSequential Equilibriardquo Econometrica 50 863ndash894

LEHMANN E L (1988) ldquoComparing Location Experimentsrdquo The Annals of Statistics 16521ndash533

LEVIN J (2001) ldquoInformation and the Market for Lemonsrdquo RAND Journal of Economics32 657ndash666

23

LIBGOBER J AND X MU (2019) ldquoInformational Robustness in Intertemporal PricingrdquoWorking paper

MAKRIS M AND L RENOU (2018) ldquoInformation Design in Multi-Stage Gamesrdquo Workingpaper

MILGROM P R AND R J WEBER (1985) ldquoDistributional Strategies for Games with In-complete Informationrdquo Mathematics of Operations Research 10 619ndash632

MUSSA M AND S ROSEN (1978) ldquoMonopoly and Product Qualityrdquo Journal of EconomicTheory 18 301ndash317

RAVID D A-K ROESLER AND B SZENTES (2019) ldquoLearning Before Trading On theInefficiency of Ignoring Free Informationrdquo Working paper

ROESLER A-K AND B SZENTES (2017) ldquoBuyer-Optimal Learning and Monopoly Pric-ingrdquo American Economic Review 107 2072ndash80

TERSTIEGE S AND C WASSER (2019) ldquoBuyer-optimal Extensionproof InformationrdquoWorking paper

Appendices omitted from this version but are available on request

24

A constant c(middot) captures an environment in which there no uncertainty about Sellerrsquos costthis is the canonical monopoly pricing problem when Seller is uninformed about v andthird-degree price discrimination when Seller has some partial information while Buyer isbetter informed

Summary of results For any given environment (ie Sellerrsquos cost function and the dis-tribution of Buyerrsquos values) we seek to identify the possible market outcomes Specificallywe are interested in the ex-ante expected payoffs that obtain given sequentially rational be-havior in an equilibrium under some information structure3 We provide three results eachof which covers a different class of information structures Our main theorems are Theo-rem 1Theorem 1lowast which impose no restrictions on information and Theorem 2 whichapplies when Buyer is better informed than Seller in the sense of Blackwell (1953) in factTheorem 2 applies more broadly as elaborated later Theorem 3 concerns a fully-informedBuyer (ie who knows his value v) We view each of these three cases as intellectuallysalient and economically relevant Plainly these payoff sets must be ordered by set inclu-sion Theorem 1rsquos is the largest Theorem 2rsquos is intermediate and Theorem 3rsquos the smallestFigure 1 below summarizes

Akerlof

Full or No Information

Fully-Informed BuyerABC

Uninformed Seller(More Informed Buyer)

ADE

All Info StructuresAFG

Expected SurplusFrontier

C

E

G

0πb

Maxv-E[c(v)]0 F

A

B

D

πs

Figure 1 Outcome under different restrictions on information structures

In the figurersquos axes πb and πs represent respectively Buyerrsquos and Sellerrsquos ex-ante ex-

3 As detailed in Section 2 an information structure specifies a joint distribution of private signals foreach party conditional on the value v This induces an extensive-form game of incomplete informationOur primary solution concept is weak Perfect Bayesian equilibrium we also address refinements for ourconstructive arguments

2

pected utilities or payoffs (for readability we often drop the ldquoexpectedrdquo qualifier) Theno-trade payoffs are normalized to zero The three triangles AFG ADE and ABC depictTheorems 1ndash3 respectively That payoffs must lie within AFG is straightforward Buyercan guarantee himself a payoff of zero by not purchasing Seller can guarantee herself notonly a payoff of zero (by posting any price p gt v which will not be accepted) but alsov minus E[c(v)] (by pricing at or just below v which will be accepted) and the sum of pay-offs cannot exceed the trading surplus E[v minus c(v)] We refer to the first two constraints asindividual rationality and the third as feasibility

Theorem 1 says that every feasible and individually-rational payoff pair can be imple-mented ie obtains in an equilibrium under some information structure It is immediatethat point A obtains whens both parties learn v (full information) or neither party has anyinformation (no information) More interestingly at the point G trade occurs with probabil-ity one and Buyer obtains the entire surplus despite Seller posting the price While perhapssurprising this outcome obtains with sparse information structures For simplicitly sup-pose E[c(v)|v gt v] le maxvE[c(v)] Then Buyer can be uninformed while Seller learnswhether v = v or v gt v In equilibrium Seller prices at p = maxvE[c(v)] regardless ofher signal and Buyer purchases If Seller were to deviate to a higher price Buyer wouldreject because he believes v = v Subsection 31 explains how a single information struc-ture in fact implements every point in the triangle AFG Theorem 1lowast there discusses how aricher information structure using imperfectly-correlated signals ensures implementationin Kreps and Wilsonrsquos (1982) sequential equilibrium in discretized versions of the model

Turning to Figure 1rsquos triangle ADE Theorem 2 establishes that the payoff pair in anyequilibrium when Buyer is better informed than Seller arises in an equilibrium of an(other)information structure in which Seller is uninformed4 In other words there is no loss ofgenerality in studying an uninformed Seller so long as Buyer is better informed Whenc(middot) is increasing such information generates a game with adverse selection when c(middot)is decreasing there is favorable selection Sellerrsquos payoff along the line segment DE isthe lowest payoff she can get in any information structure in which she is uninformed5

Theorem 2 further establishes that any point within the ADE triangle can be implementedwith some such information structure by suitably varying Buyerrsquos information In fact weshow that higher slices of the triangle (ie those corresponding to larger Sellerrsquos payoff)

4 We stipulate that a better-informed Buyer does not update his value from Sellerrsquos price (even off theequilibrium path) in line with the ldquono signaling what you donrsquot knowrdquo requirement (Fudenberg and Tirole1991) that is standard in versions of Perfect Bayesian Equilibrium and implied by sequential equilibrium

5 It is because Seller cannot commit to the price as a function of her signal that she can be harmed (iereceive a payoff lower than that on the DE segment) with more information However Theorem 2 assuresthat Seller is not harmed so long as Buyer is better informed

3

can always be implemented by reducing Buyerrsquos information in the sense of Blackwell(1953) We also explain in Subsection 32 why the triangle ADE actually characterizes allpayoffs that can obtain when Buyer does not update from Sellerrsquos price even if Buyer isnot better informed than Seller

Finally the ABC triangle in Figure 1 depicts Theorem 3 which characterizes all payoffpairs when Buyer is fully informed ie learns v We use the term ldquoAkerlofrdquo to describe afully-informed Buyer and an uninformed Seller as these information structures are stan-dard in the adverse-selection literature the corresponding payoff pair is marked as suchin the figure Depending on the environmentrsquos primitives the Akerlof point can be any-where on the segment BC including at the extreme points Any feasible payoff pair thatsatisfies Buyerrsquos individual rationality and gives Seller at least her Akerlof payoff can beimplemented with a fully-informed Buyer by suitably varying Sellerrsquos information

An implication of Theorems 1ndash3 is that it is without loss in terms of ex-ante equilibriumpayoffs to focus on information structures in which Buyer is fully informed if and onlyif Sellerrsquos Akerlof payoff coincides with her individual rationality constraint This coinci-dence occurs if and only if the Akerlof market can have full trade (Seller prices at p = v

and gets payoff v minus E[c(v)] ge 0) or no trade (the price is p ge v and both partiesrsquo payoffsare 0) As detailed in Remark 4 of Subsection 33 in all other cases the point B in Fig-ure 1 is distinct from the point D (and hence also F ) which means that Buyer can obtaina higher payoff with less-than-full information while keeping Seller uninformed Further-more under a reasonable condition if Sellerrsquos individual rationality constraint is zero (iev le E[c(v)]) then point D is also distinct from F see Remark 3 in Subsection 32 WhenD and F are distinct maximizing Buyerrsquos payoff ie achieving point G requires Seller tohave some information Buyer does not and an equilibrium with price-dependent beliefsafter conditioning on his signal Buyer must update about v from the price either on or offthe equilibrium path

Related literature Our perspective and results are most closely related to BergemannBrooks and Morris (2015) Roesler and Szentes (2017) and Makris and Renou (2018 Sec-tion 4) These papersmdashonly the relevant section of the third papermdashstudy the monopolypricing problem in which there is uncertainty only about Buyerrsquos valuation This is thespecial case of our interdependent-values model with a constant function c(v) le v6

6 The online appendix of Roesler and Szentes (2017) relaxes the assumption that there are always gainsfrom trade but maintains no uncertainty about Sellerrsquos cost Related to Roesler and Szentes (2017) are alsoDu (2018) and Libgober and Mu (2019) who consider worst-case profit guarantees for Seller in static and dy-namic environments respectively Terstiege and Wasser (2019) qualify Roesler and Szentes (2017) by allowing

4

Bergemann et al (2015) assume Buyer is fully informed and hence can only vary Sellerrsquosinformation Our Theorem 3 corresponding to triangle ABC in Figure 1 is a generaliza-tion of their main result to our environment the key step in our proof methodology is toconstruct ldquoincentive compatible distributionsrdquo which reduces to their ldquoextreme marketsrdquoin the monopoly-pricing environment Another contribution of our result is to ensure ap-proximately unique implementation in a sense explained in Subsection 33

Roesler and Szentes (2017) assume Seller is uninformed and only vary Buyerrsquos infor-mation For the monopoly-pricing environment they derive one part of our Theorem 2viz they identify the triangle ADE in Figure 1 as the implementable set given Seller isuninformed Even for this result our methodology is quite different from theirs becausewe do not stipulate a linear c(middot) function our methodology delivers new insights includ-ing that noted in Remark 2 of Subsection 32 When we specialize to a linear c(middot) we canobtain a sharper characterization of the point E which extends Roesler and Szentesrsquo char-acterization of Buyer-optimal information to an interdependent-values environment seeProposition 2 in Subsection 43

While our main interest is in interdependent values our results provide new insightseven for monopoly pricing Theorem 2 implies that the Roesler and Szentes (2017) boundsare without loss so long as Buyer is better informed than Seller or more generally in equi-libria in which Buyerrsquos belief is price independent (after conditioning on his own signal)On the other hand Theorem 1 establishes that any feasible and individually rational pay-off pair can be implemented absent these restrictions in particular Buyer may even get allthe surplus This latter point has a parallel with Makris and Renou (2018) As an applica-tion of their general results on ldquorevelation principlesrdquo for information design in multi-stagegames Makris and Renoursquos (2018) Proposition 1 deduces an analog of our Theorem 1 forthe (independent values) monopoly pricing problem We share with Makris and Renou anemphasis on sequential rationality7 we differ in showing that a single information struc-ture implements all payoffs in the relevant triangle and also in establishing in Theorem 1lowast

off-the-equilibrium-path belief consistency in the sense of sequential equilibrium (Krepsand Wilson 1982) in discretizations

Other authors have studied different aspects of more specific changes of information inadverse-selection settings maintaining that one side of the market is better informed than

Seller to supply Buyer with additional information although Seller cannot have any private information ofher own

7 Makris and Renou use an apparatus of ldquosequential Bayes correlated equilibriumrdquo which we do notThey draw a contrast with Bergemann and Morrisrsquos (2016) Bayes correlated equilibrium In our approachnote that Sellerrsquos individual rationality constraint described earlier hinges when E[c(v)] lt v on Buyerrsquosbehavior being sequentially rational even off the equilibrium path

5

the other Levin (2001) identifies conditions under which the volume of trade decreaseswhen one party is kept uninformed and the otherrsquos information become more effective inthe sense of Lehmann (1988) see also Kessler (2001) Assuming a linear payoff structureBar-Isaac Jewitt and Leaver (2018) consider how certain changes in Gaussian informationaffect the volume of trade surplus and a certain quantification of adverse selection

Finally Garcia Teper and Tsur (2018) solve for socially optimal information provisionmdashrather than characterizing the range of market outcomesmdashin an insurance setting with ad-verse selection owing to a cross-subsidization motive full information disclosure is typi-cally not optimal

The rest of the paper proceeds as follows We introduce our model equilibrium con-cept(s) and certain classes of information structures in Section 2 Section 3 presents themain results implementable payoffs when the information structure is arbitrary or varieswithin canonical classes Section 4 contains discussion and extensions Proofs and someadditional material are contained in the Appendices

2 Model

21 Primitives

Seller may sell an indivisible good to Buyer Buyerrsquos value for the good is v isin V sub Rwhere V is a compact (finite or infinite) set with v equiv minV lt maxV equiv v The valuev is drawn from a probability measure micro with support V Sellerrsquos cost of production isgiven by a function c(v)8 We assume c V rarr R is continuous v minus c(v) ge 0 for all vand E[v minus c(v)] gt 0 So trading surplus is nonnegative for all v and positive for a positivemeasure of v (Throughout expectations are with respect to the prior measure micro unlessindicated otherwise lsquopositiversquo means lsquostrictly positiversquo and similarly elsewhere) Note thatfunction c(v) need not be monotonic We call Γ = (c micro) an environment We refer to anenvironment with a constant c(middot) function as that of monopoly pricing

An information structure consists of signal spaces for each party and a joint signal dis-tribution (We abuse notation and sometimes write lsquodistributionrsquo even though lsquomeasurersquowould be more precise) Formally there is a probability space (ΩF P ) complete and sep-arable metric spaces Ts and Tb (with measurable structure given by their Borel sigma alge-bras) and an integrable function X Ω rarr TstimesTbtimesV We hereafter suppress the probabilityspace and define with an abuse of notation P (D) = P (Xminus1(D)) for any D isin Ts times Tb times V

measurable Each realization of random variable X is a triplet (tb ts v) where tb isin Tb

8 Subsection 41 discusses Sellerrsquos cost being stochastic even conditional on v

6

is Buyerrsquos signal and ts isin Ts is Sellerrsquos signal For i isin s b v let Pi denote the corre-sponding marginal distribution of P on dimension of Ti with the convention Tv equiv V Werequire Pv = micro this is the iterated expectation or Bayes plausibility requirement Denotean information structure by τ

The environment Γ and information structure τ define the following game

1 The random variables (tb ts v) are realized Signal tb is privately observed by Buyerand signal ts privately observed by Seller Neither party observes v

2 Seller posts a price p isin R

3 Buyer accepts or rejects the price If Buyer accepts his von-Neumann Morgensternpayoff is v minus p and Sellerrsquos is p minus c(v) If Buyer rejects both partiesrsquo payoffs arenormalized to 0

Note that because the signal spaces Tb and Ts are abstract and the two partiesrsquo signalscan be arbitrarily correlated conditional on v there is no loss of generality in assuming thateach party privately observes their own signal For example public information can becaptured by perfectly correlating (components of) tb and ts

22 Strategies and Equilibria

In the game defined by (Γ τ) denote Sellerrsquos strategy by σ and Buyerrsquos by α FollowingMilgrom and Weber (1985) we define σ as a distributional strategy σ is a joint distributionon R times Ts whose marginal distribution on Ts must be the Sellerrsquos signal distribution Soσ(middot|ts) is Sellerrsquos price distribution given her signal ts9 Buyerrsquos strategy α R times Tb rarr[0 1] maps each price-signal pair (p tb) into a trading probability A strategy profile (σα)

induces expected utilities for Buyer and Seller (πb πs) in the natural way

πb =

983133(v minus p)α(tb p)σ(dp|ts)P (dts dtb dv)

πs =

983133(pminus c(v))α(tb p)σ(dp|ts)P (dts dtb dv)

Our baseline equilibrium concept is weak Perfect Bayesian equilibrium Since Sellerrsquosaction is not preceded by Buyerrsquos we can dispense with specifying beliefs for Seller ForBuyer it suffices to focus on his belief about the value v given his signal and the price wedenote this distribution by ν(v|p tb)

9 Here σ(middot|ts) is the regular conditional distribution which exists and is unique almost everywhere becauseTs is a standard Borel space (Durrett 1995 pp 229ndash230) Similarly for subsequent such notation we dropldquoalmost everywhererdquo qualifiers unless essential

7

Definition 1 A strategy profile (σα) and a belief ν(v|p tb) is a weak perfect Bayesian equilib-rium (wPBE) of game (Γ τ) if

1 Buyer plays optimally at every information set given his belief

α(p tb) =

983099983103

9831011 if Eν(v|ptb)[v] gt p

0 if Eν(v|ptb)[v] lt p

2 Seller plays optimally

σ isin argmaxσ

983133(pminus c(v))α(p tb)σ(dp|ts)P (dts dtb dv)

3 Beliefs satisfy Bayes rule on path for every measurable D sube Rtimes Ts times Tb times V

983133

D

ν(dv|p tb)σ(dp|ts)P (dts dtb V ) =

983133

D

σ(dp|ts)P (dts dtb dv)

Notice that we have formulated Sellerrsquos optimality requirement ex ante but Buyerrsquos ateach information set The latter is needed to capture sequential rationality The formeris for (notational) convenience this choice is inconsequential because Seller moves beforeBuyer

Hereafter ldquoequilibriumrdquo without qualification refers to a wPBE As is well understoodwPBE permits significant latitude in beliefs off the equilibrium path We will discuss re-finements when appropriate

23 Implementable Payoffs and Canonical Information Structures

We now define the set of implementable equilibrium outcomesmdashthat is the payoffs thatobtain in some equilibrium under some information structuremdashand some canonical classesof information structures

For a game (Γ τ) let the equilibrium payoff set be

Π(Γ τ) equiv (πb πs) exist wPBE of (Γ τ) with payoffs (πb πs)

Denote the class of all information structures by T and define

Π(Γ) equiv983134

τisinT

Π(Γ τ)

8

That is for environment Γ Π(Γ) is the set of all equilibrium payoff pairs that obtain undersome information structure

Uninformed Seller An information structure has uninformed Seller if Ts is a singletonSellerrsquos own signal contains no information about Buyerrsquos value v and hence neither aboutSellerrsquos cost c(v) When discussing such information structures we write the associateddistribution as just P (tb v) and Sellerrsquos strategy as just σ(p) omitting the argument ts inboth cases The class of all uninformed-Seller information structure is denoted Tus

Fully-informed Buyer An information structure has fully-informed Buyer if Buyerrsquos signalfully reveals his value v Formally this holds if Tb = V and the conditional distribution onV P (middot|tb) satisfies P (tb|tb) = 1 We denote the class of fully-informed-Buyer informationstructures Tfb

More-informed Buyer An information structure has more-informed Buyer if Buyer hasmore information than Seller Formally this holds when v and ts are independent condi-tional on tb ie for any measurable Ds sub Ts and Dv sub V P (DscapDv|tb) = P (Ds|tb)P (Dv|tb)10

Another way to interpret this requirement is that random variable tb must be statisticallysufficient for ts with respect to v ie tb is more informative than ts about v in the sense ofBlackwell (1953) We denote the class of more-informed-Buyer information structures byTmb Naturally information structures with uninformed Seller or fully-informed Buyer arecases of more-informed Buyer both Tus and Tfb are subclasses of Tmb

No updating from price For more-informed-Buyer information structures it is desirableto impose further requirements on Buyerrsquos equilibrium belief Since Sellerrsquos price can onlydepend on her own signal and this signal contains no additional information about v givenBuyerrsquos signal the price is statistically uninformative about v given Buyerrsquos signal Con-sequently Buyerrsquos posterior belief should be price independent once his signal has beenconditioned upon Formally regardless of the price p the equilibrium belief ν(middot|p tb) mustsatisfy 983133

ν(dv|p tb)P (dtb Ts V ) =

983133P (dtb Ts dv)

We refer to this condition as price-independent belief Note that the condition is meaningfulregardless of whether Buyer is more informed than Seller In a more-informed-Buyer in-formation structure price-independent belief would be implied by the ldquono signaling what

10 We write sub to mean ldquoweak subsetrdquo

9

you donrsquot knowrdquo requirement (Fudenberg and Tirole 1991) frequently imposed in versionsof perfect Bayesian equilibrium and the concept of sequential equilibrium (Kreps and Wil-son 1982) in finite versions of our setting11

Some notation will be helpful Define

Πlowast(Γ τ) equiv (πb πs) exist wPBE of (Γ τ) with price-independent belief and payoffs(πb πs)

Πlowast(Γ) equiv983134

τisinT

Πlowast(Γ τ)

Πlowasti (Γ) equiv

983134

τisinTi

Πlowast(Γ τ) for i = us fbmb

So Πlowast and Πlowast are analogous to the implementable payoff sets Π and Π defined earlier butrestricted to equilibria with price-independent beliefs Πlowast

us Πlowastfb and Πlowast

mb are the imple-mentable payoff sets when further restricted to uninformed-Seller fully-informed-Buyerand more-informed-Buyer information structures Plainly for any environment Γ

Πlowastus(Γ) cupΠlowast

fb(Γ) sub Πlowastmb(Γ) sub Πlowast(Γ) sub Π(Γ)

3 Main Results

Our goal is to characterize equilibrium payoff pairs across information structures in anarbitrary environment Γ In particular we seek to characterize the five sets Π(Γ) Πlowast(Γ)Πlowast

mb(Γ) Πlowastus(Γ) and Πlowast

fb(Γ) Let

S(Γ) equiv E[v minus c(v)]

be the (expected) surplus from trade in environment Γ This quantity will play an impor-tant role

31 All Information Structures

Define Sellerrsquos payoff guarantee as

πs(Γ) equiv max v minus E [c(v)] 0

11 An example clarifying our terminology may be helpful If both Seller and Buyer are fully informed of vthen the natural equilibriummdashthe unique sequential equilibrium in a finite version of the gamemdashhas Sellerpricing at p = v and Buyerrsquos belief being degenerate on v regardless of Sellerrsquos price This equilibrium hasprice-independent belief even though ex-ante Sellerrsquos price are Buyerrsquos belief are perfectly correlated Thepoint is that Buyerrsquos belief does not depend on price conditional on his signal

10

To interpret this observe that it is optimal for Buyer to accept the price v no matter hisbelief Therefore Seller can guarantee herself the (expected) profit v minus E[c(v)] no matterwhat the information structure is More precisely she can guarantee v minus E[c(v)] minus ε forany ε gt 0 since sequential rationality requires Buyer to accept any price v minus ε SimilarlySeller can also guarantee zero profit offering a price p gt v Hence Sellerrsquos payoff in anyequilibrium of any information structure must be at least the payoff guarantee πs(Γ) above

On the other hand Buyer can guarantee himself the payoff πb = 0 by rejecting all pricesIt follows that the implementable set Π(Γ) must satisfy three simple constraints 1) Sellerrsquosldquoindividual rationalityrdquo constraint πs ge πs(Γ) 2) Buyerrsquos ldquoindividual rationalityrdquo con-straint πb ge 0 and 3) the feasibility constraint πb + πs le S(Γ)

Our first result is that these individual rationality and feasibility constraints are alsosufficient for a payoff pair to be implementable

Theorem 1 Consider all information structures and equilibria

Π(Γ) =

983099983105983103

983105983101

πb ge 0

(πb πs) πs ge πs(Γ)

πb + πs le S(Γ)

983100983105983104

983105983102

Theorem 1 says that the set Π(Γ) corresponds to the triangle AFG in Figure 1 In par-ticular Buyer can receive the entire surplus beyond Sellerrsquos payoff guarantee This is sur-prising as Seller has substantial bargaining power Note that when v le E[c(v)] an entirelyreasonable condition Sellerrsquos payoff guarantee is zero12

The proof of Theorem 1 is in fact straightforward Suppose for expositional simplicityv ge E[c(v)] Fix the trivial information structure in which neither player receives any infor-mation and consider the following family of strategy profiles Seller randomizes betweentwo prices some pl isin [vE[v]] and ph = E[v] Buyer accepts pl with probability one andaccepts ph with probability α(ph) where α(ph) isin [0 1] is specified to make Seller indifferentbetween the two prices That is α(ph)(ph minus E[c(v)]) = pl minus E[c(v)] The expected payoffsfrom this strategy profile are

πb = σ(pl)(E[v]minus pl) and πs = pl minus E[c(v)]

As pl traverses the interval [vE[v]] Sellerrsquos payoff πs traverses [πs(Γ) S(Γ)] Given any

12 In monopoly pricing with c(middot) = v Sellerrsquos payoff guarantee of zero is lower than the revenue guaranteeidentified in Du (2018 Section 5) which is typically positive Dursquos notion is different from ours

11

pl Buyerrsquos payoff πb traverses [0 S(Γ)minusπs] as σ(pl) traverses [0 1] Therefore the proposedstrategy profiles induce all the payoff pairs stated in Theorem 1

We are left to specify beliefs ν for Buyer After prices pl and ph Buyer holds the priorbelief micro After any other (necessarily off-path) price Buyerrsquos belief is that v = v and soBuyer rejects all prices p isin [vinfin)pl ph It is straightforward to confirm that the specified(σα ν) constitute a wPBE

To get more insight into this construction consider its implication for the monopolypricing with c(middot) = v The equilibrium with pl = v (hence α(ph) = 0) and σ(pl) = 1

corresponds to the monopolist deterministically pricing at v and Buyer purchasing Giventhat both sides of the market receive no information why doesnrsquot the monopolist deviate toany price in (vE[v]) The reason is that in this equilibrium the consumer will then not buybecause he updates to v = v Such belief is compatible with wPBE because the equilibriumconcept places no restrictions on off-path beliefs This may seem like a game-theoreticmisdirection Buyerrsquos belief is not consistent with ldquono signaling what you donrsquot knowrdquoPut differently since we have a (weakly) more-informed Buyer information structure weought to impose the price-independent belief condition described in Subsection 23 thatwould imply Buyer must purchase at any price p lt E[v]

But the message of Theorem 1 does not rely on the permissiveness of wPBE To illus-trate continue with the above monopoly-pricing environment and suppose v has positiveprior probability Consider Buyer remaining uninformed but Seller learning whether v = v

or v gt v Now Buyerrsquos off-path belief that v = v is consistent with ldquono signaling whatyou donrsquot knowrdquo More generally using richer information structures we can prove thatany payoff pair identified in Theorem 1 can be approximately implemented as a sequentialequilibrium (Kreps and Wilson 1982) in a suitably discretized game

Theorem 1lowast Fix any ε gt 0 There is a finite information structure and a finite price grid inducinga game with a set of sequential equilibrium payoffs that is an ε-net of Π(Γ)13

In fact the proof of Theorem 1lowast establishes even more the sequential equilibria in thediscretized games satisfy a natural version of the D1 refinement (Cho and Kreps 1987) Werelegate the logic to the Appendix but mention here that we use imperfectly-correlatedsignals for Buyer and Seller14

13 An information structure is finite if the signal spaces Tb and Ts are finite Sequential equilibrium isdefined in the obvious way for the ldquoinducedrdquo finite game where Nature directly draws (tb ts) rather thanfirst drawing v and playersrsquo payoffs are defined directly using (tb ts) For X sub R2 and ε gt 0 the set A sub Xis an ε-net of X if for each x isin X there is a isin A such that 983042xminus a983042 lt ε where 983042middot983042 is the Euclidean distance

14 The idea behind D1 is to ask for any off-path price whether one type of Seller would deviate for any

12

Even when our environment is specialized to monopoly pricing it is worth highlightingtwo contrasts between Theorem 11lowast and results of Bergemann et al (2015) and Roeslerand Szentes (2017) First we find that by not restricting the monopolist to be uninformedthe implementable payoff set typically expands rather dramatically trade can be efficientwith the monopolist securing none of the surplus beyond her payoff guarantee (πs) whichmay be zero (Roesler and Szentes establish implicitly that the implementable set withan uninformed monopolist is a superset of Bergemann et alrsquos which has a fully-informedconsumer) We will see in Subsection 32 that what is crucial to this expansion is price-dependent beliefs In particular the proof of Theorem 1lowast uses an information structure inwhich Buyer is not better informed than Seller mdash if he were then sequential equilibriumwould imply price-independent beliefs Second Theorem 1lowast establishes that for a givenε gt 0 a single information structure (and price grid) can be used to approximate the entirepayoff set Π(Γ) analogously to the construction described after Theorem 1 that used asingle information structure15 Bergemann et al (2015) and Roesler and Szentes (2017) onthe other hand vary information structures to span their payoff sets

32 More-informed Buyer and Price-independent Beliefs

In some economic settings it is plausible that Buyer is more informed than Seller Howdoes a restriction to such information structures ie τ isin Tmb affect the implementablepayoff set It turns out that what is in fact crucial is price-independent beliefs We have ex-plained earlier why it is desirable to impose this condition when Buyer is more informedbut that the condition is well defined even otherwise If Buyer is not more informed thanSeller price-independent beliefs ought to be viewed as an equilibrium restriction in gen-eral To reduce repetition readers should be bear in mind for the rest of this subsectionthe qualifier ldquowith price-independent beliefsrdquo applies unless stated explicitly otherwise

It is useful to define

πuss (Γ) equiv inf πs exist(πb πs) isin Πlowast

us(Γ) (1)

as the lowest payoff that an uninformed Seller can obtain no matter Buyerrsquos information(among equilibria with price-independent beliefs) Plainly πus

s (Γ) ge πs(Γ) In monopoly

Buyer mixed response that another type would Our construction has multiple Buyer types that are imper-fectly correlated with Seller types So different types of Seller have different beliefs about Buyer types Thisblunts the power of dominance considerations to the point where D1 does not exclude any Seller type fromthe support of Buyerrsquos off-path belief

15 In fact if one lets the price grid vary with ε then a single information structure implements exactlyrather than approximately in sequential equilibrium all payoffs ε-away from the boundary of Π(Γ) See in the Appendix for a formal statement

13

pricing with V = [v v] and c(middot) = v Roesler and Szentesrsquos (2017) characterization of theconsumer-optimal information structure identifies πus

s establishing that πuss gt πs If there

is no trade due to adverse selection when Seller is uniformed and Buyer has some informa-tion then πus

s = πs = 0 We do not have a general explicit formula for πuss Subsection 43

provides it for linear c(middot) Nonetheless we establish next that (i) the only additional restric-tion on equilibrium payoffs imposed by price-independent beliefs is a lower bound of πus

s

for Seller and (ii) uninformed-Seller information structures implement all such payoffs

Theorem 2 Consider equilibria with price-independent beliefs

1 Πlowast(Γ) = Πlowastmb(Γ) = Πlowast

us(Γ)

2 Πlowastus(Γ) = (πb πs) isin Π(Γ) πs ge πus

s (Γ)

3 For any (πb πs) isin Πlowastus(Γ) with πs gt πus

s (Γ) there is τ isin Tus with Π(Γ τ) = (πb πs)

Remark 1 We believe the substance of Theorem 2 would hold using discretizations and se-quential equilibria analogous to Theorem 1lowast As previously noted sequential equilibriumimplies price-independent beliefs when Buyer is more informed than Seller

To digest Theorem 2 note that Πlowast(Γ) sup Πlowastmb(Γ) sup Πlowast

us(Γ) is trivial So part 1 of thetheorem amounts to establishing the reverse inclusions The intuition for thismdashgiven part2rsquos characterization of Πlowast

usmdashis fairly straightforward with price-independent beliefs ad-ditional information cannot harm Seller even though it could alter the set of equilibria SoSellerrsquos lowest payoff obtains when she is uninformed

The characterization in part 2 of payoffs with an uninformed Seller corresponds to thetriangle ADE in Figure 1 Part 3 of the theorem assures ldquounique implementationrdquo of allimplementable payoffs satisfying πs gt πus

s (Γ) That is for any such payoff pair there is anuninformed-Seller information structure such that all equilibria (with price-independentbeliefs) induce exactly that payoff pair Unique implementation is appealing for multiplereasons one of which is that it obviates concerns about which among multiple payoff-distinct equilibria is more reasonable For example Ravid Roesler and Szentes (2019)bring to the fore the issue of equilibrium multiplicity in Roesler and Szentesrsquos (2017) infor-mation structure

Let us describe how we obtain the characterization of Πlowastus(Γ) and unique implementa-

tion There are two steps The first ensures that there is some information structure callit τ lowast isin Tus that implements Sellerrsquos payoff πus

s (Γ) That is we ensure that the infimum

14

in (1) is in fact a minimum16 While this argument is technical knowing τ lowast exists is usefulin what follows The second and economically insightful step is to construct informa-tion structures that implement every point in the triangle Πlowast

us(Γ) by suitably garbling theinformation structure τ lowast The construction is illustrated in Figure 2 Consider the distri-bution of Buyerrsquos posterior mean of his valuation v in information structure τ lowast (Givenprice-independent beliefs Buyerrsquos posterior mean is sufficient for his decision) For sim-plicity suppose this posterior-mean distribution has a density as depicted by the red curvein Figure 2 Fix any (πb πs) isin Πlowast

us(Γ)

First there is some number zlowast such that πb + πs is the total surplus from trading onlywhen Buyerrsquos posterior mean is greater than zlowast Next there is some price plowast ge zlowast suchthat Sellerrsquos payoff is πs if all these trades were to occur at price plowast17 Note that plowast must beno larger than the expected Buyer posterior mean conditional on that being above zlowast forotherwise πb lt 0 We claim that the information structure τ lowast can be garbled so that plowast isan equilibrium price and trade occurs only when Buyerrsquos posterior mean is greater thanzlowast The garbling is illustrated in Figure 2 by the blue distribution There is one signal that

pzv

ν(v)

Figure 2 Construction of information structure in Theorem 2

Buyer receives when the original posterior mean is between zlowast and plowast and also receiveswith some probability when the original posterior mean is above plowast The probability ischosen to make the posterior mean from this signal exactly plowast Apart from this one newsignal Buyer receives the original signal in τ lowast Plainly this is a garbling of τ lowast and hence isfeasible

16 The difficulty is in establishing suitable continuity Uninformed-Seller information structures can beviewed as probability measures over Buyerrsquos beliefs with convergence in the sense of the weak topologyThis topology ensures continuity with respect to probability measures of expectations of continuous orat least Lipschitz (and bounded) functions However Sellerrsquos expected payoff is not the expectation of aLipschitz function as Sellerrsquos profit is truncated at the price she charges

17 That plowast ge zlowast follows from πs ge πuss (Γ) as πs(Γ) itself is weakly larger than Sellerrsquos payoff from posting

price zlowast (and thus trading with the same set of Buyer posterior means) under information structure τlowast

15

Figure 2 makes clear why the new information structure has an equilibrium with priceplowast and Buyer breaking indifference in favor of trading (i) Sellerrsquos profit from posting anyprice below zlowast is the same as under τ lowast and hence no larger than πus

s (Γ) (ii) similarly Sellerrsquosprofit from posting any price above plowast is no higher than some fraction of πus

s (Γ) and (iii)any price between zlowast and plowast is worse that price plowast Moreover since Sellerrsquos profit fromoffering any price other than plowast is no more than πus

s (Γ) it follows that when πs gt πuss (Γ)

Buyer must break indifference as specified for Seller to have an optimal price and theequilibrium payoffs are unique

Remark 2 The above logic establishes that given any τ isin Tus that implements some (πb πs)τ can be garbled to uniquely implement any (πprime

b πprimes) isin Π(Γ) such that πprime

s gt πs That isan uninformed Sellerrsquos payoff can always be strictly raised and Buyerrsquos payoff reduced(strictly so long as it was not already zero) by garbling Buyerrsquos information

Remark 3 According to Theorem 11lowast and Theorem 2 uninformed-Seller information struc-tures cannot implement all implementable payoff pairs in an environment Γ if and only ifπuss (Γ) gt πs(Γ) This inequality fails in environment Γ if πus

s (Γ) = 0 since that impliesπuss (Γ) = πs(Γ) = 0 An example is when there is no trade due to adverse selection when

Seller is uniformed and Buyer has some information On the other hand it is sufficient forthe inequality that

v le E[c(v)] and forallv isin V c(v) lt v (2)

To see why (2) implies πuss (Γ) gt πs(Γ) notice that in any uninformed-Seller information

structure Seller can price at slightly less than Buyerrsquos highest posterior mean valuationand guarantee trade with only (a neighborhood of) that Buyer type If c(v) lt v for all v thisgives Seller a positive expected payoff and hence πus

s (Γ) gt 018 But the first inequality in (2)is equivalent to πs(Γ) = 0 We observe that Condition (2) is compatible with severe adverseselection resulting in very little trade when Seller is uninformed and Buyer (partially orfully) informed

33 Fully-Informed Buyer

We now turn to the third canonical class of information structures Buyer is fully in-formed of his value v As this is a special case of more-informed Buyer we maintain price-independent beliefs throughout this subsection

18 More precisely as V is compact c(v) lt v for all v implies there exists ε gt 0 such that v minus c(v) gt εGiven any uninformed-Seller information structure let mv be the highest posterior mean valuation in thesupport of the posterior means induced by Buyerrsquos signals So there is positive probability of Buyer signalswith posterior mean valuations at least mv minus ε2 By pricing at mv minus ε2 Sellerrsquos expected cost conditionalon trade is bounded above by mv minus ε and hence Sellerrsquos profit conditional on trade is at least ε2 gt 0 Itfollows that πus

s gt 0

16

Faced with a fully informed Buyer and any sequentially rational Buyer strategy an un-informed Seller can guarantee a profit level

πfbs (Γ) equiv sup

p

983133 v

p

(pminus c(v))micro(dv)

regardless of her information Plainly πfbs (Γ) ge πus

s (Γ) In monopoly pricing with c(middot) = vRoesler and Szentes (2017) have shown that πfb

s gt πuss if there is no trade due to adverse

selection when Buyer is fully informed and Seller is uniformed then πfbs = πus

s = 0 Belowwe establish that when Buyer is fully informed πfb

s is the only additional constraint onequilibrium payoffs Let Bε(πb πs) denote an ε-ball around (πb πs)

Theorem 3 Consider a fully informed Buyer and price-independent beliefs

1 Πlowastfb(Γ) = (πb πs) isin Π(Γ) πs ge πfb

s (Γ)

2 For any (πb πs) isin Πlowastfb(Γ) and any ε gt 0 there is τ isin Tfb with Π(Γ τ) sub Bε(πb πs)

The payoff set characterized in Theorem 3 corresponds to the triangle ABE in Figure 1Part 2 of the theorem establishes ldquoapproximately unique implementationrdquo for any point inthe triangle there is an information structure under which all equilibria yield payoff pairsin a neighborhood of that point

Here is the idea behind part 1 of Theorem 3 When Buyer is fully informed an informa-tion structure can be viewed as dividing vrsquos prior distribution micro into a set of microi that aver-age to micro with Seller informed of which microi she faces Theorem 3 is proven by establishingthat we can divide micro suitably so that against each microi Seller is indifferent between pricingat all prices in the support of microi including the price corresponding to πfb

s in that environ-ment We call such a microi a (Seller-price) incentive compatible distribution or ICD (Bergemannet al (2015) call their analogous construct an extreme market) The Appendix provides aldquogreedyrdquo algorithm to compute ICDs19

We can sketch how the algorithm works and construct a set of ICDs that average to theprior Suppose V = v1 v2 vK with vi lt vi+1 for i isin 1 K minus 1 and c(v) lt v forall v Given any small-enough mass of vK there is a unique mass of type vKminus1 that makesSeller indifferent between charging price vK and vKminus1 (If the mass is too low Seller prefersvK if it is too high she prefers vKminus1) Iterating down to keep Seller indifferent betweenall prices pins down an ICD Choose the maximum mass of type vK for which this worksRemove that ICD and repeat

19 The algorithm is defined for finite V and we take limits to handle the infinite case

17

Crucially whenever an ICD is removed the price corresponding to πfbs (Γ) remains op-

timal in the remaining ldquomarketrdquo this follows from the ICDrsquos defining property of Sellerindifference and an accounting identity Therefore Sellerrsquos profit in this segmentation ofICDs remains πfb

s (Γ) Moreover it is also optimal for Seller to always (ie for each microi) priceso that there is full trade or no trade Hence Buyerrsquos expected payoff can be either 0 or theentire surplus less πfb

s (Γ) It follows that the fully-informed Buyer information structuredefined by this set of ICDs implements point B and C in Figure 1 The entire triangle ABC

can then be implemented by convexification randomizing over this information structure(and two equilibria) and full information (where Seller obtains all the surplus)

The above logic of incentive compatible distributions reduces to that of Bergemann et al(2015) in the context of monopoly pricing Even in the monopoly pricing case Theorem 3part 2 strengthens the conclusions of Bergemann et al (2015) by ensuring approximatelyunique implementation

Remark 4 Theorems 1ndash3 imply that fully-informed-Buyer information structures imple-ment all implementable payoff pairs if and only if πfb

s (Γ) = πs(Γ) In that case trianglesAFG and ABC coincide in Figure 1 It follows that πfb

s (Γ) = πs(Γ) only when a fully-informed Buyer and uninformed Seller can result in full trade (v ge E[c(v)] and Sellerprices at v) or no trade (v le E[c(v)] and Seller prices at some p ge v) Interestinglywhen πfb

s (Γ) gt πs(Γ) fully-informed-Buyer information structures cannot even implementall payoff pairs implementable by uninformed-Seller information structures ie trianglesABC and ADE in Figure 1 are distinct That is when πfb

s (Γ) gt πs(Γ) there is an uninformed-Seller information structure that implements some πs lt πfb

s (Γ)20 Theorems 2ndash3 further im-ply in this case that Buyer can obtain a strictly higher payoff when he is not fully informed(and Seller is uninformed)

4 Discussion

41 Multidimensionality

Suppose Buyer and Sellerrsquos cost and valuation pair (c v) is a two-dimensional randomvariable distributed according to joint distribution micro with a compact support The ex-tension of our maintained assumption of commonly known gains from trade is for all

20 Pick any pprime gt v such that pprimeminusE[c(v)] lt πfbs (Γ) Following the construction described after Theorem 2 we

can mix all valuations v le pprime with a fraction λ gt 0 of valuations v gt pprime so that the mixture has posterior meanexactly pprime The remaining fraction 1 minus λ of valuations above pprime are revealed to Buyer With this uninformed-Seller information structure consider any equilibrium in which Buyer purchases when indifferent (Suchan equilibrium with price-independent belief exists) Sellerrsquos profit is at most (1 minus λ)πfb

s (Γ) from any pricep gt pprime and pprime minus E[c(v)] from price p = pprime Hence Sellerrsquos profit is strictly less than πfb

s (Γ)

18

(c v) isin Supp(micro) v ge c and E[v minus c] gt 0 An information structure is now a joint distribu-tion P (tb ts v c) whose marginal distribution on (v c) is micro

We claim that the content of our key results Theorem 1Theorem 1lowast Theorem 2 andTheorem 3 still hold To see why let v be the lowest valuation in the support of micro Sellerrsquosindividual rationality constraint is now maxv minus E[c] 0 as she can guarantee this by set-ting either a sufficiently high price or a price (arbitrarily close to) v regardless of her signalWe can define a cost function c(vprime) equiv Emicro[c|v = vprime] ge vprime This results in an environment sat-isfying all the maintained assumptions of our baseline model except that c(middot) may not becontinuous Recall that such continuity is not needed for proving Theorem 1 Theorem 1lowastBoth Theorem 2 and Theorem 3 use continuity of c(middot) to guarantee that Eν [c(v)] is a contin-uous function of ν isin ∆(V ) for a key convergence result However in the two-dimensionaltype environment Eν [c] is still a continuous function of ν isin ∆(V times C) Note that Theo-rem 3 uses upper semi-continuity of Sellerrsquos profit in price boundedness of c and c le v issufficient for such upper semi-continuity

42 Negative Trading Surplus

Returning to our baseline model we next discuss cases in which trade sometimes gen-erates negative surplus That is we drop the assumption that c(v) le v we do not requireE[v minus c(v)] gt 0 either Define Sλ(Γ) for λ isin [1infin) as

Sλ(Γ) equiv983133 v

v

[v minus c(v) + λ(v minus v)]+ micro(dv)

where [middot]+ equiv max middot 0 It is readily verified that S1(Γ) = E [[v minus c(v)]+] and limλrarr0

Sλ(Γ)λ =

E[v] minus v Allowing negative trading surplus does not affect our definition of wPBE Sothe notation Π(Γ) and πs(Γ) still have the same meanings as before The next propositionshows that Π(Γ) is now characterized by three constraints as before the two individualrationality constraints πs ge πs(Γ) and πb ge 0 and different now a Pareto frontier definedby all Sλ(Γ)

Proposition 1 Consider all information structures and equilibria when trade can generate negativesurplus

Π(Γ) =

983099983105983103

983105983101

πb ge 0

(πb πs) πs ge πs(Γ)

λπb + πs le Sλ(Γ) forallλ ge 1

983100983105983104

983105983102

Proposition 1 shows that when the trade can be inefficient it may be impossible to lower

19

Sellerrsquos profit without losing total surplus (ie πb + πs) This result is intuitive at the ex-tremes Suppose for simplicity v gt E[c(v)] If πs = πs(Γ) then since πs(Γ) = v minus E[c(v)]Seller must find it optimal to price at v and trade with probability one including whentrade generates negative surplus So Buyerrsquos payoff is E[v]minus v = limλrarrinfin Sλ(Γ)λ and totalsurplus is E[v minus c(v)] which includes the maximum amount of inefficient trade On theother hand an information structure that publicly reveals (only) whether trade is efficient(ie whether v ge c(v)) can implement efficient trade with all the surplus accruing to SellerIn this case Seller obtains S1(Γ) while Buyer obtains zero The proposition establishes thatin between S1(Γ) and πs(Γ) information structures and equilibria that lower Sellerrsquos profitmust reduce total surplus as the Pareto frontier Sλ(Γ) is a concave function with maximalslope minus1

The function capturing the Pareto frontier Sλ(Γ) is the maximal weighted sum of Buyerand Seller payoff assuming that Seller will trade at the low price p = v whenever it createspositive weighted total payoff Since we place higher weight on buyer (λ ge 1) S1(Γ) isthe upper bound for the weighted total payoff In the proof of Proposition 1 we showthat these bounds are tight mdash any payoff pair on the frontier can be implemented by someinformation structure The key observation is that whenever v minus c(v) + λ(v minus v) lt 0 itfollows that vminus c(v) lt (1minusλ)(vminus v) le 0 That is it is incentive compatible for Seller to notsell when it is common knowledge that v creates negative weighted total payoff Thereforewe can simply release a public signal indicating the sign of weighted total payoff and ifpositive induce Seller to sell at v which implies all bounds Sλ(Γ)

43 Linear Cost

When Sellerrsquos cost function c(v) is linear we can explicitly characterize an informationstructure that implements Sellerrsquos minimum implementable payoff πus

s (Γ) when Seller isuninformed (or per Theorem 2 when Buyer is better informed) subject to price-independentbeliefs We note that given the discussion in Subsection 41 a linear c(v) subsumes richerenvironments in which conditional expectations are linear including Gaussian environ-ments (cf Bar-Isaac et al 2018)

Condition 1 c(v) = λv + γ for some λ isin R

Let F (v) be the cumulative distribution function (CDF) corresponding to the prior mea-sure micro Let D(micro) be the set of all distributions whose CDF G satisfies

983133

V

vdG(v) =

983133

V

vdF (v) and983133 v

v

G(s)ds le983133 v

v

F (s)ds forallv isin V

20

That is D(micro) contains all distributions that are mean-preserving-contractions (MPC) of microIt is well known that D(micro) characterizes the set of distributions of Buyer posterior meansthat can be generated by any (uninformed-Seller) information structure

Definition 2 G is an incentive compatible distribution (ICD) if

983133

sgep

(pminus c(s))G(ds) = constant ge 0 forallp isin Supp(G)

That is when Buyerrsquos (mean) valuation is distributed according to an ICD G Seller isindifferent among all prices in the support of G assuming Buyer buys when indifferent Wefocus on a special family of ICDs whose supports are intervals [vlowast v

lowast] Given Condition 1such ICDs have analytical an expression

G(v) =

983099983105983105983103

983105983105983101

1minus983061

(1minus λ)v minus γ

(1minus λ)vlowast minus γ

983062 1λminus1

if λ ∕= 1

1minus evminusvlowast

γ if λ = 1

(3)

Given any vlowast and the above G(v) an upper bound vlowast is pinned down by the conditionEG[v] = EF [v] So a family of ICDs is parametrized by a single parameter vlowast and hence anICD is denoted by CDF 983144Gvlowast(v) with density 983144gvlowast(v) It is not difficult to verify that 983144Gvlowast(v) ismonotonic in vlowast for v in the support Consequently all ICDs are ordered according to theMPC order For vlowast isin [vE[v]] when vlowast increases the ICD increases in the MPC order

Proposition 2 Suppose Condition 1 is satisfied and let plowast equiv min983153vlowast983055983055 983144Gvlowast isin D(micro)

983154 It holds that

πuss (Γ) = plowast minus Emicro[c(v)]

Proposition 2 states that an uninformed Sellerrsquos minimum payoff (with price-independentbeliefs) is characterized by the ICDs that are MPCs of the prior distribution micro By defini-tion of ICDs Sellerrsquos profit when facing ICD 983144Gvlowast is vlowast minus E[v] Sellerrsquos minimum payoff isimplemented by the most dispersed ICD that is a MPC of the prior distribution

In proving Proposition 2 the key step is to show that given the prior micro garbling Buyerrsquosinformation such that the posterior mean distribution is an ICD makes Seller weakly worseoff It is then without loss to only consider ICDs to implement πus

s (Γ) We can thus workwith a tractable one-dimensional problem Consider the set D(micro) Suppose we find G(v)

such that (i) G isin D(micro) (ii) G is an ICD and (iii) existp such that983125 p

vG(s)ds =

983125 p

vF (s)ds Such

a G is the most-dispersed ICD that is a mean preserving contraction of F Consider the

21

following two manipulations

λ

983133 v

p

F (s)ds =

983133 v

p

(pminus c(s))dF (s)minus (1minus F (v)) (pminus c(v)) + λ(v minus p) (4)

λ

983133 v

p

G(s)ds =

983133 v

p

(pminus c(s))dG(s)minus (1minusG(v)) (pminus c(v)) + λ(v minus p) (5)

First by property (iii) above the LHS of Equation 4 equals the LHS of Equation 5 Secondsince

983125 v

p(F (s) minus G(s))ds is minimized at v this implies F (p) le G(p) Therefore the first

term on RHS of Equation 4 must be larger than that of Equation 5 Notice that the firstterm on RHS is Sellerrsquos profit when offering price p This implies that the profit fromoffering p given Buyer mean-valuation distribution G(v) is lower than that given F (v)which is lower than Sellerrsquos maximum profit given F (v) On the other hand by the ICDproperty p is already Sellerrsquos optimal price given G(v) The optimal profit given valuationdistribution F must be no lower than that of G Therefore we have proved that to minimizeSellerrsquos profit it is without loss to consider only ICDs within the set D(micro) which is a onedimensional subspace The detailed proof that such ICDs exist is relegated to the appendix

The monopoly-pricing environment with c(middot) = v is covered by Condition 1 with λ = 0

and γ = v The distribution G in (3) then reduces to that identified by Roesler and Szentes(2017)

If the prior micro has binary support then the cost function c(v) is linear In this special casewe can solve for πus

s (Γ) explicitly

Corollary 1 Suppose micro has binary support V = v1 v2 Let λ = c(v2)minusc(v1)v2minusv1

and let p be theunique solution to

(pminus c(p))1

λminus1 (pminus E[c(v)]) = (v2 minus c(v2))λ

λminus1 (6)

Then πuss (Γ) = max p v2minus E[c(v)]

References

AKERLOF G A (1970) ldquoThe Market for rdquoLemonsrdquo Quality Uncertainty and the MarketMechanismrdquo Quarterly Journal of Economics 84 488ndash500

BAGWELL K AND M H RIORDAN (1991) ldquoHigh and Declining Prices Signal ProductQualityrdquo American Economic Review 81 224ndash239

BAR-ISAAC H I JEWITT AND C LEAVER (2018) ldquoAdverse Selection Efficiency and theStructure of Informationrdquo Unpublished

22

BERGEMANN D B BROOKS AND S MORRIS (2015) ldquoThe Limits of Price Discrimina-tionrdquo American Economic Review 105 921ndash57

BERGEMANN D AND S MORRIS (2016) ldquoBayes Correlated Equilibrium and the Compar-ison of Information Structures in Gamesrdquo Theoretical Economics 11 487ndash522

BLACKWELL D (1953) ldquoEquivalent Comparisons of Experimentsrdquo Annals of MathematicalStatistics 24 265ndash272

CHO I-K AND D KREPS (1987) ldquoSignaling Games and Stable Equilibriardquo Quarterly Jour-nal of Economics 102 179ndash221

DU S (2018) ldquoRobust Mechanisms Under Common Valuationrdquo Econometrica 86 1569ndash1588

DURRETT R (1995) Probability Theory and Examples Duxbury Press 2nd ed

EINAV L AND A FINKELSTEIN (2011) ldquoSelection in Insurance Markets Theory and Em-pirics in Picturesrdquo Journal of Economic Perspectives 25 115ndash38

FUDENBERG D AND J TIROLE (1991) ldquoPerfect Bayesian Equilibrium and Sequential Equi-libriumrdquo Journal of Economic Theory 53 236ndash260

GARCIA D R TEPER AND M TSUR (2018) ldquoInformation Design in Insurance MarketsSelling Peaches in a Market for Lemonsrdquo Unpublished

GREENWALD B C (1986) ldquoAdverse Selection in the Labour Marketrdquo Review of EconomicStudies 53 325ndash347

JOVANOVIC B (1982) ldquoFavorable Selection with Asymmetric Informationrdquo QuarterlyJournal of Economics 97 535ndash539

KESSLER A S (2001) ldquoRevisiting the Lemons Marketrdquo International Economic Review 4225ndash41

KREPS D AND R WILSON (1982) ldquoSequential Equilibriardquo Econometrica 50 863ndash894

LEHMANN E L (1988) ldquoComparing Location Experimentsrdquo The Annals of Statistics 16521ndash533

LEVIN J (2001) ldquoInformation and the Market for Lemonsrdquo RAND Journal of Economics32 657ndash666

23

LIBGOBER J AND X MU (2019) ldquoInformational Robustness in Intertemporal PricingrdquoWorking paper

MAKRIS M AND L RENOU (2018) ldquoInformation Design in Multi-Stage Gamesrdquo Workingpaper

MILGROM P R AND R J WEBER (1985) ldquoDistributional Strategies for Games with In-complete Informationrdquo Mathematics of Operations Research 10 619ndash632

MUSSA M AND S ROSEN (1978) ldquoMonopoly and Product Qualityrdquo Journal of EconomicTheory 18 301ndash317

RAVID D A-K ROESLER AND B SZENTES (2019) ldquoLearning Before Trading On theInefficiency of Ignoring Free Informationrdquo Working paper

ROESLER A-K AND B SZENTES (2017) ldquoBuyer-Optimal Learning and Monopoly Pric-ingrdquo American Economic Review 107 2072ndash80

TERSTIEGE S AND C WASSER (2019) ldquoBuyer-optimal Extensionproof InformationrdquoWorking paper

Appendices omitted from this version but are available on request

24

pected utilities or payoffs (for readability we often drop the ldquoexpectedrdquo qualifier) Theno-trade payoffs are normalized to zero The three triangles AFG ADE and ABC depictTheorems 1ndash3 respectively That payoffs must lie within AFG is straightforward Buyercan guarantee himself a payoff of zero by not purchasing Seller can guarantee herself notonly a payoff of zero (by posting any price p gt v which will not be accepted) but alsov minus E[c(v)] (by pricing at or just below v which will be accepted) and the sum of pay-offs cannot exceed the trading surplus E[v minus c(v)] We refer to the first two constraints asindividual rationality and the third as feasibility

Theorem 1 says that every feasible and individually-rational payoff pair can be imple-mented ie obtains in an equilibrium under some information structure It is immediatethat point A obtains whens both parties learn v (full information) or neither party has anyinformation (no information) More interestingly at the point G trade occurs with probabil-ity one and Buyer obtains the entire surplus despite Seller posting the price While perhapssurprising this outcome obtains with sparse information structures For simplicitly sup-pose E[c(v)|v gt v] le maxvE[c(v)] Then Buyer can be uninformed while Seller learnswhether v = v or v gt v In equilibrium Seller prices at p = maxvE[c(v)] regardless ofher signal and Buyer purchases If Seller were to deviate to a higher price Buyer wouldreject because he believes v = v Subsection 31 explains how a single information struc-ture in fact implements every point in the triangle AFG Theorem 1lowast there discusses how aricher information structure using imperfectly-correlated signals ensures implementationin Kreps and Wilsonrsquos (1982) sequential equilibrium in discretized versions of the model

Turning to Figure 1rsquos triangle ADE Theorem 2 establishes that the payoff pair in anyequilibrium when Buyer is better informed than Seller arises in an equilibrium of an(other)information structure in which Seller is uninformed4 In other words there is no loss ofgenerality in studying an uninformed Seller so long as Buyer is better informed Whenc(middot) is increasing such information generates a game with adverse selection when c(middot)is decreasing there is favorable selection Sellerrsquos payoff along the line segment DE isthe lowest payoff she can get in any information structure in which she is uninformed5

Theorem 2 further establishes that any point within the ADE triangle can be implementedwith some such information structure by suitably varying Buyerrsquos information In fact weshow that higher slices of the triangle (ie those corresponding to larger Sellerrsquos payoff)

4 We stipulate that a better-informed Buyer does not update his value from Sellerrsquos price (even off theequilibrium path) in line with the ldquono signaling what you donrsquot knowrdquo requirement (Fudenberg and Tirole1991) that is standard in versions of Perfect Bayesian Equilibrium and implied by sequential equilibrium

5 It is because Seller cannot commit to the price as a function of her signal that she can be harmed (iereceive a payoff lower than that on the DE segment) with more information However Theorem 2 assuresthat Seller is not harmed so long as Buyer is better informed

3

can always be implemented by reducing Buyerrsquos information in the sense of Blackwell(1953) We also explain in Subsection 32 why the triangle ADE actually characterizes allpayoffs that can obtain when Buyer does not update from Sellerrsquos price even if Buyer isnot better informed than Seller

Finally the ABC triangle in Figure 1 depicts Theorem 3 which characterizes all payoffpairs when Buyer is fully informed ie learns v We use the term ldquoAkerlofrdquo to describe afully-informed Buyer and an uninformed Seller as these information structures are stan-dard in the adverse-selection literature the corresponding payoff pair is marked as suchin the figure Depending on the environmentrsquos primitives the Akerlof point can be any-where on the segment BC including at the extreme points Any feasible payoff pair thatsatisfies Buyerrsquos individual rationality and gives Seller at least her Akerlof payoff can beimplemented with a fully-informed Buyer by suitably varying Sellerrsquos information

An implication of Theorems 1ndash3 is that it is without loss in terms of ex-ante equilibriumpayoffs to focus on information structures in which Buyer is fully informed if and onlyif Sellerrsquos Akerlof payoff coincides with her individual rationality constraint This coinci-dence occurs if and only if the Akerlof market can have full trade (Seller prices at p = v

and gets payoff v minus E[c(v)] ge 0) or no trade (the price is p ge v and both partiesrsquo payoffsare 0) As detailed in Remark 4 of Subsection 33 in all other cases the point B in Fig-ure 1 is distinct from the point D (and hence also F ) which means that Buyer can obtaina higher payoff with less-than-full information while keeping Seller uninformed Further-more under a reasonable condition if Sellerrsquos individual rationality constraint is zero (iev le E[c(v)]) then point D is also distinct from F see Remark 3 in Subsection 32 WhenD and F are distinct maximizing Buyerrsquos payoff ie achieving point G requires Seller tohave some information Buyer does not and an equilibrium with price-dependent beliefsafter conditioning on his signal Buyer must update about v from the price either on or offthe equilibrium path

Related literature Our perspective and results are most closely related to BergemannBrooks and Morris (2015) Roesler and Szentes (2017) and Makris and Renou (2018 Sec-tion 4) These papersmdashonly the relevant section of the third papermdashstudy the monopolypricing problem in which there is uncertainty only about Buyerrsquos valuation This is thespecial case of our interdependent-values model with a constant function c(v) le v6

6 The online appendix of Roesler and Szentes (2017) relaxes the assumption that there are always gainsfrom trade but maintains no uncertainty about Sellerrsquos cost Related to Roesler and Szentes (2017) are alsoDu (2018) and Libgober and Mu (2019) who consider worst-case profit guarantees for Seller in static and dy-namic environments respectively Terstiege and Wasser (2019) qualify Roesler and Szentes (2017) by allowing

4

Bergemann et al (2015) assume Buyer is fully informed and hence can only vary Sellerrsquosinformation Our Theorem 3 corresponding to triangle ABC in Figure 1 is a generaliza-tion of their main result to our environment the key step in our proof methodology is toconstruct ldquoincentive compatible distributionsrdquo which reduces to their ldquoextreme marketsrdquoin the monopoly-pricing environment Another contribution of our result is to ensure ap-proximately unique implementation in a sense explained in Subsection 33

Roesler and Szentes (2017) assume Seller is uninformed and only vary Buyerrsquos infor-mation For the monopoly-pricing environment they derive one part of our Theorem 2viz they identify the triangle ADE in Figure 1 as the implementable set given Seller isuninformed Even for this result our methodology is quite different from theirs becausewe do not stipulate a linear c(middot) function our methodology delivers new insights includ-ing that noted in Remark 2 of Subsection 32 When we specialize to a linear c(middot) we canobtain a sharper characterization of the point E which extends Roesler and Szentesrsquo char-acterization of Buyer-optimal information to an interdependent-values environment seeProposition 2 in Subsection 43

While our main interest is in interdependent values our results provide new insightseven for monopoly pricing Theorem 2 implies that the Roesler and Szentes (2017) boundsare without loss so long as Buyer is better informed than Seller or more generally in equi-libria in which Buyerrsquos belief is price independent (after conditioning on his own signal)On the other hand Theorem 1 establishes that any feasible and individually rational pay-off pair can be implemented absent these restrictions in particular Buyer may even get allthe surplus This latter point has a parallel with Makris and Renou (2018) As an applica-tion of their general results on ldquorevelation principlesrdquo for information design in multi-stagegames Makris and Renoursquos (2018) Proposition 1 deduces an analog of our Theorem 1 forthe (independent values) monopoly pricing problem We share with Makris and Renou anemphasis on sequential rationality7 we differ in showing that a single information struc-ture implements all payoffs in the relevant triangle and also in establishing in Theorem 1lowast

off-the-equilibrium-path belief consistency in the sense of sequential equilibrium (Krepsand Wilson 1982) in discretizations

Other authors have studied different aspects of more specific changes of information inadverse-selection settings maintaining that one side of the market is better informed than

Seller to supply Buyer with additional information although Seller cannot have any private information ofher own

7 Makris and Renou use an apparatus of ldquosequential Bayes correlated equilibriumrdquo which we do notThey draw a contrast with Bergemann and Morrisrsquos (2016) Bayes correlated equilibrium In our approachnote that Sellerrsquos individual rationality constraint described earlier hinges when E[c(v)] lt v on Buyerrsquosbehavior being sequentially rational even off the equilibrium path

5

the other Levin (2001) identifies conditions under which the volume of trade decreaseswhen one party is kept uninformed and the otherrsquos information become more effective inthe sense of Lehmann (1988) see also Kessler (2001) Assuming a linear payoff structureBar-Isaac Jewitt and Leaver (2018) consider how certain changes in Gaussian informationaffect the volume of trade surplus and a certain quantification of adverse selection

Finally Garcia Teper and Tsur (2018) solve for socially optimal information provisionmdashrather than characterizing the range of market outcomesmdashin an insurance setting with ad-verse selection owing to a cross-subsidization motive full information disclosure is typi-cally not optimal

The rest of the paper proceeds as follows We introduce our model equilibrium con-cept(s) and certain classes of information structures in Section 2 Section 3 presents themain results implementable payoffs when the information structure is arbitrary or varieswithin canonical classes Section 4 contains discussion and extensions Proofs and someadditional material are contained in the Appendices

2 Model

21 Primitives

Seller may sell an indivisible good to Buyer Buyerrsquos value for the good is v isin V sub Rwhere V is a compact (finite or infinite) set with v equiv minV lt maxV equiv v The valuev is drawn from a probability measure micro with support V Sellerrsquos cost of production isgiven by a function c(v)8 We assume c V rarr R is continuous v minus c(v) ge 0 for all vand E[v minus c(v)] gt 0 So trading surplus is nonnegative for all v and positive for a positivemeasure of v (Throughout expectations are with respect to the prior measure micro unlessindicated otherwise lsquopositiversquo means lsquostrictly positiversquo and similarly elsewhere) Note thatfunction c(v) need not be monotonic We call Γ = (c micro) an environment We refer to anenvironment with a constant c(middot) function as that of monopoly pricing

An information structure consists of signal spaces for each party and a joint signal dis-tribution (We abuse notation and sometimes write lsquodistributionrsquo even though lsquomeasurersquowould be more precise) Formally there is a probability space (ΩF P ) complete and sep-arable metric spaces Ts and Tb (with measurable structure given by their Borel sigma alge-bras) and an integrable function X Ω rarr TstimesTbtimesV We hereafter suppress the probabilityspace and define with an abuse of notation P (D) = P (Xminus1(D)) for any D isin Ts times Tb times V

measurable Each realization of random variable X is a triplet (tb ts v) where tb isin Tb

8 Subsection 41 discusses Sellerrsquos cost being stochastic even conditional on v

6

is Buyerrsquos signal and ts isin Ts is Sellerrsquos signal For i isin s b v let Pi denote the corre-sponding marginal distribution of P on dimension of Ti with the convention Tv equiv V Werequire Pv = micro this is the iterated expectation or Bayes plausibility requirement Denotean information structure by τ

The environment Γ and information structure τ define the following game

1 The random variables (tb ts v) are realized Signal tb is privately observed by Buyerand signal ts privately observed by Seller Neither party observes v

2 Seller posts a price p isin R

3 Buyer accepts or rejects the price If Buyer accepts his von-Neumann Morgensternpayoff is v minus p and Sellerrsquos is p minus c(v) If Buyer rejects both partiesrsquo payoffs arenormalized to 0

Note that because the signal spaces Tb and Ts are abstract and the two partiesrsquo signalscan be arbitrarily correlated conditional on v there is no loss of generality in assuming thateach party privately observes their own signal For example public information can becaptured by perfectly correlating (components of) tb and ts

22 Strategies and Equilibria

In the game defined by (Γ τ) denote Sellerrsquos strategy by σ and Buyerrsquos by α FollowingMilgrom and Weber (1985) we define σ as a distributional strategy σ is a joint distributionon R times Ts whose marginal distribution on Ts must be the Sellerrsquos signal distribution Soσ(middot|ts) is Sellerrsquos price distribution given her signal ts9 Buyerrsquos strategy α R times Tb rarr[0 1] maps each price-signal pair (p tb) into a trading probability A strategy profile (σα)

induces expected utilities for Buyer and Seller (πb πs) in the natural way

πb =

983133(v minus p)α(tb p)σ(dp|ts)P (dts dtb dv)

πs =

983133(pminus c(v))α(tb p)σ(dp|ts)P (dts dtb dv)

Our baseline equilibrium concept is weak Perfect Bayesian equilibrium Since Sellerrsquosaction is not preceded by Buyerrsquos we can dispense with specifying beliefs for Seller ForBuyer it suffices to focus on his belief about the value v given his signal and the price wedenote this distribution by ν(v|p tb)

9 Here σ(middot|ts) is the regular conditional distribution which exists and is unique almost everywhere becauseTs is a standard Borel space (Durrett 1995 pp 229ndash230) Similarly for subsequent such notation we dropldquoalmost everywhererdquo qualifiers unless essential

7

Definition 1 A strategy profile (σα) and a belief ν(v|p tb) is a weak perfect Bayesian equilib-rium (wPBE) of game (Γ τ) if

1 Buyer plays optimally at every information set given his belief

α(p tb) =

983099983103

9831011 if Eν(v|ptb)[v] gt p

0 if Eν(v|ptb)[v] lt p

2 Seller plays optimally

σ isin argmaxσ

983133(pminus c(v))α(p tb)σ(dp|ts)P (dts dtb dv)

3 Beliefs satisfy Bayes rule on path for every measurable D sube Rtimes Ts times Tb times V

983133

D

ν(dv|p tb)σ(dp|ts)P (dts dtb V ) =

983133

D

σ(dp|ts)P (dts dtb dv)

Notice that we have formulated Sellerrsquos optimality requirement ex ante but Buyerrsquos ateach information set The latter is needed to capture sequential rationality The formeris for (notational) convenience this choice is inconsequential because Seller moves beforeBuyer

Hereafter ldquoequilibriumrdquo without qualification refers to a wPBE As is well understoodwPBE permits significant latitude in beliefs off the equilibrium path We will discuss re-finements when appropriate

23 Implementable Payoffs and Canonical Information Structures

We now define the set of implementable equilibrium outcomesmdashthat is the payoffs thatobtain in some equilibrium under some information structuremdashand some canonical classesof information structures

For a game (Γ τ) let the equilibrium payoff set be

Π(Γ τ) equiv (πb πs) exist wPBE of (Γ τ) with payoffs (πb πs)

Denote the class of all information structures by T and define

Π(Γ) equiv983134

τisinT

Π(Γ τ)

8

That is for environment Γ Π(Γ) is the set of all equilibrium payoff pairs that obtain undersome information structure

Uninformed Seller An information structure has uninformed Seller if Ts is a singletonSellerrsquos own signal contains no information about Buyerrsquos value v and hence neither aboutSellerrsquos cost c(v) When discussing such information structures we write the associateddistribution as just P (tb v) and Sellerrsquos strategy as just σ(p) omitting the argument ts inboth cases The class of all uninformed-Seller information structure is denoted Tus

Fully-informed Buyer An information structure has fully-informed Buyer if Buyerrsquos signalfully reveals his value v Formally this holds if Tb = V and the conditional distribution onV P (middot|tb) satisfies P (tb|tb) = 1 We denote the class of fully-informed-Buyer informationstructures Tfb

More-informed Buyer An information structure has more-informed Buyer if Buyer hasmore information than Seller Formally this holds when v and ts are independent condi-tional on tb ie for any measurable Ds sub Ts and Dv sub V P (DscapDv|tb) = P (Ds|tb)P (Dv|tb)10

Another way to interpret this requirement is that random variable tb must be statisticallysufficient for ts with respect to v ie tb is more informative than ts about v in the sense ofBlackwell (1953) We denote the class of more-informed-Buyer information structures byTmb Naturally information structures with uninformed Seller or fully-informed Buyer arecases of more-informed Buyer both Tus and Tfb are subclasses of Tmb

No updating from price For more-informed-Buyer information structures it is desirableto impose further requirements on Buyerrsquos equilibrium belief Since Sellerrsquos price can onlydepend on her own signal and this signal contains no additional information about v givenBuyerrsquos signal the price is statistically uninformative about v given Buyerrsquos signal Con-sequently Buyerrsquos posterior belief should be price independent once his signal has beenconditioned upon Formally regardless of the price p the equilibrium belief ν(middot|p tb) mustsatisfy 983133

ν(dv|p tb)P (dtb Ts V ) =

983133P (dtb Ts dv)

We refer to this condition as price-independent belief Note that the condition is meaningfulregardless of whether Buyer is more informed than Seller In a more-informed-Buyer in-formation structure price-independent belief would be implied by the ldquono signaling what

10 We write sub to mean ldquoweak subsetrdquo

9

you donrsquot knowrdquo requirement (Fudenberg and Tirole 1991) frequently imposed in versionsof perfect Bayesian equilibrium and the concept of sequential equilibrium (Kreps and Wil-son 1982) in finite versions of our setting11

Some notation will be helpful Define

Πlowast(Γ τ) equiv (πb πs) exist wPBE of (Γ τ) with price-independent belief and payoffs(πb πs)

Πlowast(Γ) equiv983134

τisinT

Πlowast(Γ τ)

Πlowasti (Γ) equiv

983134

τisinTi

Πlowast(Γ τ) for i = us fbmb

So Πlowast and Πlowast are analogous to the implementable payoff sets Π and Π defined earlier butrestricted to equilibria with price-independent beliefs Πlowast

us Πlowastfb and Πlowast

mb are the imple-mentable payoff sets when further restricted to uninformed-Seller fully-informed-Buyerand more-informed-Buyer information structures Plainly for any environment Γ

Πlowastus(Γ) cupΠlowast

fb(Γ) sub Πlowastmb(Γ) sub Πlowast(Γ) sub Π(Γ)

3 Main Results

Our goal is to characterize equilibrium payoff pairs across information structures in anarbitrary environment Γ In particular we seek to characterize the five sets Π(Γ) Πlowast(Γ)Πlowast

mb(Γ) Πlowastus(Γ) and Πlowast

fb(Γ) Let

S(Γ) equiv E[v minus c(v)]

be the (expected) surplus from trade in environment Γ This quantity will play an impor-tant role

31 All Information Structures

Define Sellerrsquos payoff guarantee as

πs(Γ) equiv max v minus E [c(v)] 0

11 An example clarifying our terminology may be helpful If both Seller and Buyer are fully informed of vthen the natural equilibriummdashthe unique sequential equilibrium in a finite version of the gamemdashhas Sellerpricing at p = v and Buyerrsquos belief being degenerate on v regardless of Sellerrsquos price This equilibrium hasprice-independent belief even though ex-ante Sellerrsquos price are Buyerrsquos belief are perfectly correlated Thepoint is that Buyerrsquos belief does not depend on price conditional on his signal

10

To interpret this observe that it is optimal for Buyer to accept the price v no matter hisbelief Therefore Seller can guarantee herself the (expected) profit v minus E[c(v)] no matterwhat the information structure is More precisely she can guarantee v minus E[c(v)] minus ε forany ε gt 0 since sequential rationality requires Buyer to accept any price v minus ε SimilarlySeller can also guarantee zero profit offering a price p gt v Hence Sellerrsquos payoff in anyequilibrium of any information structure must be at least the payoff guarantee πs(Γ) above

On the other hand Buyer can guarantee himself the payoff πb = 0 by rejecting all pricesIt follows that the implementable set Π(Γ) must satisfy three simple constraints 1) Sellerrsquosldquoindividual rationalityrdquo constraint πs ge πs(Γ) 2) Buyerrsquos ldquoindividual rationalityrdquo con-straint πb ge 0 and 3) the feasibility constraint πb + πs le S(Γ)

Our first result is that these individual rationality and feasibility constraints are alsosufficient for a payoff pair to be implementable

Theorem 1 Consider all information structures and equilibria

Π(Γ) =

983099983105983103

983105983101

πb ge 0

(πb πs) πs ge πs(Γ)

πb + πs le S(Γ)

983100983105983104

983105983102

Theorem 1 says that the set Π(Γ) corresponds to the triangle AFG in Figure 1 In par-ticular Buyer can receive the entire surplus beyond Sellerrsquos payoff guarantee This is sur-prising as Seller has substantial bargaining power Note that when v le E[c(v)] an entirelyreasonable condition Sellerrsquos payoff guarantee is zero12

The proof of Theorem 1 is in fact straightforward Suppose for expositional simplicityv ge E[c(v)] Fix the trivial information structure in which neither player receives any infor-mation and consider the following family of strategy profiles Seller randomizes betweentwo prices some pl isin [vE[v]] and ph = E[v] Buyer accepts pl with probability one andaccepts ph with probability α(ph) where α(ph) isin [0 1] is specified to make Seller indifferentbetween the two prices That is α(ph)(ph minus E[c(v)]) = pl minus E[c(v)] The expected payoffsfrom this strategy profile are

πb = σ(pl)(E[v]minus pl) and πs = pl minus E[c(v)]

As pl traverses the interval [vE[v]] Sellerrsquos payoff πs traverses [πs(Γ) S(Γ)] Given any

12 In monopoly pricing with c(middot) = v Sellerrsquos payoff guarantee of zero is lower than the revenue guaranteeidentified in Du (2018 Section 5) which is typically positive Dursquos notion is different from ours

11

pl Buyerrsquos payoff πb traverses [0 S(Γ)minusπs] as σ(pl) traverses [0 1] Therefore the proposedstrategy profiles induce all the payoff pairs stated in Theorem 1

We are left to specify beliefs ν for Buyer After prices pl and ph Buyer holds the priorbelief micro After any other (necessarily off-path) price Buyerrsquos belief is that v = v and soBuyer rejects all prices p isin [vinfin)pl ph It is straightforward to confirm that the specified(σα ν) constitute a wPBE

To get more insight into this construction consider its implication for the monopolypricing with c(middot) = v The equilibrium with pl = v (hence α(ph) = 0) and σ(pl) = 1

corresponds to the monopolist deterministically pricing at v and Buyer purchasing Giventhat both sides of the market receive no information why doesnrsquot the monopolist deviate toany price in (vE[v]) The reason is that in this equilibrium the consumer will then not buybecause he updates to v = v Such belief is compatible with wPBE because the equilibriumconcept places no restrictions on off-path beliefs This may seem like a game-theoreticmisdirection Buyerrsquos belief is not consistent with ldquono signaling what you donrsquot knowrdquoPut differently since we have a (weakly) more-informed Buyer information structure weought to impose the price-independent belief condition described in Subsection 23 thatwould imply Buyer must purchase at any price p lt E[v]

But the message of Theorem 1 does not rely on the permissiveness of wPBE To illus-trate continue with the above monopoly-pricing environment and suppose v has positiveprior probability Consider Buyer remaining uninformed but Seller learning whether v = v

or v gt v Now Buyerrsquos off-path belief that v = v is consistent with ldquono signaling whatyou donrsquot knowrdquo More generally using richer information structures we can prove thatany payoff pair identified in Theorem 1 can be approximately implemented as a sequentialequilibrium (Kreps and Wilson 1982) in a suitably discretized game

Theorem 1lowast Fix any ε gt 0 There is a finite information structure and a finite price grid inducinga game with a set of sequential equilibrium payoffs that is an ε-net of Π(Γ)13

In fact the proof of Theorem 1lowast establishes even more the sequential equilibria in thediscretized games satisfy a natural version of the D1 refinement (Cho and Kreps 1987) Werelegate the logic to the Appendix but mention here that we use imperfectly-correlatedsignals for Buyer and Seller14

13 An information structure is finite if the signal spaces Tb and Ts are finite Sequential equilibrium isdefined in the obvious way for the ldquoinducedrdquo finite game where Nature directly draws (tb ts) rather thanfirst drawing v and playersrsquo payoffs are defined directly using (tb ts) For X sub R2 and ε gt 0 the set A sub Xis an ε-net of X if for each x isin X there is a isin A such that 983042xminus a983042 lt ε where 983042middot983042 is the Euclidean distance

14 The idea behind D1 is to ask for any off-path price whether one type of Seller would deviate for any

12

Even when our environment is specialized to monopoly pricing it is worth highlightingtwo contrasts between Theorem 11lowast and results of Bergemann et al (2015) and Roeslerand Szentes (2017) First we find that by not restricting the monopolist to be uninformedthe implementable payoff set typically expands rather dramatically trade can be efficientwith the monopolist securing none of the surplus beyond her payoff guarantee (πs) whichmay be zero (Roesler and Szentes establish implicitly that the implementable set withan uninformed monopolist is a superset of Bergemann et alrsquos which has a fully-informedconsumer) We will see in Subsection 32 that what is crucial to this expansion is price-dependent beliefs In particular the proof of Theorem 1lowast uses an information structure inwhich Buyer is not better informed than Seller mdash if he were then sequential equilibriumwould imply price-independent beliefs Second Theorem 1lowast establishes that for a givenε gt 0 a single information structure (and price grid) can be used to approximate the entirepayoff set Π(Γ) analogously to the construction described after Theorem 1 that used asingle information structure15 Bergemann et al (2015) and Roesler and Szentes (2017) onthe other hand vary information structures to span their payoff sets

32 More-informed Buyer and Price-independent Beliefs

In some economic settings it is plausible that Buyer is more informed than Seller Howdoes a restriction to such information structures ie τ isin Tmb affect the implementablepayoff set It turns out that what is in fact crucial is price-independent beliefs We have ex-plained earlier why it is desirable to impose this condition when Buyer is more informedbut that the condition is well defined even otherwise If Buyer is not more informed thanSeller price-independent beliefs ought to be viewed as an equilibrium restriction in gen-eral To reduce repetition readers should be bear in mind for the rest of this subsectionthe qualifier ldquowith price-independent beliefsrdquo applies unless stated explicitly otherwise

It is useful to define

πuss (Γ) equiv inf πs exist(πb πs) isin Πlowast

us(Γ) (1)

as the lowest payoff that an uninformed Seller can obtain no matter Buyerrsquos information(among equilibria with price-independent beliefs) Plainly πus

s (Γ) ge πs(Γ) In monopoly

Buyer mixed response that another type would Our construction has multiple Buyer types that are imper-fectly correlated with Seller types So different types of Seller have different beliefs about Buyer types Thisblunts the power of dominance considerations to the point where D1 does not exclude any Seller type fromthe support of Buyerrsquos off-path belief

15 In fact if one lets the price grid vary with ε then a single information structure implements exactlyrather than approximately in sequential equilibrium all payoffs ε-away from the boundary of Π(Γ) See in the Appendix for a formal statement

13

pricing with V = [v v] and c(middot) = v Roesler and Szentesrsquos (2017) characterization of theconsumer-optimal information structure identifies πus

s establishing that πuss gt πs If there

is no trade due to adverse selection when Seller is uniformed and Buyer has some informa-tion then πus

s = πs = 0 We do not have a general explicit formula for πuss Subsection 43

provides it for linear c(middot) Nonetheless we establish next that (i) the only additional restric-tion on equilibrium payoffs imposed by price-independent beliefs is a lower bound of πus

s

for Seller and (ii) uninformed-Seller information structures implement all such payoffs

Theorem 2 Consider equilibria with price-independent beliefs

1 Πlowast(Γ) = Πlowastmb(Γ) = Πlowast

us(Γ)

2 Πlowastus(Γ) = (πb πs) isin Π(Γ) πs ge πus

s (Γ)

3 For any (πb πs) isin Πlowastus(Γ) with πs gt πus

s (Γ) there is τ isin Tus with Π(Γ τ) = (πb πs)

Remark 1 We believe the substance of Theorem 2 would hold using discretizations and se-quential equilibria analogous to Theorem 1lowast As previously noted sequential equilibriumimplies price-independent beliefs when Buyer is more informed than Seller

To digest Theorem 2 note that Πlowast(Γ) sup Πlowastmb(Γ) sup Πlowast

us(Γ) is trivial So part 1 of thetheorem amounts to establishing the reverse inclusions The intuition for thismdashgiven part2rsquos characterization of Πlowast

usmdashis fairly straightforward with price-independent beliefs ad-ditional information cannot harm Seller even though it could alter the set of equilibria SoSellerrsquos lowest payoff obtains when she is uninformed

The characterization in part 2 of payoffs with an uninformed Seller corresponds to thetriangle ADE in Figure 1 Part 3 of the theorem assures ldquounique implementationrdquo of allimplementable payoffs satisfying πs gt πus

s (Γ) That is for any such payoff pair there is anuninformed-Seller information structure such that all equilibria (with price-independentbeliefs) induce exactly that payoff pair Unique implementation is appealing for multiplereasons one of which is that it obviates concerns about which among multiple payoff-distinct equilibria is more reasonable For example Ravid Roesler and Szentes (2019)bring to the fore the issue of equilibrium multiplicity in Roesler and Szentesrsquos (2017) infor-mation structure

Let us describe how we obtain the characterization of Πlowastus(Γ) and unique implementa-

tion There are two steps The first ensures that there is some information structure callit τ lowast isin Tus that implements Sellerrsquos payoff πus

s (Γ) That is we ensure that the infimum

14

in (1) is in fact a minimum16 While this argument is technical knowing τ lowast exists is usefulin what follows The second and economically insightful step is to construct informa-tion structures that implement every point in the triangle Πlowast

us(Γ) by suitably garbling theinformation structure τ lowast The construction is illustrated in Figure 2 Consider the distri-bution of Buyerrsquos posterior mean of his valuation v in information structure τ lowast (Givenprice-independent beliefs Buyerrsquos posterior mean is sufficient for his decision) For sim-plicity suppose this posterior-mean distribution has a density as depicted by the red curvein Figure 2 Fix any (πb πs) isin Πlowast

us(Γ)

First there is some number zlowast such that πb + πs is the total surplus from trading onlywhen Buyerrsquos posterior mean is greater than zlowast Next there is some price plowast ge zlowast suchthat Sellerrsquos payoff is πs if all these trades were to occur at price plowast17 Note that plowast must beno larger than the expected Buyer posterior mean conditional on that being above zlowast forotherwise πb lt 0 We claim that the information structure τ lowast can be garbled so that plowast isan equilibrium price and trade occurs only when Buyerrsquos posterior mean is greater thanzlowast The garbling is illustrated in Figure 2 by the blue distribution There is one signal that

pzv

ν(v)

Figure 2 Construction of information structure in Theorem 2

Buyer receives when the original posterior mean is between zlowast and plowast and also receiveswith some probability when the original posterior mean is above plowast The probability ischosen to make the posterior mean from this signal exactly plowast Apart from this one newsignal Buyer receives the original signal in τ lowast Plainly this is a garbling of τ lowast and hence isfeasible

16 The difficulty is in establishing suitable continuity Uninformed-Seller information structures can beviewed as probability measures over Buyerrsquos beliefs with convergence in the sense of the weak topologyThis topology ensures continuity with respect to probability measures of expectations of continuous orat least Lipschitz (and bounded) functions However Sellerrsquos expected payoff is not the expectation of aLipschitz function as Sellerrsquos profit is truncated at the price she charges

17 That plowast ge zlowast follows from πs ge πuss (Γ) as πs(Γ) itself is weakly larger than Sellerrsquos payoff from posting

price zlowast (and thus trading with the same set of Buyer posterior means) under information structure τlowast

15

Figure 2 makes clear why the new information structure has an equilibrium with priceplowast and Buyer breaking indifference in favor of trading (i) Sellerrsquos profit from posting anyprice below zlowast is the same as under τ lowast and hence no larger than πus

s (Γ) (ii) similarly Sellerrsquosprofit from posting any price above plowast is no higher than some fraction of πus

s (Γ) and (iii)any price between zlowast and plowast is worse that price plowast Moreover since Sellerrsquos profit fromoffering any price other than plowast is no more than πus

s (Γ) it follows that when πs gt πuss (Γ)

Buyer must break indifference as specified for Seller to have an optimal price and theequilibrium payoffs are unique

Remark 2 The above logic establishes that given any τ isin Tus that implements some (πb πs)τ can be garbled to uniquely implement any (πprime

b πprimes) isin Π(Γ) such that πprime

s gt πs That isan uninformed Sellerrsquos payoff can always be strictly raised and Buyerrsquos payoff reduced(strictly so long as it was not already zero) by garbling Buyerrsquos information

Remark 3 According to Theorem 11lowast and Theorem 2 uninformed-Seller information struc-tures cannot implement all implementable payoff pairs in an environment Γ if and only ifπuss (Γ) gt πs(Γ) This inequality fails in environment Γ if πus

s (Γ) = 0 since that impliesπuss (Γ) = πs(Γ) = 0 An example is when there is no trade due to adverse selection when

Seller is uniformed and Buyer has some information On the other hand it is sufficient forthe inequality that

v le E[c(v)] and forallv isin V c(v) lt v (2)

To see why (2) implies πuss (Γ) gt πs(Γ) notice that in any uninformed-Seller information

structure Seller can price at slightly less than Buyerrsquos highest posterior mean valuationand guarantee trade with only (a neighborhood of) that Buyer type If c(v) lt v for all v thisgives Seller a positive expected payoff and hence πus

s (Γ) gt 018 But the first inequality in (2)is equivalent to πs(Γ) = 0 We observe that Condition (2) is compatible with severe adverseselection resulting in very little trade when Seller is uninformed and Buyer (partially orfully) informed

33 Fully-Informed Buyer

We now turn to the third canonical class of information structures Buyer is fully in-formed of his value v As this is a special case of more-informed Buyer we maintain price-independent beliefs throughout this subsection

18 More precisely as V is compact c(v) lt v for all v implies there exists ε gt 0 such that v minus c(v) gt εGiven any uninformed-Seller information structure let mv be the highest posterior mean valuation in thesupport of the posterior means induced by Buyerrsquos signals So there is positive probability of Buyer signalswith posterior mean valuations at least mv minus ε2 By pricing at mv minus ε2 Sellerrsquos expected cost conditionalon trade is bounded above by mv minus ε and hence Sellerrsquos profit conditional on trade is at least ε2 gt 0 Itfollows that πus

s gt 0

16

Faced with a fully informed Buyer and any sequentially rational Buyer strategy an un-informed Seller can guarantee a profit level

πfbs (Γ) equiv sup

p

983133 v

p

(pminus c(v))micro(dv)

regardless of her information Plainly πfbs (Γ) ge πus

s (Γ) In monopoly pricing with c(middot) = vRoesler and Szentes (2017) have shown that πfb

s gt πuss if there is no trade due to adverse

selection when Buyer is fully informed and Seller is uniformed then πfbs = πus

s = 0 Belowwe establish that when Buyer is fully informed πfb

s is the only additional constraint onequilibrium payoffs Let Bε(πb πs) denote an ε-ball around (πb πs)

Theorem 3 Consider a fully informed Buyer and price-independent beliefs

1 Πlowastfb(Γ) = (πb πs) isin Π(Γ) πs ge πfb

s (Γ)

2 For any (πb πs) isin Πlowastfb(Γ) and any ε gt 0 there is τ isin Tfb with Π(Γ τ) sub Bε(πb πs)

The payoff set characterized in Theorem 3 corresponds to the triangle ABE in Figure 1Part 2 of the theorem establishes ldquoapproximately unique implementationrdquo for any point inthe triangle there is an information structure under which all equilibria yield payoff pairsin a neighborhood of that point

Here is the idea behind part 1 of Theorem 3 When Buyer is fully informed an informa-tion structure can be viewed as dividing vrsquos prior distribution micro into a set of microi that aver-age to micro with Seller informed of which microi she faces Theorem 3 is proven by establishingthat we can divide micro suitably so that against each microi Seller is indifferent between pricingat all prices in the support of microi including the price corresponding to πfb

s in that environ-ment We call such a microi a (Seller-price) incentive compatible distribution or ICD (Bergemannet al (2015) call their analogous construct an extreme market) The Appendix provides aldquogreedyrdquo algorithm to compute ICDs19

We can sketch how the algorithm works and construct a set of ICDs that average to theprior Suppose V = v1 v2 vK with vi lt vi+1 for i isin 1 K minus 1 and c(v) lt v forall v Given any small-enough mass of vK there is a unique mass of type vKminus1 that makesSeller indifferent between charging price vK and vKminus1 (If the mass is too low Seller prefersvK if it is too high she prefers vKminus1) Iterating down to keep Seller indifferent betweenall prices pins down an ICD Choose the maximum mass of type vK for which this worksRemove that ICD and repeat

19 The algorithm is defined for finite V and we take limits to handle the infinite case

17

Crucially whenever an ICD is removed the price corresponding to πfbs (Γ) remains op-

timal in the remaining ldquomarketrdquo this follows from the ICDrsquos defining property of Sellerindifference and an accounting identity Therefore Sellerrsquos profit in this segmentation ofICDs remains πfb

s (Γ) Moreover it is also optimal for Seller to always (ie for each microi) priceso that there is full trade or no trade Hence Buyerrsquos expected payoff can be either 0 or theentire surplus less πfb

s (Γ) It follows that the fully-informed Buyer information structuredefined by this set of ICDs implements point B and C in Figure 1 The entire triangle ABC

can then be implemented by convexification randomizing over this information structure(and two equilibria) and full information (where Seller obtains all the surplus)

The above logic of incentive compatible distributions reduces to that of Bergemann et al(2015) in the context of monopoly pricing Even in the monopoly pricing case Theorem 3part 2 strengthens the conclusions of Bergemann et al (2015) by ensuring approximatelyunique implementation

Remark 4 Theorems 1ndash3 imply that fully-informed-Buyer information structures imple-ment all implementable payoff pairs if and only if πfb

s (Γ) = πs(Γ) In that case trianglesAFG and ABC coincide in Figure 1 It follows that πfb

s (Γ) = πs(Γ) only when a fully-informed Buyer and uninformed Seller can result in full trade (v ge E[c(v)] and Sellerprices at v) or no trade (v le E[c(v)] and Seller prices at some p ge v) Interestinglywhen πfb

s (Γ) gt πs(Γ) fully-informed-Buyer information structures cannot even implementall payoff pairs implementable by uninformed-Seller information structures ie trianglesABC and ADE in Figure 1 are distinct That is when πfb

s (Γ) gt πs(Γ) there is an uninformed-Seller information structure that implements some πs lt πfb

s (Γ)20 Theorems 2ndash3 further im-ply in this case that Buyer can obtain a strictly higher payoff when he is not fully informed(and Seller is uninformed)

4 Discussion

41 Multidimensionality

Suppose Buyer and Sellerrsquos cost and valuation pair (c v) is a two-dimensional randomvariable distributed according to joint distribution micro with a compact support The ex-tension of our maintained assumption of commonly known gains from trade is for all

20 Pick any pprime gt v such that pprimeminusE[c(v)] lt πfbs (Γ) Following the construction described after Theorem 2 we

can mix all valuations v le pprime with a fraction λ gt 0 of valuations v gt pprime so that the mixture has posterior meanexactly pprime The remaining fraction 1 minus λ of valuations above pprime are revealed to Buyer With this uninformed-Seller information structure consider any equilibrium in which Buyer purchases when indifferent (Suchan equilibrium with price-independent belief exists) Sellerrsquos profit is at most (1 minus λ)πfb

s (Γ) from any pricep gt pprime and pprime minus E[c(v)] from price p = pprime Hence Sellerrsquos profit is strictly less than πfb

s (Γ)

18

(c v) isin Supp(micro) v ge c and E[v minus c] gt 0 An information structure is now a joint distribu-tion P (tb ts v c) whose marginal distribution on (v c) is micro

We claim that the content of our key results Theorem 1Theorem 1lowast Theorem 2 andTheorem 3 still hold To see why let v be the lowest valuation in the support of micro Sellerrsquosindividual rationality constraint is now maxv minus E[c] 0 as she can guarantee this by set-ting either a sufficiently high price or a price (arbitrarily close to) v regardless of her signalWe can define a cost function c(vprime) equiv Emicro[c|v = vprime] ge vprime This results in an environment sat-isfying all the maintained assumptions of our baseline model except that c(middot) may not becontinuous Recall that such continuity is not needed for proving Theorem 1 Theorem 1lowastBoth Theorem 2 and Theorem 3 use continuity of c(middot) to guarantee that Eν [c(v)] is a contin-uous function of ν isin ∆(V ) for a key convergence result However in the two-dimensionaltype environment Eν [c] is still a continuous function of ν isin ∆(V times C) Note that Theo-rem 3 uses upper semi-continuity of Sellerrsquos profit in price boundedness of c and c le v issufficient for such upper semi-continuity

42 Negative Trading Surplus

Returning to our baseline model we next discuss cases in which trade sometimes gen-erates negative surplus That is we drop the assumption that c(v) le v we do not requireE[v minus c(v)] gt 0 either Define Sλ(Γ) for λ isin [1infin) as

Sλ(Γ) equiv983133 v

v

[v minus c(v) + λ(v minus v)]+ micro(dv)

where [middot]+ equiv max middot 0 It is readily verified that S1(Γ) = E [[v minus c(v)]+] and limλrarr0

Sλ(Γ)λ =

E[v] minus v Allowing negative trading surplus does not affect our definition of wPBE Sothe notation Π(Γ) and πs(Γ) still have the same meanings as before The next propositionshows that Π(Γ) is now characterized by three constraints as before the two individualrationality constraints πs ge πs(Γ) and πb ge 0 and different now a Pareto frontier definedby all Sλ(Γ)

Proposition 1 Consider all information structures and equilibria when trade can generate negativesurplus

Π(Γ) =

983099983105983103

983105983101

πb ge 0

(πb πs) πs ge πs(Γ)

λπb + πs le Sλ(Γ) forallλ ge 1

983100983105983104

983105983102

Proposition 1 shows that when the trade can be inefficient it may be impossible to lower

19

Sellerrsquos profit without losing total surplus (ie πb + πs) This result is intuitive at the ex-tremes Suppose for simplicity v gt E[c(v)] If πs = πs(Γ) then since πs(Γ) = v minus E[c(v)]Seller must find it optimal to price at v and trade with probability one including whentrade generates negative surplus So Buyerrsquos payoff is E[v]minus v = limλrarrinfin Sλ(Γ)λ and totalsurplus is E[v minus c(v)] which includes the maximum amount of inefficient trade On theother hand an information structure that publicly reveals (only) whether trade is efficient(ie whether v ge c(v)) can implement efficient trade with all the surplus accruing to SellerIn this case Seller obtains S1(Γ) while Buyer obtains zero The proposition establishes thatin between S1(Γ) and πs(Γ) information structures and equilibria that lower Sellerrsquos profitmust reduce total surplus as the Pareto frontier Sλ(Γ) is a concave function with maximalslope minus1

The function capturing the Pareto frontier Sλ(Γ) is the maximal weighted sum of Buyerand Seller payoff assuming that Seller will trade at the low price p = v whenever it createspositive weighted total payoff Since we place higher weight on buyer (λ ge 1) S1(Γ) isthe upper bound for the weighted total payoff In the proof of Proposition 1 we showthat these bounds are tight mdash any payoff pair on the frontier can be implemented by someinformation structure The key observation is that whenever v minus c(v) + λ(v minus v) lt 0 itfollows that vminus c(v) lt (1minusλ)(vminus v) le 0 That is it is incentive compatible for Seller to notsell when it is common knowledge that v creates negative weighted total payoff Thereforewe can simply release a public signal indicating the sign of weighted total payoff and ifpositive induce Seller to sell at v which implies all bounds Sλ(Γ)

43 Linear Cost

When Sellerrsquos cost function c(v) is linear we can explicitly characterize an informationstructure that implements Sellerrsquos minimum implementable payoff πus

s (Γ) when Seller isuninformed (or per Theorem 2 when Buyer is better informed) subject to price-independentbeliefs We note that given the discussion in Subsection 41 a linear c(v) subsumes richerenvironments in which conditional expectations are linear including Gaussian environ-ments (cf Bar-Isaac et al 2018)

Condition 1 c(v) = λv + γ for some λ isin R

Let F (v) be the cumulative distribution function (CDF) corresponding to the prior mea-sure micro Let D(micro) be the set of all distributions whose CDF G satisfies

983133

V

vdG(v) =

983133

V

vdF (v) and983133 v

v

G(s)ds le983133 v

v

F (s)ds forallv isin V

20

That is D(micro) contains all distributions that are mean-preserving-contractions (MPC) of microIt is well known that D(micro) characterizes the set of distributions of Buyer posterior meansthat can be generated by any (uninformed-Seller) information structure

Definition 2 G is an incentive compatible distribution (ICD) if

983133

sgep

(pminus c(s))G(ds) = constant ge 0 forallp isin Supp(G)

That is when Buyerrsquos (mean) valuation is distributed according to an ICD G Seller isindifferent among all prices in the support of G assuming Buyer buys when indifferent Wefocus on a special family of ICDs whose supports are intervals [vlowast v

lowast] Given Condition 1such ICDs have analytical an expression

G(v) =

983099983105983105983103

983105983105983101

1minus983061

(1minus λ)v minus γ

(1minus λ)vlowast minus γ

983062 1λminus1

if λ ∕= 1

1minus evminusvlowast

γ if λ = 1

(3)

Given any vlowast and the above G(v) an upper bound vlowast is pinned down by the conditionEG[v] = EF [v] So a family of ICDs is parametrized by a single parameter vlowast and hence anICD is denoted by CDF 983144Gvlowast(v) with density 983144gvlowast(v) It is not difficult to verify that 983144Gvlowast(v) ismonotonic in vlowast for v in the support Consequently all ICDs are ordered according to theMPC order For vlowast isin [vE[v]] when vlowast increases the ICD increases in the MPC order

Proposition 2 Suppose Condition 1 is satisfied and let plowast equiv min983153vlowast983055983055 983144Gvlowast isin D(micro)

983154 It holds that

πuss (Γ) = plowast minus Emicro[c(v)]

Proposition 2 states that an uninformed Sellerrsquos minimum payoff (with price-independentbeliefs) is characterized by the ICDs that are MPCs of the prior distribution micro By defini-tion of ICDs Sellerrsquos profit when facing ICD 983144Gvlowast is vlowast minus E[v] Sellerrsquos minimum payoff isimplemented by the most dispersed ICD that is a MPC of the prior distribution

In proving Proposition 2 the key step is to show that given the prior micro garbling Buyerrsquosinformation such that the posterior mean distribution is an ICD makes Seller weakly worseoff It is then without loss to only consider ICDs to implement πus

s (Γ) We can thus workwith a tractable one-dimensional problem Consider the set D(micro) Suppose we find G(v)

such that (i) G isin D(micro) (ii) G is an ICD and (iii) existp such that983125 p

vG(s)ds =

983125 p

vF (s)ds Such

a G is the most-dispersed ICD that is a mean preserving contraction of F Consider the

21

following two manipulations

λ

983133 v

p

F (s)ds =

983133 v

p

(pminus c(s))dF (s)minus (1minus F (v)) (pminus c(v)) + λ(v minus p) (4)

λ

983133 v

p

G(s)ds =

983133 v

p

(pminus c(s))dG(s)minus (1minusG(v)) (pminus c(v)) + λ(v minus p) (5)

First by property (iii) above the LHS of Equation 4 equals the LHS of Equation 5 Secondsince

983125 v

p(F (s) minus G(s))ds is minimized at v this implies F (p) le G(p) Therefore the first

term on RHS of Equation 4 must be larger than that of Equation 5 Notice that the firstterm on RHS is Sellerrsquos profit when offering price p This implies that the profit fromoffering p given Buyer mean-valuation distribution G(v) is lower than that given F (v)which is lower than Sellerrsquos maximum profit given F (v) On the other hand by the ICDproperty p is already Sellerrsquos optimal price given G(v) The optimal profit given valuationdistribution F must be no lower than that of G Therefore we have proved that to minimizeSellerrsquos profit it is without loss to consider only ICDs within the set D(micro) which is a onedimensional subspace The detailed proof that such ICDs exist is relegated to the appendix

The monopoly-pricing environment with c(middot) = v is covered by Condition 1 with λ = 0

and γ = v The distribution G in (3) then reduces to that identified by Roesler and Szentes(2017)

If the prior micro has binary support then the cost function c(v) is linear In this special casewe can solve for πus

s (Γ) explicitly

Corollary 1 Suppose micro has binary support V = v1 v2 Let λ = c(v2)minusc(v1)v2minusv1

and let p be theunique solution to

(pminus c(p))1

λminus1 (pminus E[c(v)]) = (v2 minus c(v2))λ

λminus1 (6)

Then πuss (Γ) = max p v2minus E[c(v)]

References

AKERLOF G A (1970) ldquoThe Market for rdquoLemonsrdquo Quality Uncertainty and the MarketMechanismrdquo Quarterly Journal of Economics 84 488ndash500

BAGWELL K AND M H RIORDAN (1991) ldquoHigh and Declining Prices Signal ProductQualityrdquo American Economic Review 81 224ndash239

BAR-ISAAC H I JEWITT AND C LEAVER (2018) ldquoAdverse Selection Efficiency and theStructure of Informationrdquo Unpublished

22

BERGEMANN D B BROOKS AND S MORRIS (2015) ldquoThe Limits of Price Discrimina-tionrdquo American Economic Review 105 921ndash57

BERGEMANN D AND S MORRIS (2016) ldquoBayes Correlated Equilibrium and the Compar-ison of Information Structures in Gamesrdquo Theoretical Economics 11 487ndash522

BLACKWELL D (1953) ldquoEquivalent Comparisons of Experimentsrdquo Annals of MathematicalStatistics 24 265ndash272

CHO I-K AND D KREPS (1987) ldquoSignaling Games and Stable Equilibriardquo Quarterly Jour-nal of Economics 102 179ndash221

DU S (2018) ldquoRobust Mechanisms Under Common Valuationrdquo Econometrica 86 1569ndash1588

DURRETT R (1995) Probability Theory and Examples Duxbury Press 2nd ed

EINAV L AND A FINKELSTEIN (2011) ldquoSelection in Insurance Markets Theory and Em-pirics in Picturesrdquo Journal of Economic Perspectives 25 115ndash38

FUDENBERG D AND J TIROLE (1991) ldquoPerfect Bayesian Equilibrium and Sequential Equi-libriumrdquo Journal of Economic Theory 53 236ndash260

GARCIA D R TEPER AND M TSUR (2018) ldquoInformation Design in Insurance MarketsSelling Peaches in a Market for Lemonsrdquo Unpublished

GREENWALD B C (1986) ldquoAdverse Selection in the Labour Marketrdquo Review of EconomicStudies 53 325ndash347

JOVANOVIC B (1982) ldquoFavorable Selection with Asymmetric Informationrdquo QuarterlyJournal of Economics 97 535ndash539

KESSLER A S (2001) ldquoRevisiting the Lemons Marketrdquo International Economic Review 4225ndash41

KREPS D AND R WILSON (1982) ldquoSequential Equilibriardquo Econometrica 50 863ndash894

LEHMANN E L (1988) ldquoComparing Location Experimentsrdquo The Annals of Statistics 16521ndash533

LEVIN J (2001) ldquoInformation and the Market for Lemonsrdquo RAND Journal of Economics32 657ndash666

23

LIBGOBER J AND X MU (2019) ldquoInformational Robustness in Intertemporal PricingrdquoWorking paper

MAKRIS M AND L RENOU (2018) ldquoInformation Design in Multi-Stage Gamesrdquo Workingpaper

MILGROM P R AND R J WEBER (1985) ldquoDistributional Strategies for Games with In-complete Informationrdquo Mathematics of Operations Research 10 619ndash632

MUSSA M AND S ROSEN (1978) ldquoMonopoly and Product Qualityrdquo Journal of EconomicTheory 18 301ndash317

RAVID D A-K ROESLER AND B SZENTES (2019) ldquoLearning Before Trading On theInefficiency of Ignoring Free Informationrdquo Working paper

ROESLER A-K AND B SZENTES (2017) ldquoBuyer-Optimal Learning and Monopoly Pric-ingrdquo American Economic Review 107 2072ndash80

TERSTIEGE S AND C WASSER (2019) ldquoBuyer-optimal Extensionproof InformationrdquoWorking paper

Appendices omitted from this version but are available on request

24

can always be implemented by reducing Buyerrsquos information in the sense of Blackwell(1953) We also explain in Subsection 32 why the triangle ADE actually characterizes allpayoffs that can obtain when Buyer does not update from Sellerrsquos price even if Buyer isnot better informed than Seller

Finally the ABC triangle in Figure 1 depicts Theorem 3 which characterizes all payoffpairs when Buyer is fully informed ie learns v We use the term ldquoAkerlofrdquo to describe afully-informed Buyer and an uninformed Seller as these information structures are stan-dard in the adverse-selection literature the corresponding payoff pair is marked as suchin the figure Depending on the environmentrsquos primitives the Akerlof point can be any-where on the segment BC including at the extreme points Any feasible payoff pair thatsatisfies Buyerrsquos individual rationality and gives Seller at least her Akerlof payoff can beimplemented with a fully-informed Buyer by suitably varying Sellerrsquos information

An implication of Theorems 1ndash3 is that it is without loss in terms of ex-ante equilibriumpayoffs to focus on information structures in which Buyer is fully informed if and onlyif Sellerrsquos Akerlof payoff coincides with her individual rationality constraint This coinci-dence occurs if and only if the Akerlof market can have full trade (Seller prices at p = v

and gets payoff v minus E[c(v)] ge 0) or no trade (the price is p ge v and both partiesrsquo payoffsare 0) As detailed in Remark 4 of Subsection 33 in all other cases the point B in Fig-ure 1 is distinct from the point D (and hence also F ) which means that Buyer can obtaina higher payoff with less-than-full information while keeping Seller uninformed Further-more under a reasonable condition if Sellerrsquos individual rationality constraint is zero (iev le E[c(v)]) then point D is also distinct from F see Remark 3 in Subsection 32 WhenD and F are distinct maximizing Buyerrsquos payoff ie achieving point G requires Seller tohave some information Buyer does not and an equilibrium with price-dependent beliefsafter conditioning on his signal Buyer must update about v from the price either on or offthe equilibrium path

Related literature Our perspective and results are most closely related to BergemannBrooks and Morris (2015) Roesler and Szentes (2017) and Makris and Renou (2018 Sec-tion 4) These papersmdashonly the relevant section of the third papermdashstudy the monopolypricing problem in which there is uncertainty only about Buyerrsquos valuation This is thespecial case of our interdependent-values model with a constant function c(v) le v6

6 The online appendix of Roesler and Szentes (2017) relaxes the assumption that there are always gainsfrom trade but maintains no uncertainty about Sellerrsquos cost Related to Roesler and Szentes (2017) are alsoDu (2018) and Libgober and Mu (2019) who consider worst-case profit guarantees for Seller in static and dy-namic environments respectively Terstiege and Wasser (2019) qualify Roesler and Szentes (2017) by allowing

4

Bergemann et al (2015) assume Buyer is fully informed and hence can only vary Sellerrsquosinformation Our Theorem 3 corresponding to triangle ABC in Figure 1 is a generaliza-tion of their main result to our environment the key step in our proof methodology is toconstruct ldquoincentive compatible distributionsrdquo which reduces to their ldquoextreme marketsrdquoin the monopoly-pricing environment Another contribution of our result is to ensure ap-proximately unique implementation in a sense explained in Subsection 33

Roesler and Szentes (2017) assume Seller is uninformed and only vary Buyerrsquos infor-mation For the monopoly-pricing environment they derive one part of our Theorem 2viz they identify the triangle ADE in Figure 1 as the implementable set given Seller isuninformed Even for this result our methodology is quite different from theirs becausewe do not stipulate a linear c(middot) function our methodology delivers new insights includ-ing that noted in Remark 2 of Subsection 32 When we specialize to a linear c(middot) we canobtain a sharper characterization of the point E which extends Roesler and Szentesrsquo char-acterization of Buyer-optimal information to an interdependent-values environment seeProposition 2 in Subsection 43

While our main interest is in interdependent values our results provide new insightseven for monopoly pricing Theorem 2 implies that the Roesler and Szentes (2017) boundsare without loss so long as Buyer is better informed than Seller or more generally in equi-libria in which Buyerrsquos belief is price independent (after conditioning on his own signal)On the other hand Theorem 1 establishes that any feasible and individually rational pay-off pair can be implemented absent these restrictions in particular Buyer may even get allthe surplus This latter point has a parallel with Makris and Renou (2018) As an applica-tion of their general results on ldquorevelation principlesrdquo for information design in multi-stagegames Makris and Renoursquos (2018) Proposition 1 deduces an analog of our Theorem 1 forthe (independent values) monopoly pricing problem We share with Makris and Renou anemphasis on sequential rationality7 we differ in showing that a single information struc-ture implements all payoffs in the relevant triangle and also in establishing in Theorem 1lowast

off-the-equilibrium-path belief consistency in the sense of sequential equilibrium (Krepsand Wilson 1982) in discretizations

Other authors have studied different aspects of more specific changes of information inadverse-selection settings maintaining that one side of the market is better informed than

Seller to supply Buyer with additional information although Seller cannot have any private information ofher own

7 Makris and Renou use an apparatus of ldquosequential Bayes correlated equilibriumrdquo which we do notThey draw a contrast with Bergemann and Morrisrsquos (2016) Bayes correlated equilibrium In our approachnote that Sellerrsquos individual rationality constraint described earlier hinges when E[c(v)] lt v on Buyerrsquosbehavior being sequentially rational even off the equilibrium path

5

the other Levin (2001) identifies conditions under which the volume of trade decreaseswhen one party is kept uninformed and the otherrsquos information become more effective inthe sense of Lehmann (1988) see also Kessler (2001) Assuming a linear payoff structureBar-Isaac Jewitt and Leaver (2018) consider how certain changes in Gaussian informationaffect the volume of trade surplus and a certain quantification of adverse selection

Finally Garcia Teper and Tsur (2018) solve for socially optimal information provisionmdashrather than characterizing the range of market outcomesmdashin an insurance setting with ad-verse selection owing to a cross-subsidization motive full information disclosure is typi-cally not optimal

The rest of the paper proceeds as follows We introduce our model equilibrium con-cept(s) and certain classes of information structures in Section 2 Section 3 presents themain results implementable payoffs when the information structure is arbitrary or varieswithin canonical classes Section 4 contains discussion and extensions Proofs and someadditional material are contained in the Appendices

2 Model

21 Primitives

Seller may sell an indivisible good to Buyer Buyerrsquos value for the good is v isin V sub Rwhere V is a compact (finite or infinite) set with v equiv minV lt maxV equiv v The valuev is drawn from a probability measure micro with support V Sellerrsquos cost of production isgiven by a function c(v)8 We assume c V rarr R is continuous v minus c(v) ge 0 for all vand E[v minus c(v)] gt 0 So trading surplus is nonnegative for all v and positive for a positivemeasure of v (Throughout expectations are with respect to the prior measure micro unlessindicated otherwise lsquopositiversquo means lsquostrictly positiversquo and similarly elsewhere) Note thatfunction c(v) need not be monotonic We call Γ = (c micro) an environment We refer to anenvironment with a constant c(middot) function as that of monopoly pricing

An information structure consists of signal spaces for each party and a joint signal dis-tribution (We abuse notation and sometimes write lsquodistributionrsquo even though lsquomeasurersquowould be more precise) Formally there is a probability space (ΩF P ) complete and sep-arable metric spaces Ts and Tb (with measurable structure given by their Borel sigma alge-bras) and an integrable function X Ω rarr TstimesTbtimesV We hereafter suppress the probabilityspace and define with an abuse of notation P (D) = P (Xminus1(D)) for any D isin Ts times Tb times V

measurable Each realization of random variable X is a triplet (tb ts v) where tb isin Tb

8 Subsection 41 discusses Sellerrsquos cost being stochastic even conditional on v

6

is Buyerrsquos signal and ts isin Ts is Sellerrsquos signal For i isin s b v let Pi denote the corre-sponding marginal distribution of P on dimension of Ti with the convention Tv equiv V Werequire Pv = micro this is the iterated expectation or Bayes plausibility requirement Denotean information structure by τ

The environment Γ and information structure τ define the following game

1 The random variables (tb ts v) are realized Signal tb is privately observed by Buyerand signal ts privately observed by Seller Neither party observes v

2 Seller posts a price p isin R

3 Buyer accepts or rejects the price If Buyer accepts his von-Neumann Morgensternpayoff is v minus p and Sellerrsquos is p minus c(v) If Buyer rejects both partiesrsquo payoffs arenormalized to 0

Note that because the signal spaces Tb and Ts are abstract and the two partiesrsquo signalscan be arbitrarily correlated conditional on v there is no loss of generality in assuming thateach party privately observes their own signal For example public information can becaptured by perfectly correlating (components of) tb and ts

22 Strategies and Equilibria

In the game defined by (Γ τ) denote Sellerrsquos strategy by σ and Buyerrsquos by α FollowingMilgrom and Weber (1985) we define σ as a distributional strategy σ is a joint distributionon R times Ts whose marginal distribution on Ts must be the Sellerrsquos signal distribution Soσ(middot|ts) is Sellerrsquos price distribution given her signal ts9 Buyerrsquos strategy α R times Tb rarr[0 1] maps each price-signal pair (p tb) into a trading probability A strategy profile (σα)

induces expected utilities for Buyer and Seller (πb πs) in the natural way

πb =

983133(v minus p)α(tb p)σ(dp|ts)P (dts dtb dv)

πs =

983133(pminus c(v))α(tb p)σ(dp|ts)P (dts dtb dv)

Our baseline equilibrium concept is weak Perfect Bayesian equilibrium Since Sellerrsquosaction is not preceded by Buyerrsquos we can dispense with specifying beliefs for Seller ForBuyer it suffices to focus on his belief about the value v given his signal and the price wedenote this distribution by ν(v|p tb)

9 Here σ(middot|ts) is the regular conditional distribution which exists and is unique almost everywhere becauseTs is a standard Borel space (Durrett 1995 pp 229ndash230) Similarly for subsequent such notation we dropldquoalmost everywhererdquo qualifiers unless essential

7

Definition 1 A strategy profile (σα) and a belief ν(v|p tb) is a weak perfect Bayesian equilib-rium (wPBE) of game (Γ τ) if

1 Buyer plays optimally at every information set given his belief

α(p tb) =

983099983103

9831011 if Eν(v|ptb)[v] gt p

0 if Eν(v|ptb)[v] lt p

2 Seller plays optimally

σ isin argmaxσ

983133(pminus c(v))α(p tb)σ(dp|ts)P (dts dtb dv)

3 Beliefs satisfy Bayes rule on path for every measurable D sube Rtimes Ts times Tb times V

983133

D

ν(dv|p tb)σ(dp|ts)P (dts dtb V ) =

983133

D

σ(dp|ts)P (dts dtb dv)

Notice that we have formulated Sellerrsquos optimality requirement ex ante but Buyerrsquos ateach information set The latter is needed to capture sequential rationality The formeris for (notational) convenience this choice is inconsequential because Seller moves beforeBuyer

Hereafter ldquoequilibriumrdquo without qualification refers to a wPBE As is well understoodwPBE permits significant latitude in beliefs off the equilibrium path We will discuss re-finements when appropriate

23 Implementable Payoffs and Canonical Information Structures

We now define the set of implementable equilibrium outcomesmdashthat is the payoffs thatobtain in some equilibrium under some information structuremdashand some canonical classesof information structures

For a game (Γ τ) let the equilibrium payoff set be

Π(Γ τ) equiv (πb πs) exist wPBE of (Γ τ) with payoffs (πb πs)

Denote the class of all information structures by T and define

Π(Γ) equiv983134

τisinT

Π(Γ τ)

8

That is for environment Γ Π(Γ) is the set of all equilibrium payoff pairs that obtain undersome information structure

Uninformed Seller An information structure has uninformed Seller if Ts is a singletonSellerrsquos own signal contains no information about Buyerrsquos value v and hence neither aboutSellerrsquos cost c(v) When discussing such information structures we write the associateddistribution as just P (tb v) and Sellerrsquos strategy as just σ(p) omitting the argument ts inboth cases The class of all uninformed-Seller information structure is denoted Tus

Fully-informed Buyer An information structure has fully-informed Buyer if Buyerrsquos signalfully reveals his value v Formally this holds if Tb = V and the conditional distribution onV P (middot|tb) satisfies P (tb|tb) = 1 We denote the class of fully-informed-Buyer informationstructures Tfb

More-informed Buyer An information structure has more-informed Buyer if Buyer hasmore information than Seller Formally this holds when v and ts are independent condi-tional on tb ie for any measurable Ds sub Ts and Dv sub V P (DscapDv|tb) = P (Ds|tb)P (Dv|tb)10

Another way to interpret this requirement is that random variable tb must be statisticallysufficient for ts with respect to v ie tb is more informative than ts about v in the sense ofBlackwell (1953) We denote the class of more-informed-Buyer information structures byTmb Naturally information structures with uninformed Seller or fully-informed Buyer arecases of more-informed Buyer both Tus and Tfb are subclasses of Tmb

No updating from price For more-informed-Buyer information structures it is desirableto impose further requirements on Buyerrsquos equilibrium belief Since Sellerrsquos price can onlydepend on her own signal and this signal contains no additional information about v givenBuyerrsquos signal the price is statistically uninformative about v given Buyerrsquos signal Con-sequently Buyerrsquos posterior belief should be price independent once his signal has beenconditioned upon Formally regardless of the price p the equilibrium belief ν(middot|p tb) mustsatisfy 983133

ν(dv|p tb)P (dtb Ts V ) =

983133P (dtb Ts dv)

We refer to this condition as price-independent belief Note that the condition is meaningfulregardless of whether Buyer is more informed than Seller In a more-informed-Buyer in-formation structure price-independent belief would be implied by the ldquono signaling what

10 We write sub to mean ldquoweak subsetrdquo

9

you donrsquot knowrdquo requirement (Fudenberg and Tirole 1991) frequently imposed in versionsof perfect Bayesian equilibrium and the concept of sequential equilibrium (Kreps and Wil-son 1982) in finite versions of our setting11

Some notation will be helpful Define

Πlowast(Γ τ) equiv (πb πs) exist wPBE of (Γ τ) with price-independent belief and payoffs(πb πs)

Πlowast(Γ) equiv983134

τisinT

Πlowast(Γ τ)

Πlowasti (Γ) equiv

983134

τisinTi

Πlowast(Γ τ) for i = us fbmb

So Πlowast and Πlowast are analogous to the implementable payoff sets Π and Π defined earlier butrestricted to equilibria with price-independent beliefs Πlowast

us Πlowastfb and Πlowast

mb are the imple-mentable payoff sets when further restricted to uninformed-Seller fully-informed-Buyerand more-informed-Buyer information structures Plainly for any environment Γ

Πlowastus(Γ) cupΠlowast

fb(Γ) sub Πlowastmb(Γ) sub Πlowast(Γ) sub Π(Γ)

3 Main Results

Our goal is to characterize equilibrium payoff pairs across information structures in anarbitrary environment Γ In particular we seek to characterize the five sets Π(Γ) Πlowast(Γ)Πlowast

mb(Γ) Πlowastus(Γ) and Πlowast

fb(Γ) Let

S(Γ) equiv E[v minus c(v)]

be the (expected) surplus from trade in environment Γ This quantity will play an impor-tant role

31 All Information Structures

Define Sellerrsquos payoff guarantee as

πs(Γ) equiv max v minus E [c(v)] 0

11 An example clarifying our terminology may be helpful If both Seller and Buyer are fully informed of vthen the natural equilibriummdashthe unique sequential equilibrium in a finite version of the gamemdashhas Sellerpricing at p = v and Buyerrsquos belief being degenerate on v regardless of Sellerrsquos price This equilibrium hasprice-independent belief even though ex-ante Sellerrsquos price are Buyerrsquos belief are perfectly correlated Thepoint is that Buyerrsquos belief does not depend on price conditional on his signal

10

To interpret this observe that it is optimal for Buyer to accept the price v no matter hisbelief Therefore Seller can guarantee herself the (expected) profit v minus E[c(v)] no matterwhat the information structure is More precisely she can guarantee v minus E[c(v)] minus ε forany ε gt 0 since sequential rationality requires Buyer to accept any price v minus ε SimilarlySeller can also guarantee zero profit offering a price p gt v Hence Sellerrsquos payoff in anyequilibrium of any information structure must be at least the payoff guarantee πs(Γ) above

On the other hand Buyer can guarantee himself the payoff πb = 0 by rejecting all pricesIt follows that the implementable set Π(Γ) must satisfy three simple constraints 1) Sellerrsquosldquoindividual rationalityrdquo constraint πs ge πs(Γ) 2) Buyerrsquos ldquoindividual rationalityrdquo con-straint πb ge 0 and 3) the feasibility constraint πb + πs le S(Γ)

Our first result is that these individual rationality and feasibility constraints are alsosufficient for a payoff pair to be implementable

Theorem 1 Consider all information structures and equilibria

Π(Γ) =

983099983105983103

983105983101

πb ge 0

(πb πs) πs ge πs(Γ)

πb + πs le S(Γ)

983100983105983104

983105983102

Theorem 1 says that the set Π(Γ) corresponds to the triangle AFG in Figure 1 In par-ticular Buyer can receive the entire surplus beyond Sellerrsquos payoff guarantee This is sur-prising as Seller has substantial bargaining power Note that when v le E[c(v)] an entirelyreasonable condition Sellerrsquos payoff guarantee is zero12

The proof of Theorem 1 is in fact straightforward Suppose for expositional simplicityv ge E[c(v)] Fix the trivial information structure in which neither player receives any infor-mation and consider the following family of strategy profiles Seller randomizes betweentwo prices some pl isin [vE[v]] and ph = E[v] Buyer accepts pl with probability one andaccepts ph with probability α(ph) where α(ph) isin [0 1] is specified to make Seller indifferentbetween the two prices That is α(ph)(ph minus E[c(v)]) = pl minus E[c(v)] The expected payoffsfrom this strategy profile are

πb = σ(pl)(E[v]minus pl) and πs = pl minus E[c(v)]

As pl traverses the interval [vE[v]] Sellerrsquos payoff πs traverses [πs(Γ) S(Γ)] Given any

12 In monopoly pricing with c(middot) = v Sellerrsquos payoff guarantee of zero is lower than the revenue guaranteeidentified in Du (2018 Section 5) which is typically positive Dursquos notion is different from ours

11

pl Buyerrsquos payoff πb traverses [0 S(Γ)minusπs] as σ(pl) traverses [0 1] Therefore the proposedstrategy profiles induce all the payoff pairs stated in Theorem 1

We are left to specify beliefs ν for Buyer After prices pl and ph Buyer holds the priorbelief micro After any other (necessarily off-path) price Buyerrsquos belief is that v = v and soBuyer rejects all prices p isin [vinfin)pl ph It is straightforward to confirm that the specified(σα ν) constitute a wPBE

To get more insight into this construction consider its implication for the monopolypricing with c(middot) = v The equilibrium with pl = v (hence α(ph) = 0) and σ(pl) = 1

corresponds to the monopolist deterministically pricing at v and Buyer purchasing Giventhat both sides of the market receive no information why doesnrsquot the monopolist deviate toany price in (vE[v]) The reason is that in this equilibrium the consumer will then not buybecause he updates to v = v Such belief is compatible with wPBE because the equilibriumconcept places no restrictions on off-path beliefs This may seem like a game-theoreticmisdirection Buyerrsquos belief is not consistent with ldquono signaling what you donrsquot knowrdquoPut differently since we have a (weakly) more-informed Buyer information structure weought to impose the price-independent belief condition described in Subsection 23 thatwould imply Buyer must purchase at any price p lt E[v]

But the message of Theorem 1 does not rely on the permissiveness of wPBE To illus-trate continue with the above monopoly-pricing environment and suppose v has positiveprior probability Consider Buyer remaining uninformed but Seller learning whether v = v

or v gt v Now Buyerrsquos off-path belief that v = v is consistent with ldquono signaling whatyou donrsquot knowrdquo More generally using richer information structures we can prove thatany payoff pair identified in Theorem 1 can be approximately implemented as a sequentialequilibrium (Kreps and Wilson 1982) in a suitably discretized game

Theorem 1lowast Fix any ε gt 0 There is a finite information structure and a finite price grid inducinga game with a set of sequential equilibrium payoffs that is an ε-net of Π(Γ)13

In fact the proof of Theorem 1lowast establishes even more the sequential equilibria in thediscretized games satisfy a natural version of the D1 refinement (Cho and Kreps 1987) Werelegate the logic to the Appendix but mention here that we use imperfectly-correlatedsignals for Buyer and Seller14

13 An information structure is finite if the signal spaces Tb and Ts are finite Sequential equilibrium isdefined in the obvious way for the ldquoinducedrdquo finite game where Nature directly draws (tb ts) rather thanfirst drawing v and playersrsquo payoffs are defined directly using (tb ts) For X sub R2 and ε gt 0 the set A sub Xis an ε-net of X if for each x isin X there is a isin A such that 983042xminus a983042 lt ε where 983042middot983042 is the Euclidean distance

14 The idea behind D1 is to ask for any off-path price whether one type of Seller would deviate for any

12

Even when our environment is specialized to monopoly pricing it is worth highlightingtwo contrasts between Theorem 11lowast and results of Bergemann et al (2015) and Roeslerand Szentes (2017) First we find that by not restricting the monopolist to be uninformedthe implementable payoff set typically expands rather dramatically trade can be efficientwith the monopolist securing none of the surplus beyond her payoff guarantee (πs) whichmay be zero (Roesler and Szentes establish implicitly that the implementable set withan uninformed monopolist is a superset of Bergemann et alrsquos which has a fully-informedconsumer) We will see in Subsection 32 that what is crucial to this expansion is price-dependent beliefs In particular the proof of Theorem 1lowast uses an information structure inwhich Buyer is not better informed than Seller mdash if he were then sequential equilibriumwould imply price-independent beliefs Second Theorem 1lowast establishes that for a givenε gt 0 a single information structure (and price grid) can be used to approximate the entirepayoff set Π(Γ) analogously to the construction described after Theorem 1 that used asingle information structure15 Bergemann et al (2015) and Roesler and Szentes (2017) onthe other hand vary information structures to span their payoff sets

32 More-informed Buyer and Price-independent Beliefs

In some economic settings it is plausible that Buyer is more informed than Seller Howdoes a restriction to such information structures ie τ isin Tmb affect the implementablepayoff set It turns out that what is in fact crucial is price-independent beliefs We have ex-plained earlier why it is desirable to impose this condition when Buyer is more informedbut that the condition is well defined even otherwise If Buyer is not more informed thanSeller price-independent beliefs ought to be viewed as an equilibrium restriction in gen-eral To reduce repetition readers should be bear in mind for the rest of this subsectionthe qualifier ldquowith price-independent beliefsrdquo applies unless stated explicitly otherwise

It is useful to define

πuss (Γ) equiv inf πs exist(πb πs) isin Πlowast

us(Γ) (1)

as the lowest payoff that an uninformed Seller can obtain no matter Buyerrsquos information(among equilibria with price-independent beliefs) Plainly πus

s (Γ) ge πs(Γ) In monopoly

Buyer mixed response that another type would Our construction has multiple Buyer types that are imper-fectly correlated with Seller types So different types of Seller have different beliefs about Buyer types Thisblunts the power of dominance considerations to the point where D1 does not exclude any Seller type fromthe support of Buyerrsquos off-path belief

15 In fact if one lets the price grid vary with ε then a single information structure implements exactlyrather than approximately in sequential equilibrium all payoffs ε-away from the boundary of Π(Γ) See in the Appendix for a formal statement

13

pricing with V = [v v] and c(middot) = v Roesler and Szentesrsquos (2017) characterization of theconsumer-optimal information structure identifies πus

s establishing that πuss gt πs If there

is no trade due to adverse selection when Seller is uniformed and Buyer has some informa-tion then πus

s = πs = 0 We do not have a general explicit formula for πuss Subsection 43

provides it for linear c(middot) Nonetheless we establish next that (i) the only additional restric-tion on equilibrium payoffs imposed by price-independent beliefs is a lower bound of πus

s

for Seller and (ii) uninformed-Seller information structures implement all such payoffs

Theorem 2 Consider equilibria with price-independent beliefs

1 Πlowast(Γ) = Πlowastmb(Γ) = Πlowast

us(Γ)

2 Πlowastus(Γ) = (πb πs) isin Π(Γ) πs ge πus

s (Γ)

3 For any (πb πs) isin Πlowastus(Γ) with πs gt πus

s (Γ) there is τ isin Tus with Π(Γ τ) = (πb πs)

Remark 1 We believe the substance of Theorem 2 would hold using discretizations and se-quential equilibria analogous to Theorem 1lowast As previously noted sequential equilibriumimplies price-independent beliefs when Buyer is more informed than Seller

To digest Theorem 2 note that Πlowast(Γ) sup Πlowastmb(Γ) sup Πlowast

us(Γ) is trivial So part 1 of thetheorem amounts to establishing the reverse inclusions The intuition for thismdashgiven part2rsquos characterization of Πlowast

usmdashis fairly straightforward with price-independent beliefs ad-ditional information cannot harm Seller even though it could alter the set of equilibria SoSellerrsquos lowest payoff obtains when she is uninformed

The characterization in part 2 of payoffs with an uninformed Seller corresponds to thetriangle ADE in Figure 1 Part 3 of the theorem assures ldquounique implementationrdquo of allimplementable payoffs satisfying πs gt πus

s (Γ) That is for any such payoff pair there is anuninformed-Seller information structure such that all equilibria (with price-independentbeliefs) induce exactly that payoff pair Unique implementation is appealing for multiplereasons one of which is that it obviates concerns about which among multiple payoff-distinct equilibria is more reasonable For example Ravid Roesler and Szentes (2019)bring to the fore the issue of equilibrium multiplicity in Roesler and Szentesrsquos (2017) infor-mation structure

Let us describe how we obtain the characterization of Πlowastus(Γ) and unique implementa-

tion There are two steps The first ensures that there is some information structure callit τ lowast isin Tus that implements Sellerrsquos payoff πus

s (Γ) That is we ensure that the infimum

14

in (1) is in fact a minimum16 While this argument is technical knowing τ lowast exists is usefulin what follows The second and economically insightful step is to construct informa-tion structures that implement every point in the triangle Πlowast

us(Γ) by suitably garbling theinformation structure τ lowast The construction is illustrated in Figure 2 Consider the distri-bution of Buyerrsquos posterior mean of his valuation v in information structure τ lowast (Givenprice-independent beliefs Buyerrsquos posterior mean is sufficient for his decision) For sim-plicity suppose this posterior-mean distribution has a density as depicted by the red curvein Figure 2 Fix any (πb πs) isin Πlowast

us(Γ)

First there is some number zlowast such that πb + πs is the total surplus from trading onlywhen Buyerrsquos posterior mean is greater than zlowast Next there is some price plowast ge zlowast suchthat Sellerrsquos payoff is πs if all these trades were to occur at price plowast17 Note that plowast must beno larger than the expected Buyer posterior mean conditional on that being above zlowast forotherwise πb lt 0 We claim that the information structure τ lowast can be garbled so that plowast isan equilibrium price and trade occurs only when Buyerrsquos posterior mean is greater thanzlowast The garbling is illustrated in Figure 2 by the blue distribution There is one signal that

pzv

ν(v)

Figure 2 Construction of information structure in Theorem 2

Buyer receives when the original posterior mean is between zlowast and plowast and also receiveswith some probability when the original posterior mean is above plowast The probability ischosen to make the posterior mean from this signal exactly plowast Apart from this one newsignal Buyer receives the original signal in τ lowast Plainly this is a garbling of τ lowast and hence isfeasible

16 The difficulty is in establishing suitable continuity Uninformed-Seller information structures can beviewed as probability measures over Buyerrsquos beliefs with convergence in the sense of the weak topologyThis topology ensures continuity with respect to probability measures of expectations of continuous orat least Lipschitz (and bounded) functions However Sellerrsquos expected payoff is not the expectation of aLipschitz function as Sellerrsquos profit is truncated at the price she charges

17 That plowast ge zlowast follows from πs ge πuss (Γ) as πs(Γ) itself is weakly larger than Sellerrsquos payoff from posting

price zlowast (and thus trading with the same set of Buyer posterior means) under information structure τlowast

15

Figure 2 makes clear why the new information structure has an equilibrium with priceplowast and Buyer breaking indifference in favor of trading (i) Sellerrsquos profit from posting anyprice below zlowast is the same as under τ lowast and hence no larger than πus

s (Γ) (ii) similarly Sellerrsquosprofit from posting any price above plowast is no higher than some fraction of πus

s (Γ) and (iii)any price between zlowast and plowast is worse that price plowast Moreover since Sellerrsquos profit fromoffering any price other than plowast is no more than πus

s (Γ) it follows that when πs gt πuss (Γ)

Buyer must break indifference as specified for Seller to have an optimal price and theequilibrium payoffs are unique

Remark 2 The above logic establishes that given any τ isin Tus that implements some (πb πs)τ can be garbled to uniquely implement any (πprime

b πprimes) isin Π(Γ) such that πprime

s gt πs That isan uninformed Sellerrsquos payoff can always be strictly raised and Buyerrsquos payoff reduced(strictly so long as it was not already zero) by garbling Buyerrsquos information

Remark 3 According to Theorem 11lowast and Theorem 2 uninformed-Seller information struc-tures cannot implement all implementable payoff pairs in an environment Γ if and only ifπuss (Γ) gt πs(Γ) This inequality fails in environment Γ if πus

s (Γ) = 0 since that impliesπuss (Γ) = πs(Γ) = 0 An example is when there is no trade due to adverse selection when

Seller is uniformed and Buyer has some information On the other hand it is sufficient forthe inequality that

v le E[c(v)] and forallv isin V c(v) lt v (2)

To see why (2) implies πuss (Γ) gt πs(Γ) notice that in any uninformed-Seller information

structure Seller can price at slightly less than Buyerrsquos highest posterior mean valuationand guarantee trade with only (a neighborhood of) that Buyer type If c(v) lt v for all v thisgives Seller a positive expected payoff and hence πus

s (Γ) gt 018 But the first inequality in (2)is equivalent to πs(Γ) = 0 We observe that Condition (2) is compatible with severe adverseselection resulting in very little trade when Seller is uninformed and Buyer (partially orfully) informed

33 Fully-Informed Buyer

We now turn to the third canonical class of information structures Buyer is fully in-formed of his value v As this is a special case of more-informed Buyer we maintain price-independent beliefs throughout this subsection

18 More precisely as V is compact c(v) lt v for all v implies there exists ε gt 0 such that v minus c(v) gt εGiven any uninformed-Seller information structure let mv be the highest posterior mean valuation in thesupport of the posterior means induced by Buyerrsquos signals So there is positive probability of Buyer signalswith posterior mean valuations at least mv minus ε2 By pricing at mv minus ε2 Sellerrsquos expected cost conditionalon trade is bounded above by mv minus ε and hence Sellerrsquos profit conditional on trade is at least ε2 gt 0 Itfollows that πus

s gt 0

16

Faced with a fully informed Buyer and any sequentially rational Buyer strategy an un-informed Seller can guarantee a profit level

πfbs (Γ) equiv sup

p

983133 v

p

(pminus c(v))micro(dv)

regardless of her information Plainly πfbs (Γ) ge πus

s (Γ) In monopoly pricing with c(middot) = vRoesler and Szentes (2017) have shown that πfb

s gt πuss if there is no trade due to adverse

selection when Buyer is fully informed and Seller is uniformed then πfbs = πus

s = 0 Belowwe establish that when Buyer is fully informed πfb

s is the only additional constraint onequilibrium payoffs Let Bε(πb πs) denote an ε-ball around (πb πs)

Theorem 3 Consider a fully informed Buyer and price-independent beliefs

1 Πlowastfb(Γ) = (πb πs) isin Π(Γ) πs ge πfb

s (Γ)

2 For any (πb πs) isin Πlowastfb(Γ) and any ε gt 0 there is τ isin Tfb with Π(Γ τ) sub Bε(πb πs)

The payoff set characterized in Theorem 3 corresponds to the triangle ABE in Figure 1Part 2 of the theorem establishes ldquoapproximately unique implementationrdquo for any point inthe triangle there is an information structure under which all equilibria yield payoff pairsin a neighborhood of that point

Here is the idea behind part 1 of Theorem 3 When Buyer is fully informed an informa-tion structure can be viewed as dividing vrsquos prior distribution micro into a set of microi that aver-age to micro with Seller informed of which microi she faces Theorem 3 is proven by establishingthat we can divide micro suitably so that against each microi Seller is indifferent between pricingat all prices in the support of microi including the price corresponding to πfb

s in that environ-ment We call such a microi a (Seller-price) incentive compatible distribution or ICD (Bergemannet al (2015) call their analogous construct an extreme market) The Appendix provides aldquogreedyrdquo algorithm to compute ICDs19

We can sketch how the algorithm works and construct a set of ICDs that average to theprior Suppose V = v1 v2 vK with vi lt vi+1 for i isin 1 K minus 1 and c(v) lt v forall v Given any small-enough mass of vK there is a unique mass of type vKminus1 that makesSeller indifferent between charging price vK and vKminus1 (If the mass is too low Seller prefersvK if it is too high she prefers vKminus1) Iterating down to keep Seller indifferent betweenall prices pins down an ICD Choose the maximum mass of type vK for which this worksRemove that ICD and repeat

19 The algorithm is defined for finite V and we take limits to handle the infinite case

17

Crucially whenever an ICD is removed the price corresponding to πfbs (Γ) remains op-

timal in the remaining ldquomarketrdquo this follows from the ICDrsquos defining property of Sellerindifference and an accounting identity Therefore Sellerrsquos profit in this segmentation ofICDs remains πfb

s (Γ) Moreover it is also optimal for Seller to always (ie for each microi) priceso that there is full trade or no trade Hence Buyerrsquos expected payoff can be either 0 or theentire surplus less πfb

s (Γ) It follows that the fully-informed Buyer information structuredefined by this set of ICDs implements point B and C in Figure 1 The entire triangle ABC

can then be implemented by convexification randomizing over this information structure(and two equilibria) and full information (where Seller obtains all the surplus)

The above logic of incentive compatible distributions reduces to that of Bergemann et al(2015) in the context of monopoly pricing Even in the monopoly pricing case Theorem 3part 2 strengthens the conclusions of Bergemann et al (2015) by ensuring approximatelyunique implementation

Remark 4 Theorems 1ndash3 imply that fully-informed-Buyer information structures imple-ment all implementable payoff pairs if and only if πfb

s (Γ) = πs(Γ) In that case trianglesAFG and ABC coincide in Figure 1 It follows that πfb

s (Γ) = πs(Γ) only when a fully-informed Buyer and uninformed Seller can result in full trade (v ge E[c(v)] and Sellerprices at v) or no trade (v le E[c(v)] and Seller prices at some p ge v) Interestinglywhen πfb

s (Γ) gt πs(Γ) fully-informed-Buyer information structures cannot even implementall payoff pairs implementable by uninformed-Seller information structures ie trianglesABC and ADE in Figure 1 are distinct That is when πfb

s (Γ) gt πs(Γ) there is an uninformed-Seller information structure that implements some πs lt πfb

s (Γ)20 Theorems 2ndash3 further im-ply in this case that Buyer can obtain a strictly higher payoff when he is not fully informed(and Seller is uninformed)

4 Discussion

41 Multidimensionality

Suppose Buyer and Sellerrsquos cost and valuation pair (c v) is a two-dimensional randomvariable distributed according to joint distribution micro with a compact support The ex-tension of our maintained assumption of commonly known gains from trade is for all

20 Pick any pprime gt v such that pprimeminusE[c(v)] lt πfbs (Γ) Following the construction described after Theorem 2 we

can mix all valuations v le pprime with a fraction λ gt 0 of valuations v gt pprime so that the mixture has posterior meanexactly pprime The remaining fraction 1 minus λ of valuations above pprime are revealed to Buyer With this uninformed-Seller information structure consider any equilibrium in which Buyer purchases when indifferent (Suchan equilibrium with price-independent belief exists) Sellerrsquos profit is at most (1 minus λ)πfb

s (Γ) from any pricep gt pprime and pprime minus E[c(v)] from price p = pprime Hence Sellerrsquos profit is strictly less than πfb

s (Γ)

18

(c v) isin Supp(micro) v ge c and E[v minus c] gt 0 An information structure is now a joint distribu-tion P (tb ts v c) whose marginal distribution on (v c) is micro

We claim that the content of our key results Theorem 1Theorem 1lowast Theorem 2 andTheorem 3 still hold To see why let v be the lowest valuation in the support of micro Sellerrsquosindividual rationality constraint is now maxv minus E[c] 0 as she can guarantee this by set-ting either a sufficiently high price or a price (arbitrarily close to) v regardless of her signalWe can define a cost function c(vprime) equiv Emicro[c|v = vprime] ge vprime This results in an environment sat-isfying all the maintained assumptions of our baseline model except that c(middot) may not becontinuous Recall that such continuity is not needed for proving Theorem 1 Theorem 1lowastBoth Theorem 2 and Theorem 3 use continuity of c(middot) to guarantee that Eν [c(v)] is a contin-uous function of ν isin ∆(V ) for a key convergence result However in the two-dimensionaltype environment Eν [c] is still a continuous function of ν isin ∆(V times C) Note that Theo-rem 3 uses upper semi-continuity of Sellerrsquos profit in price boundedness of c and c le v issufficient for such upper semi-continuity

42 Negative Trading Surplus

Returning to our baseline model we next discuss cases in which trade sometimes gen-erates negative surplus That is we drop the assumption that c(v) le v we do not requireE[v minus c(v)] gt 0 either Define Sλ(Γ) for λ isin [1infin) as

Sλ(Γ) equiv983133 v

v

[v minus c(v) + λ(v minus v)]+ micro(dv)

where [middot]+ equiv max middot 0 It is readily verified that S1(Γ) = E [[v minus c(v)]+] and limλrarr0

Sλ(Γ)λ =

E[v] minus v Allowing negative trading surplus does not affect our definition of wPBE Sothe notation Π(Γ) and πs(Γ) still have the same meanings as before The next propositionshows that Π(Γ) is now characterized by three constraints as before the two individualrationality constraints πs ge πs(Γ) and πb ge 0 and different now a Pareto frontier definedby all Sλ(Γ)

Proposition 1 Consider all information structures and equilibria when trade can generate negativesurplus

Π(Γ) =

983099983105983103

983105983101

πb ge 0

(πb πs) πs ge πs(Γ)

λπb + πs le Sλ(Γ) forallλ ge 1

983100983105983104

983105983102

Proposition 1 shows that when the trade can be inefficient it may be impossible to lower

19

Sellerrsquos profit without losing total surplus (ie πb + πs) This result is intuitive at the ex-tremes Suppose for simplicity v gt E[c(v)] If πs = πs(Γ) then since πs(Γ) = v minus E[c(v)]Seller must find it optimal to price at v and trade with probability one including whentrade generates negative surplus So Buyerrsquos payoff is E[v]minus v = limλrarrinfin Sλ(Γ)λ and totalsurplus is E[v minus c(v)] which includes the maximum amount of inefficient trade On theother hand an information structure that publicly reveals (only) whether trade is efficient(ie whether v ge c(v)) can implement efficient trade with all the surplus accruing to SellerIn this case Seller obtains S1(Γ) while Buyer obtains zero The proposition establishes thatin between S1(Γ) and πs(Γ) information structures and equilibria that lower Sellerrsquos profitmust reduce total surplus as the Pareto frontier Sλ(Γ) is a concave function with maximalslope minus1

The function capturing the Pareto frontier Sλ(Γ) is the maximal weighted sum of Buyerand Seller payoff assuming that Seller will trade at the low price p = v whenever it createspositive weighted total payoff Since we place higher weight on buyer (λ ge 1) S1(Γ) isthe upper bound for the weighted total payoff In the proof of Proposition 1 we showthat these bounds are tight mdash any payoff pair on the frontier can be implemented by someinformation structure The key observation is that whenever v minus c(v) + λ(v minus v) lt 0 itfollows that vminus c(v) lt (1minusλ)(vminus v) le 0 That is it is incentive compatible for Seller to notsell when it is common knowledge that v creates negative weighted total payoff Thereforewe can simply release a public signal indicating the sign of weighted total payoff and ifpositive induce Seller to sell at v which implies all bounds Sλ(Γ)

43 Linear Cost

When Sellerrsquos cost function c(v) is linear we can explicitly characterize an informationstructure that implements Sellerrsquos minimum implementable payoff πus

s (Γ) when Seller isuninformed (or per Theorem 2 when Buyer is better informed) subject to price-independentbeliefs We note that given the discussion in Subsection 41 a linear c(v) subsumes richerenvironments in which conditional expectations are linear including Gaussian environ-ments (cf Bar-Isaac et al 2018)

Condition 1 c(v) = λv + γ for some λ isin R

Let F (v) be the cumulative distribution function (CDF) corresponding to the prior mea-sure micro Let D(micro) be the set of all distributions whose CDF G satisfies

983133

V

vdG(v) =

983133

V

vdF (v) and983133 v

v

G(s)ds le983133 v

v

F (s)ds forallv isin V

20

That is D(micro) contains all distributions that are mean-preserving-contractions (MPC) of microIt is well known that D(micro) characterizes the set of distributions of Buyer posterior meansthat can be generated by any (uninformed-Seller) information structure

Definition 2 G is an incentive compatible distribution (ICD) if

983133

sgep

(pminus c(s))G(ds) = constant ge 0 forallp isin Supp(G)

That is when Buyerrsquos (mean) valuation is distributed according to an ICD G Seller isindifferent among all prices in the support of G assuming Buyer buys when indifferent Wefocus on a special family of ICDs whose supports are intervals [vlowast v

lowast] Given Condition 1such ICDs have analytical an expression

G(v) =

983099983105983105983103

983105983105983101

1minus983061

(1minus λ)v minus γ

(1minus λ)vlowast minus γ

983062 1λminus1

if λ ∕= 1

1minus evminusvlowast

γ if λ = 1

(3)

Given any vlowast and the above G(v) an upper bound vlowast is pinned down by the conditionEG[v] = EF [v] So a family of ICDs is parametrized by a single parameter vlowast and hence anICD is denoted by CDF 983144Gvlowast(v) with density 983144gvlowast(v) It is not difficult to verify that 983144Gvlowast(v) ismonotonic in vlowast for v in the support Consequently all ICDs are ordered according to theMPC order For vlowast isin [vE[v]] when vlowast increases the ICD increases in the MPC order

Proposition 2 Suppose Condition 1 is satisfied and let plowast equiv min983153vlowast983055983055 983144Gvlowast isin D(micro)

983154 It holds that

πuss (Γ) = plowast minus Emicro[c(v)]

Proposition 2 states that an uninformed Sellerrsquos minimum payoff (with price-independentbeliefs) is characterized by the ICDs that are MPCs of the prior distribution micro By defini-tion of ICDs Sellerrsquos profit when facing ICD 983144Gvlowast is vlowast minus E[v] Sellerrsquos minimum payoff isimplemented by the most dispersed ICD that is a MPC of the prior distribution

In proving Proposition 2 the key step is to show that given the prior micro garbling Buyerrsquosinformation such that the posterior mean distribution is an ICD makes Seller weakly worseoff It is then without loss to only consider ICDs to implement πus

s (Γ) We can thus workwith a tractable one-dimensional problem Consider the set D(micro) Suppose we find G(v)

such that (i) G isin D(micro) (ii) G is an ICD and (iii) existp such that983125 p

vG(s)ds =

983125 p

vF (s)ds Such

a G is the most-dispersed ICD that is a mean preserving contraction of F Consider the

21

following two manipulations

λ

983133 v

p

F (s)ds =

983133 v

p

(pminus c(s))dF (s)minus (1minus F (v)) (pminus c(v)) + λ(v minus p) (4)

λ

983133 v

p

G(s)ds =

983133 v

p

(pminus c(s))dG(s)minus (1minusG(v)) (pminus c(v)) + λ(v minus p) (5)

First by property (iii) above the LHS of Equation 4 equals the LHS of Equation 5 Secondsince

983125 v

p(F (s) minus G(s))ds is minimized at v this implies F (p) le G(p) Therefore the first

term on RHS of Equation 4 must be larger than that of Equation 5 Notice that the firstterm on RHS is Sellerrsquos profit when offering price p This implies that the profit fromoffering p given Buyer mean-valuation distribution G(v) is lower than that given F (v)which is lower than Sellerrsquos maximum profit given F (v) On the other hand by the ICDproperty p is already Sellerrsquos optimal price given G(v) The optimal profit given valuationdistribution F must be no lower than that of G Therefore we have proved that to minimizeSellerrsquos profit it is without loss to consider only ICDs within the set D(micro) which is a onedimensional subspace The detailed proof that such ICDs exist is relegated to the appendix

The monopoly-pricing environment with c(middot) = v is covered by Condition 1 with λ = 0

and γ = v The distribution G in (3) then reduces to that identified by Roesler and Szentes(2017)

If the prior micro has binary support then the cost function c(v) is linear In this special casewe can solve for πus

s (Γ) explicitly

Corollary 1 Suppose micro has binary support V = v1 v2 Let λ = c(v2)minusc(v1)v2minusv1

and let p be theunique solution to

(pminus c(p))1

λminus1 (pminus E[c(v)]) = (v2 minus c(v2))λ

λminus1 (6)

Then πuss (Γ) = max p v2minus E[c(v)]

References

AKERLOF G A (1970) ldquoThe Market for rdquoLemonsrdquo Quality Uncertainty and the MarketMechanismrdquo Quarterly Journal of Economics 84 488ndash500

BAGWELL K AND M H RIORDAN (1991) ldquoHigh and Declining Prices Signal ProductQualityrdquo American Economic Review 81 224ndash239

BAR-ISAAC H I JEWITT AND C LEAVER (2018) ldquoAdverse Selection Efficiency and theStructure of Informationrdquo Unpublished

22

BERGEMANN D B BROOKS AND S MORRIS (2015) ldquoThe Limits of Price Discrimina-tionrdquo American Economic Review 105 921ndash57

BERGEMANN D AND S MORRIS (2016) ldquoBayes Correlated Equilibrium and the Compar-ison of Information Structures in Gamesrdquo Theoretical Economics 11 487ndash522

BLACKWELL D (1953) ldquoEquivalent Comparisons of Experimentsrdquo Annals of MathematicalStatistics 24 265ndash272

CHO I-K AND D KREPS (1987) ldquoSignaling Games and Stable Equilibriardquo Quarterly Jour-nal of Economics 102 179ndash221

DU S (2018) ldquoRobust Mechanisms Under Common Valuationrdquo Econometrica 86 1569ndash1588

DURRETT R (1995) Probability Theory and Examples Duxbury Press 2nd ed

EINAV L AND A FINKELSTEIN (2011) ldquoSelection in Insurance Markets Theory and Em-pirics in Picturesrdquo Journal of Economic Perspectives 25 115ndash38

FUDENBERG D AND J TIROLE (1991) ldquoPerfect Bayesian Equilibrium and Sequential Equi-libriumrdquo Journal of Economic Theory 53 236ndash260

GARCIA D R TEPER AND M TSUR (2018) ldquoInformation Design in Insurance MarketsSelling Peaches in a Market for Lemonsrdquo Unpublished

GREENWALD B C (1986) ldquoAdverse Selection in the Labour Marketrdquo Review of EconomicStudies 53 325ndash347

JOVANOVIC B (1982) ldquoFavorable Selection with Asymmetric Informationrdquo QuarterlyJournal of Economics 97 535ndash539

KESSLER A S (2001) ldquoRevisiting the Lemons Marketrdquo International Economic Review 4225ndash41

KREPS D AND R WILSON (1982) ldquoSequential Equilibriardquo Econometrica 50 863ndash894

LEHMANN E L (1988) ldquoComparing Location Experimentsrdquo The Annals of Statistics 16521ndash533

LEVIN J (2001) ldquoInformation and the Market for Lemonsrdquo RAND Journal of Economics32 657ndash666

23

LIBGOBER J AND X MU (2019) ldquoInformational Robustness in Intertemporal PricingrdquoWorking paper

MAKRIS M AND L RENOU (2018) ldquoInformation Design in Multi-Stage Gamesrdquo Workingpaper

MILGROM P R AND R J WEBER (1985) ldquoDistributional Strategies for Games with In-complete Informationrdquo Mathematics of Operations Research 10 619ndash632

MUSSA M AND S ROSEN (1978) ldquoMonopoly and Product Qualityrdquo Journal of EconomicTheory 18 301ndash317

RAVID D A-K ROESLER AND B SZENTES (2019) ldquoLearning Before Trading On theInefficiency of Ignoring Free Informationrdquo Working paper

ROESLER A-K AND B SZENTES (2017) ldquoBuyer-Optimal Learning and Monopoly Pric-ingrdquo American Economic Review 107 2072ndash80

TERSTIEGE S AND C WASSER (2019) ldquoBuyer-optimal Extensionproof InformationrdquoWorking paper

Appendices omitted from this version but are available on request

24

Bergemann et al (2015) assume Buyer is fully informed and hence can only vary Sellerrsquosinformation Our Theorem 3 corresponding to triangle ABC in Figure 1 is a generaliza-tion of their main result to our environment the key step in our proof methodology is toconstruct ldquoincentive compatible distributionsrdquo which reduces to their ldquoextreme marketsrdquoin the monopoly-pricing environment Another contribution of our result is to ensure ap-proximately unique implementation in a sense explained in Subsection 33

Roesler and Szentes (2017) assume Seller is uninformed and only vary Buyerrsquos infor-mation For the monopoly-pricing environment they derive one part of our Theorem 2viz they identify the triangle ADE in Figure 1 as the implementable set given Seller isuninformed Even for this result our methodology is quite different from theirs becausewe do not stipulate a linear c(middot) function our methodology delivers new insights includ-ing that noted in Remark 2 of Subsection 32 When we specialize to a linear c(middot) we canobtain a sharper characterization of the point E which extends Roesler and Szentesrsquo char-acterization of Buyer-optimal information to an interdependent-values environment seeProposition 2 in Subsection 43

While our main interest is in interdependent values our results provide new insightseven for monopoly pricing Theorem 2 implies that the Roesler and Szentes (2017) boundsare without loss so long as Buyer is better informed than Seller or more generally in equi-libria in which Buyerrsquos belief is price independent (after conditioning on his own signal)On the other hand Theorem 1 establishes that any feasible and individually rational pay-off pair can be implemented absent these restrictions in particular Buyer may even get allthe surplus This latter point has a parallel with Makris and Renou (2018) As an applica-tion of their general results on ldquorevelation principlesrdquo for information design in multi-stagegames Makris and Renoursquos (2018) Proposition 1 deduces an analog of our Theorem 1 forthe (independent values) monopoly pricing problem We share with Makris and Renou anemphasis on sequential rationality7 we differ in showing that a single information struc-ture implements all payoffs in the relevant triangle and also in establishing in Theorem 1lowast

off-the-equilibrium-path belief consistency in the sense of sequential equilibrium (Krepsand Wilson 1982) in discretizations

Other authors have studied different aspects of more specific changes of information inadverse-selection settings maintaining that one side of the market is better informed than

Seller to supply Buyer with additional information although Seller cannot have any private information ofher own

7 Makris and Renou use an apparatus of ldquosequential Bayes correlated equilibriumrdquo which we do notThey draw a contrast with Bergemann and Morrisrsquos (2016) Bayes correlated equilibrium In our approachnote that Sellerrsquos individual rationality constraint described earlier hinges when E[c(v)] lt v on Buyerrsquosbehavior being sequentially rational even off the equilibrium path

5

the other Levin (2001) identifies conditions under which the volume of trade decreaseswhen one party is kept uninformed and the otherrsquos information become more effective inthe sense of Lehmann (1988) see also Kessler (2001) Assuming a linear payoff structureBar-Isaac Jewitt and Leaver (2018) consider how certain changes in Gaussian informationaffect the volume of trade surplus and a certain quantification of adverse selection

Finally Garcia Teper and Tsur (2018) solve for socially optimal information provisionmdashrather than characterizing the range of market outcomesmdashin an insurance setting with ad-verse selection owing to a cross-subsidization motive full information disclosure is typi-cally not optimal

The rest of the paper proceeds as follows We introduce our model equilibrium con-cept(s) and certain classes of information structures in Section 2 Section 3 presents themain results implementable payoffs when the information structure is arbitrary or varieswithin canonical classes Section 4 contains discussion and extensions Proofs and someadditional material are contained in the Appendices

2 Model

21 Primitives

Seller may sell an indivisible good to Buyer Buyerrsquos value for the good is v isin V sub Rwhere V is a compact (finite or infinite) set with v equiv minV lt maxV equiv v The valuev is drawn from a probability measure micro with support V Sellerrsquos cost of production isgiven by a function c(v)8 We assume c V rarr R is continuous v minus c(v) ge 0 for all vand E[v minus c(v)] gt 0 So trading surplus is nonnegative for all v and positive for a positivemeasure of v (Throughout expectations are with respect to the prior measure micro unlessindicated otherwise lsquopositiversquo means lsquostrictly positiversquo and similarly elsewhere) Note thatfunction c(v) need not be monotonic We call Γ = (c micro) an environment We refer to anenvironment with a constant c(middot) function as that of monopoly pricing

An information structure consists of signal spaces for each party and a joint signal dis-tribution (We abuse notation and sometimes write lsquodistributionrsquo even though lsquomeasurersquowould be more precise) Formally there is a probability space (ΩF P ) complete and sep-arable metric spaces Ts and Tb (with measurable structure given by their Borel sigma alge-bras) and an integrable function X Ω rarr TstimesTbtimesV We hereafter suppress the probabilityspace and define with an abuse of notation P (D) = P (Xminus1(D)) for any D isin Ts times Tb times V

measurable Each realization of random variable X is a triplet (tb ts v) where tb isin Tb

8 Subsection 41 discusses Sellerrsquos cost being stochastic even conditional on v

6

is Buyerrsquos signal and ts isin Ts is Sellerrsquos signal For i isin s b v let Pi denote the corre-sponding marginal distribution of P on dimension of Ti with the convention Tv equiv V Werequire Pv = micro this is the iterated expectation or Bayes plausibility requirement Denotean information structure by τ

The environment Γ and information structure τ define the following game

1 The random variables (tb ts v) are realized Signal tb is privately observed by Buyerand signal ts privately observed by Seller Neither party observes v

2 Seller posts a price p isin R

3 Buyer accepts or rejects the price If Buyer accepts his von-Neumann Morgensternpayoff is v minus p and Sellerrsquos is p minus c(v) If Buyer rejects both partiesrsquo payoffs arenormalized to 0

Note that because the signal spaces Tb and Ts are abstract and the two partiesrsquo signalscan be arbitrarily correlated conditional on v there is no loss of generality in assuming thateach party privately observes their own signal For example public information can becaptured by perfectly correlating (components of) tb and ts

22 Strategies and Equilibria

In the game defined by (Γ τ) denote Sellerrsquos strategy by σ and Buyerrsquos by α FollowingMilgrom and Weber (1985) we define σ as a distributional strategy σ is a joint distributionon R times Ts whose marginal distribution on Ts must be the Sellerrsquos signal distribution Soσ(middot|ts) is Sellerrsquos price distribution given her signal ts9 Buyerrsquos strategy α R times Tb rarr[0 1] maps each price-signal pair (p tb) into a trading probability A strategy profile (σα)

induces expected utilities for Buyer and Seller (πb πs) in the natural way

πb =

983133(v minus p)α(tb p)σ(dp|ts)P (dts dtb dv)

πs =

983133(pminus c(v))α(tb p)σ(dp|ts)P (dts dtb dv)

Our baseline equilibrium concept is weak Perfect Bayesian equilibrium Since Sellerrsquosaction is not preceded by Buyerrsquos we can dispense with specifying beliefs for Seller ForBuyer it suffices to focus on his belief about the value v given his signal and the price wedenote this distribution by ν(v|p tb)

9 Here σ(middot|ts) is the regular conditional distribution which exists and is unique almost everywhere becauseTs is a standard Borel space (Durrett 1995 pp 229ndash230) Similarly for subsequent such notation we dropldquoalmost everywhererdquo qualifiers unless essential

7

Definition 1 A strategy profile (σα) and a belief ν(v|p tb) is a weak perfect Bayesian equilib-rium (wPBE) of game (Γ τ) if

1 Buyer plays optimally at every information set given his belief

α(p tb) =

983099983103

9831011 if Eν(v|ptb)[v] gt p

0 if Eν(v|ptb)[v] lt p

2 Seller plays optimally

σ isin argmaxσ

983133(pminus c(v))α(p tb)σ(dp|ts)P (dts dtb dv)

3 Beliefs satisfy Bayes rule on path for every measurable D sube Rtimes Ts times Tb times V

983133

D

ν(dv|p tb)σ(dp|ts)P (dts dtb V ) =

983133

D

σ(dp|ts)P (dts dtb dv)

Notice that we have formulated Sellerrsquos optimality requirement ex ante but Buyerrsquos ateach information set The latter is needed to capture sequential rationality The formeris for (notational) convenience this choice is inconsequential because Seller moves beforeBuyer

Hereafter ldquoequilibriumrdquo without qualification refers to a wPBE As is well understoodwPBE permits significant latitude in beliefs off the equilibrium path We will discuss re-finements when appropriate

23 Implementable Payoffs and Canonical Information Structures

We now define the set of implementable equilibrium outcomesmdashthat is the payoffs thatobtain in some equilibrium under some information structuremdashand some canonical classesof information structures

For a game (Γ τ) let the equilibrium payoff set be

Π(Γ τ) equiv (πb πs) exist wPBE of (Γ τ) with payoffs (πb πs)

Denote the class of all information structures by T and define

Π(Γ) equiv983134

τisinT

Π(Γ τ)

8

That is for environment Γ Π(Γ) is the set of all equilibrium payoff pairs that obtain undersome information structure

Uninformed Seller An information structure has uninformed Seller if Ts is a singletonSellerrsquos own signal contains no information about Buyerrsquos value v and hence neither aboutSellerrsquos cost c(v) When discussing such information structures we write the associateddistribution as just P (tb v) and Sellerrsquos strategy as just σ(p) omitting the argument ts inboth cases The class of all uninformed-Seller information structure is denoted Tus

Fully-informed Buyer An information structure has fully-informed Buyer if Buyerrsquos signalfully reveals his value v Formally this holds if Tb = V and the conditional distribution onV P (middot|tb) satisfies P (tb|tb) = 1 We denote the class of fully-informed-Buyer informationstructures Tfb

More-informed Buyer An information structure has more-informed Buyer if Buyer hasmore information than Seller Formally this holds when v and ts are independent condi-tional on tb ie for any measurable Ds sub Ts and Dv sub V P (DscapDv|tb) = P (Ds|tb)P (Dv|tb)10

Another way to interpret this requirement is that random variable tb must be statisticallysufficient for ts with respect to v ie tb is more informative than ts about v in the sense ofBlackwell (1953) We denote the class of more-informed-Buyer information structures byTmb Naturally information structures with uninformed Seller or fully-informed Buyer arecases of more-informed Buyer both Tus and Tfb are subclasses of Tmb

No updating from price For more-informed-Buyer information structures it is desirableto impose further requirements on Buyerrsquos equilibrium belief Since Sellerrsquos price can onlydepend on her own signal and this signal contains no additional information about v givenBuyerrsquos signal the price is statistically uninformative about v given Buyerrsquos signal Con-sequently Buyerrsquos posterior belief should be price independent once his signal has beenconditioned upon Formally regardless of the price p the equilibrium belief ν(middot|p tb) mustsatisfy 983133

ν(dv|p tb)P (dtb Ts V ) =

983133P (dtb Ts dv)

We refer to this condition as price-independent belief Note that the condition is meaningfulregardless of whether Buyer is more informed than Seller In a more-informed-Buyer in-formation structure price-independent belief would be implied by the ldquono signaling what

10 We write sub to mean ldquoweak subsetrdquo

9

you donrsquot knowrdquo requirement (Fudenberg and Tirole 1991) frequently imposed in versionsof perfect Bayesian equilibrium and the concept of sequential equilibrium (Kreps and Wil-son 1982) in finite versions of our setting11

Some notation will be helpful Define

Πlowast(Γ τ) equiv (πb πs) exist wPBE of (Γ τ) with price-independent belief and payoffs(πb πs)

Πlowast(Γ) equiv983134

τisinT

Πlowast(Γ τ)

Πlowasti (Γ) equiv

983134

τisinTi

Πlowast(Γ τ) for i = us fbmb

So Πlowast and Πlowast are analogous to the implementable payoff sets Π and Π defined earlier butrestricted to equilibria with price-independent beliefs Πlowast

us Πlowastfb and Πlowast

mb are the imple-mentable payoff sets when further restricted to uninformed-Seller fully-informed-Buyerand more-informed-Buyer information structures Plainly for any environment Γ

Πlowastus(Γ) cupΠlowast

fb(Γ) sub Πlowastmb(Γ) sub Πlowast(Γ) sub Π(Γ)

3 Main Results

Our goal is to characterize equilibrium payoff pairs across information structures in anarbitrary environment Γ In particular we seek to characterize the five sets Π(Γ) Πlowast(Γ)Πlowast

mb(Γ) Πlowastus(Γ) and Πlowast

fb(Γ) Let

S(Γ) equiv E[v minus c(v)]

be the (expected) surplus from trade in environment Γ This quantity will play an impor-tant role

31 All Information Structures

Define Sellerrsquos payoff guarantee as

πs(Γ) equiv max v minus E [c(v)] 0

11 An example clarifying our terminology may be helpful If both Seller and Buyer are fully informed of vthen the natural equilibriummdashthe unique sequential equilibrium in a finite version of the gamemdashhas Sellerpricing at p = v and Buyerrsquos belief being degenerate on v regardless of Sellerrsquos price This equilibrium hasprice-independent belief even though ex-ante Sellerrsquos price are Buyerrsquos belief are perfectly correlated Thepoint is that Buyerrsquos belief does not depend on price conditional on his signal

10

To interpret this observe that it is optimal for Buyer to accept the price v no matter hisbelief Therefore Seller can guarantee herself the (expected) profit v minus E[c(v)] no matterwhat the information structure is More precisely she can guarantee v minus E[c(v)] minus ε forany ε gt 0 since sequential rationality requires Buyer to accept any price v minus ε SimilarlySeller can also guarantee zero profit offering a price p gt v Hence Sellerrsquos payoff in anyequilibrium of any information structure must be at least the payoff guarantee πs(Γ) above

On the other hand Buyer can guarantee himself the payoff πb = 0 by rejecting all pricesIt follows that the implementable set Π(Γ) must satisfy three simple constraints 1) Sellerrsquosldquoindividual rationalityrdquo constraint πs ge πs(Γ) 2) Buyerrsquos ldquoindividual rationalityrdquo con-straint πb ge 0 and 3) the feasibility constraint πb + πs le S(Γ)

Our first result is that these individual rationality and feasibility constraints are alsosufficient for a payoff pair to be implementable

Theorem 1 Consider all information structures and equilibria

Π(Γ) =

983099983105983103

983105983101

πb ge 0

(πb πs) πs ge πs(Γ)

πb + πs le S(Γ)

983100983105983104

983105983102

Theorem 1 says that the set Π(Γ) corresponds to the triangle AFG in Figure 1 In par-ticular Buyer can receive the entire surplus beyond Sellerrsquos payoff guarantee This is sur-prising as Seller has substantial bargaining power Note that when v le E[c(v)] an entirelyreasonable condition Sellerrsquos payoff guarantee is zero12

The proof of Theorem 1 is in fact straightforward Suppose for expositional simplicityv ge E[c(v)] Fix the trivial information structure in which neither player receives any infor-mation and consider the following family of strategy profiles Seller randomizes betweentwo prices some pl isin [vE[v]] and ph = E[v] Buyer accepts pl with probability one andaccepts ph with probability α(ph) where α(ph) isin [0 1] is specified to make Seller indifferentbetween the two prices That is α(ph)(ph minus E[c(v)]) = pl minus E[c(v)] The expected payoffsfrom this strategy profile are

πb = σ(pl)(E[v]minus pl) and πs = pl minus E[c(v)]

As pl traverses the interval [vE[v]] Sellerrsquos payoff πs traverses [πs(Γ) S(Γ)] Given any

12 In monopoly pricing with c(middot) = v Sellerrsquos payoff guarantee of zero is lower than the revenue guaranteeidentified in Du (2018 Section 5) which is typically positive Dursquos notion is different from ours

11

pl Buyerrsquos payoff πb traverses [0 S(Γ)minusπs] as σ(pl) traverses [0 1] Therefore the proposedstrategy profiles induce all the payoff pairs stated in Theorem 1

We are left to specify beliefs ν for Buyer After prices pl and ph Buyer holds the priorbelief micro After any other (necessarily off-path) price Buyerrsquos belief is that v = v and soBuyer rejects all prices p isin [vinfin)pl ph It is straightforward to confirm that the specified(σα ν) constitute a wPBE

To get more insight into this construction consider its implication for the monopolypricing with c(middot) = v The equilibrium with pl = v (hence α(ph) = 0) and σ(pl) = 1

corresponds to the monopolist deterministically pricing at v and Buyer purchasing Giventhat both sides of the market receive no information why doesnrsquot the monopolist deviate toany price in (vE[v]) The reason is that in this equilibrium the consumer will then not buybecause he updates to v = v Such belief is compatible with wPBE because the equilibriumconcept places no restrictions on off-path beliefs This may seem like a game-theoreticmisdirection Buyerrsquos belief is not consistent with ldquono signaling what you donrsquot knowrdquoPut differently since we have a (weakly) more-informed Buyer information structure weought to impose the price-independent belief condition described in Subsection 23 thatwould imply Buyer must purchase at any price p lt E[v]

But the message of Theorem 1 does not rely on the permissiveness of wPBE To illus-trate continue with the above monopoly-pricing environment and suppose v has positiveprior probability Consider Buyer remaining uninformed but Seller learning whether v = v

or v gt v Now Buyerrsquos off-path belief that v = v is consistent with ldquono signaling whatyou donrsquot knowrdquo More generally using richer information structures we can prove thatany payoff pair identified in Theorem 1 can be approximately implemented as a sequentialequilibrium (Kreps and Wilson 1982) in a suitably discretized game

Theorem 1lowast Fix any ε gt 0 There is a finite information structure and a finite price grid inducinga game with a set of sequential equilibrium payoffs that is an ε-net of Π(Γ)13

In fact the proof of Theorem 1lowast establishes even more the sequential equilibria in thediscretized games satisfy a natural version of the D1 refinement (Cho and Kreps 1987) Werelegate the logic to the Appendix but mention here that we use imperfectly-correlatedsignals for Buyer and Seller14

13 An information structure is finite if the signal spaces Tb and Ts are finite Sequential equilibrium isdefined in the obvious way for the ldquoinducedrdquo finite game where Nature directly draws (tb ts) rather thanfirst drawing v and playersrsquo payoffs are defined directly using (tb ts) For X sub R2 and ε gt 0 the set A sub Xis an ε-net of X if for each x isin X there is a isin A such that 983042xminus a983042 lt ε where 983042middot983042 is the Euclidean distance

14 The idea behind D1 is to ask for any off-path price whether one type of Seller would deviate for any

12

Even when our environment is specialized to monopoly pricing it is worth highlightingtwo contrasts between Theorem 11lowast and results of Bergemann et al (2015) and Roeslerand Szentes (2017) First we find that by not restricting the monopolist to be uninformedthe implementable payoff set typically expands rather dramatically trade can be efficientwith the monopolist securing none of the surplus beyond her payoff guarantee (πs) whichmay be zero (Roesler and Szentes establish implicitly that the implementable set withan uninformed monopolist is a superset of Bergemann et alrsquos which has a fully-informedconsumer) We will see in Subsection 32 that what is crucial to this expansion is price-dependent beliefs In particular the proof of Theorem 1lowast uses an information structure inwhich Buyer is not better informed than Seller mdash if he were then sequential equilibriumwould imply price-independent beliefs Second Theorem 1lowast establishes that for a givenε gt 0 a single information structure (and price grid) can be used to approximate the entirepayoff set Π(Γ) analogously to the construction described after Theorem 1 that used asingle information structure15 Bergemann et al (2015) and Roesler and Szentes (2017) onthe other hand vary information structures to span their payoff sets

32 More-informed Buyer and Price-independent Beliefs

In some economic settings it is plausible that Buyer is more informed than Seller Howdoes a restriction to such information structures ie τ isin Tmb affect the implementablepayoff set It turns out that what is in fact crucial is price-independent beliefs We have ex-plained earlier why it is desirable to impose this condition when Buyer is more informedbut that the condition is well defined even otherwise If Buyer is not more informed thanSeller price-independent beliefs ought to be viewed as an equilibrium restriction in gen-eral To reduce repetition readers should be bear in mind for the rest of this subsectionthe qualifier ldquowith price-independent beliefsrdquo applies unless stated explicitly otherwise

It is useful to define

πuss (Γ) equiv inf πs exist(πb πs) isin Πlowast

us(Γ) (1)

as the lowest payoff that an uninformed Seller can obtain no matter Buyerrsquos information(among equilibria with price-independent beliefs) Plainly πus

s (Γ) ge πs(Γ) In monopoly

Buyer mixed response that another type would Our construction has multiple Buyer types that are imper-fectly correlated with Seller types So different types of Seller have different beliefs about Buyer types Thisblunts the power of dominance considerations to the point where D1 does not exclude any Seller type fromthe support of Buyerrsquos off-path belief

15 In fact if one lets the price grid vary with ε then a single information structure implements exactlyrather than approximately in sequential equilibrium all payoffs ε-away from the boundary of Π(Γ) See in the Appendix for a formal statement

13

pricing with V = [v v] and c(middot) = v Roesler and Szentesrsquos (2017) characterization of theconsumer-optimal information structure identifies πus

s establishing that πuss gt πs If there

is no trade due to adverse selection when Seller is uniformed and Buyer has some informa-tion then πus

s = πs = 0 We do not have a general explicit formula for πuss Subsection 43

provides it for linear c(middot) Nonetheless we establish next that (i) the only additional restric-tion on equilibrium payoffs imposed by price-independent beliefs is a lower bound of πus

s

for Seller and (ii) uninformed-Seller information structures implement all such payoffs

Theorem 2 Consider equilibria with price-independent beliefs

1 Πlowast(Γ) = Πlowastmb(Γ) = Πlowast

us(Γ)

2 Πlowastus(Γ) = (πb πs) isin Π(Γ) πs ge πus

s (Γ)

3 For any (πb πs) isin Πlowastus(Γ) with πs gt πus

s (Γ) there is τ isin Tus with Π(Γ τ) = (πb πs)

Remark 1 We believe the substance of Theorem 2 would hold using discretizations and se-quential equilibria analogous to Theorem 1lowast As previously noted sequential equilibriumimplies price-independent beliefs when Buyer is more informed than Seller

To digest Theorem 2 note that Πlowast(Γ) sup Πlowastmb(Γ) sup Πlowast

us(Γ) is trivial So part 1 of thetheorem amounts to establishing the reverse inclusions The intuition for thismdashgiven part2rsquos characterization of Πlowast

usmdashis fairly straightforward with price-independent beliefs ad-ditional information cannot harm Seller even though it could alter the set of equilibria SoSellerrsquos lowest payoff obtains when she is uninformed

The characterization in part 2 of payoffs with an uninformed Seller corresponds to thetriangle ADE in Figure 1 Part 3 of the theorem assures ldquounique implementationrdquo of allimplementable payoffs satisfying πs gt πus

s (Γ) That is for any such payoff pair there is anuninformed-Seller information structure such that all equilibria (with price-independentbeliefs) induce exactly that payoff pair Unique implementation is appealing for multiplereasons one of which is that it obviates concerns about which among multiple payoff-distinct equilibria is more reasonable For example Ravid Roesler and Szentes (2019)bring to the fore the issue of equilibrium multiplicity in Roesler and Szentesrsquos (2017) infor-mation structure

Let us describe how we obtain the characterization of Πlowastus(Γ) and unique implementa-

tion There are two steps The first ensures that there is some information structure callit τ lowast isin Tus that implements Sellerrsquos payoff πus

s (Γ) That is we ensure that the infimum

14

in (1) is in fact a minimum16 While this argument is technical knowing τ lowast exists is usefulin what follows The second and economically insightful step is to construct informa-tion structures that implement every point in the triangle Πlowast

us(Γ) by suitably garbling theinformation structure τ lowast The construction is illustrated in Figure 2 Consider the distri-bution of Buyerrsquos posterior mean of his valuation v in information structure τ lowast (Givenprice-independent beliefs Buyerrsquos posterior mean is sufficient for his decision) For sim-plicity suppose this posterior-mean distribution has a density as depicted by the red curvein Figure 2 Fix any (πb πs) isin Πlowast

us(Γ)

First there is some number zlowast such that πb + πs is the total surplus from trading onlywhen Buyerrsquos posterior mean is greater than zlowast Next there is some price plowast ge zlowast suchthat Sellerrsquos payoff is πs if all these trades were to occur at price plowast17 Note that plowast must beno larger than the expected Buyer posterior mean conditional on that being above zlowast forotherwise πb lt 0 We claim that the information structure τ lowast can be garbled so that plowast isan equilibrium price and trade occurs only when Buyerrsquos posterior mean is greater thanzlowast The garbling is illustrated in Figure 2 by the blue distribution There is one signal that

pzv

ν(v)

Figure 2 Construction of information structure in Theorem 2

Buyer receives when the original posterior mean is between zlowast and plowast and also receiveswith some probability when the original posterior mean is above plowast The probability ischosen to make the posterior mean from this signal exactly plowast Apart from this one newsignal Buyer receives the original signal in τ lowast Plainly this is a garbling of τ lowast and hence isfeasible

16 The difficulty is in establishing suitable continuity Uninformed-Seller information structures can beviewed as probability measures over Buyerrsquos beliefs with convergence in the sense of the weak topologyThis topology ensures continuity with respect to probability measures of expectations of continuous orat least Lipschitz (and bounded) functions However Sellerrsquos expected payoff is not the expectation of aLipschitz function as Sellerrsquos profit is truncated at the price she charges

17 That plowast ge zlowast follows from πs ge πuss (Γ) as πs(Γ) itself is weakly larger than Sellerrsquos payoff from posting

price zlowast (and thus trading with the same set of Buyer posterior means) under information structure τlowast

15

Figure 2 makes clear why the new information structure has an equilibrium with priceplowast and Buyer breaking indifference in favor of trading (i) Sellerrsquos profit from posting anyprice below zlowast is the same as under τ lowast and hence no larger than πus

s (Γ) (ii) similarly Sellerrsquosprofit from posting any price above plowast is no higher than some fraction of πus

s (Γ) and (iii)any price between zlowast and plowast is worse that price plowast Moreover since Sellerrsquos profit fromoffering any price other than plowast is no more than πus

s (Γ) it follows that when πs gt πuss (Γ)

Buyer must break indifference as specified for Seller to have an optimal price and theequilibrium payoffs are unique

Remark 2 The above logic establishes that given any τ isin Tus that implements some (πb πs)τ can be garbled to uniquely implement any (πprime

b πprimes) isin Π(Γ) such that πprime

s gt πs That isan uninformed Sellerrsquos payoff can always be strictly raised and Buyerrsquos payoff reduced(strictly so long as it was not already zero) by garbling Buyerrsquos information

Remark 3 According to Theorem 11lowast and Theorem 2 uninformed-Seller information struc-tures cannot implement all implementable payoff pairs in an environment Γ if and only ifπuss (Γ) gt πs(Γ) This inequality fails in environment Γ if πus

s (Γ) = 0 since that impliesπuss (Γ) = πs(Γ) = 0 An example is when there is no trade due to adverse selection when

Seller is uniformed and Buyer has some information On the other hand it is sufficient forthe inequality that

v le E[c(v)] and forallv isin V c(v) lt v (2)

To see why (2) implies πuss (Γ) gt πs(Γ) notice that in any uninformed-Seller information

structure Seller can price at slightly less than Buyerrsquos highest posterior mean valuationand guarantee trade with only (a neighborhood of) that Buyer type If c(v) lt v for all v thisgives Seller a positive expected payoff and hence πus

s (Γ) gt 018 But the first inequality in (2)is equivalent to πs(Γ) = 0 We observe that Condition (2) is compatible with severe adverseselection resulting in very little trade when Seller is uninformed and Buyer (partially orfully) informed

33 Fully-Informed Buyer

We now turn to the third canonical class of information structures Buyer is fully in-formed of his value v As this is a special case of more-informed Buyer we maintain price-independent beliefs throughout this subsection

18 More precisely as V is compact c(v) lt v for all v implies there exists ε gt 0 such that v minus c(v) gt εGiven any uninformed-Seller information structure let mv be the highest posterior mean valuation in thesupport of the posterior means induced by Buyerrsquos signals So there is positive probability of Buyer signalswith posterior mean valuations at least mv minus ε2 By pricing at mv minus ε2 Sellerrsquos expected cost conditionalon trade is bounded above by mv minus ε and hence Sellerrsquos profit conditional on trade is at least ε2 gt 0 Itfollows that πus

s gt 0

16

Faced with a fully informed Buyer and any sequentially rational Buyer strategy an un-informed Seller can guarantee a profit level

πfbs (Γ) equiv sup

p

983133 v

p

(pminus c(v))micro(dv)

regardless of her information Plainly πfbs (Γ) ge πus

s (Γ) In monopoly pricing with c(middot) = vRoesler and Szentes (2017) have shown that πfb

s gt πuss if there is no trade due to adverse

selection when Buyer is fully informed and Seller is uniformed then πfbs = πus

s = 0 Belowwe establish that when Buyer is fully informed πfb

s is the only additional constraint onequilibrium payoffs Let Bε(πb πs) denote an ε-ball around (πb πs)

Theorem 3 Consider a fully informed Buyer and price-independent beliefs

1 Πlowastfb(Γ) = (πb πs) isin Π(Γ) πs ge πfb

s (Γ)

2 For any (πb πs) isin Πlowastfb(Γ) and any ε gt 0 there is τ isin Tfb with Π(Γ τ) sub Bε(πb πs)

The payoff set characterized in Theorem 3 corresponds to the triangle ABE in Figure 1Part 2 of the theorem establishes ldquoapproximately unique implementationrdquo for any point inthe triangle there is an information structure under which all equilibria yield payoff pairsin a neighborhood of that point

Here is the idea behind part 1 of Theorem 3 When Buyer is fully informed an informa-tion structure can be viewed as dividing vrsquos prior distribution micro into a set of microi that aver-age to micro with Seller informed of which microi she faces Theorem 3 is proven by establishingthat we can divide micro suitably so that against each microi Seller is indifferent between pricingat all prices in the support of microi including the price corresponding to πfb

s in that environ-ment We call such a microi a (Seller-price) incentive compatible distribution or ICD (Bergemannet al (2015) call their analogous construct an extreme market) The Appendix provides aldquogreedyrdquo algorithm to compute ICDs19

We can sketch how the algorithm works and construct a set of ICDs that average to theprior Suppose V = v1 v2 vK with vi lt vi+1 for i isin 1 K minus 1 and c(v) lt v forall v Given any small-enough mass of vK there is a unique mass of type vKminus1 that makesSeller indifferent between charging price vK and vKminus1 (If the mass is too low Seller prefersvK if it is too high she prefers vKminus1) Iterating down to keep Seller indifferent betweenall prices pins down an ICD Choose the maximum mass of type vK for which this worksRemove that ICD and repeat

19 The algorithm is defined for finite V and we take limits to handle the infinite case

17

Crucially whenever an ICD is removed the price corresponding to πfbs (Γ) remains op-

timal in the remaining ldquomarketrdquo this follows from the ICDrsquos defining property of Sellerindifference and an accounting identity Therefore Sellerrsquos profit in this segmentation ofICDs remains πfb

s (Γ) Moreover it is also optimal for Seller to always (ie for each microi) priceso that there is full trade or no trade Hence Buyerrsquos expected payoff can be either 0 or theentire surplus less πfb

s (Γ) It follows that the fully-informed Buyer information structuredefined by this set of ICDs implements point B and C in Figure 1 The entire triangle ABC

can then be implemented by convexification randomizing over this information structure(and two equilibria) and full information (where Seller obtains all the surplus)

The above logic of incentive compatible distributions reduces to that of Bergemann et al(2015) in the context of monopoly pricing Even in the monopoly pricing case Theorem 3part 2 strengthens the conclusions of Bergemann et al (2015) by ensuring approximatelyunique implementation

Remark 4 Theorems 1ndash3 imply that fully-informed-Buyer information structures imple-ment all implementable payoff pairs if and only if πfb

s (Γ) = πs(Γ) In that case trianglesAFG and ABC coincide in Figure 1 It follows that πfb

s (Γ) = πs(Γ) only when a fully-informed Buyer and uninformed Seller can result in full trade (v ge E[c(v)] and Sellerprices at v) or no trade (v le E[c(v)] and Seller prices at some p ge v) Interestinglywhen πfb

s (Γ) gt πs(Γ) fully-informed-Buyer information structures cannot even implementall payoff pairs implementable by uninformed-Seller information structures ie trianglesABC and ADE in Figure 1 are distinct That is when πfb

s (Γ) gt πs(Γ) there is an uninformed-Seller information structure that implements some πs lt πfb

s (Γ)20 Theorems 2ndash3 further im-ply in this case that Buyer can obtain a strictly higher payoff when he is not fully informed(and Seller is uninformed)

4 Discussion

41 Multidimensionality

Suppose Buyer and Sellerrsquos cost and valuation pair (c v) is a two-dimensional randomvariable distributed according to joint distribution micro with a compact support The ex-tension of our maintained assumption of commonly known gains from trade is for all

20 Pick any pprime gt v such that pprimeminusE[c(v)] lt πfbs (Γ) Following the construction described after Theorem 2 we

can mix all valuations v le pprime with a fraction λ gt 0 of valuations v gt pprime so that the mixture has posterior meanexactly pprime The remaining fraction 1 minus λ of valuations above pprime are revealed to Buyer With this uninformed-Seller information structure consider any equilibrium in which Buyer purchases when indifferent (Suchan equilibrium with price-independent belief exists) Sellerrsquos profit is at most (1 minus λ)πfb

s (Γ) from any pricep gt pprime and pprime minus E[c(v)] from price p = pprime Hence Sellerrsquos profit is strictly less than πfb

s (Γ)

18

(c v) isin Supp(micro) v ge c and E[v minus c] gt 0 An information structure is now a joint distribu-tion P (tb ts v c) whose marginal distribution on (v c) is micro

We claim that the content of our key results Theorem 1Theorem 1lowast Theorem 2 andTheorem 3 still hold To see why let v be the lowest valuation in the support of micro Sellerrsquosindividual rationality constraint is now maxv minus E[c] 0 as she can guarantee this by set-ting either a sufficiently high price or a price (arbitrarily close to) v regardless of her signalWe can define a cost function c(vprime) equiv Emicro[c|v = vprime] ge vprime This results in an environment sat-isfying all the maintained assumptions of our baseline model except that c(middot) may not becontinuous Recall that such continuity is not needed for proving Theorem 1 Theorem 1lowastBoth Theorem 2 and Theorem 3 use continuity of c(middot) to guarantee that Eν [c(v)] is a contin-uous function of ν isin ∆(V ) for a key convergence result However in the two-dimensionaltype environment Eν [c] is still a continuous function of ν isin ∆(V times C) Note that Theo-rem 3 uses upper semi-continuity of Sellerrsquos profit in price boundedness of c and c le v issufficient for such upper semi-continuity

42 Negative Trading Surplus

Returning to our baseline model we next discuss cases in which trade sometimes gen-erates negative surplus That is we drop the assumption that c(v) le v we do not requireE[v minus c(v)] gt 0 either Define Sλ(Γ) for λ isin [1infin) as

Sλ(Γ) equiv983133 v

v

[v minus c(v) + λ(v minus v)]+ micro(dv)

where [middot]+ equiv max middot 0 It is readily verified that S1(Γ) = E [[v minus c(v)]+] and limλrarr0

Sλ(Γ)λ =

E[v] minus v Allowing negative trading surplus does not affect our definition of wPBE Sothe notation Π(Γ) and πs(Γ) still have the same meanings as before The next propositionshows that Π(Γ) is now characterized by three constraints as before the two individualrationality constraints πs ge πs(Γ) and πb ge 0 and different now a Pareto frontier definedby all Sλ(Γ)

Proposition 1 Consider all information structures and equilibria when trade can generate negativesurplus

Π(Γ) =

983099983105983103

983105983101

πb ge 0

(πb πs) πs ge πs(Γ)

λπb + πs le Sλ(Γ) forallλ ge 1

983100983105983104

983105983102

Proposition 1 shows that when the trade can be inefficient it may be impossible to lower

19

Sellerrsquos profit without losing total surplus (ie πb + πs) This result is intuitive at the ex-tremes Suppose for simplicity v gt E[c(v)] If πs = πs(Γ) then since πs(Γ) = v minus E[c(v)]Seller must find it optimal to price at v and trade with probability one including whentrade generates negative surplus So Buyerrsquos payoff is E[v]minus v = limλrarrinfin Sλ(Γ)λ and totalsurplus is E[v minus c(v)] which includes the maximum amount of inefficient trade On theother hand an information structure that publicly reveals (only) whether trade is efficient(ie whether v ge c(v)) can implement efficient trade with all the surplus accruing to SellerIn this case Seller obtains S1(Γ) while Buyer obtains zero The proposition establishes thatin between S1(Γ) and πs(Γ) information structures and equilibria that lower Sellerrsquos profitmust reduce total surplus as the Pareto frontier Sλ(Γ) is a concave function with maximalslope minus1

The function capturing the Pareto frontier Sλ(Γ) is the maximal weighted sum of Buyerand Seller payoff assuming that Seller will trade at the low price p = v whenever it createspositive weighted total payoff Since we place higher weight on buyer (λ ge 1) S1(Γ) isthe upper bound for the weighted total payoff In the proof of Proposition 1 we showthat these bounds are tight mdash any payoff pair on the frontier can be implemented by someinformation structure The key observation is that whenever v minus c(v) + λ(v minus v) lt 0 itfollows that vminus c(v) lt (1minusλ)(vminus v) le 0 That is it is incentive compatible for Seller to notsell when it is common knowledge that v creates negative weighted total payoff Thereforewe can simply release a public signal indicating the sign of weighted total payoff and ifpositive induce Seller to sell at v which implies all bounds Sλ(Γ)

43 Linear Cost

When Sellerrsquos cost function c(v) is linear we can explicitly characterize an informationstructure that implements Sellerrsquos minimum implementable payoff πus

s (Γ) when Seller isuninformed (or per Theorem 2 when Buyer is better informed) subject to price-independentbeliefs We note that given the discussion in Subsection 41 a linear c(v) subsumes richerenvironments in which conditional expectations are linear including Gaussian environ-ments (cf Bar-Isaac et al 2018)

Condition 1 c(v) = λv + γ for some λ isin R

Let F (v) be the cumulative distribution function (CDF) corresponding to the prior mea-sure micro Let D(micro) be the set of all distributions whose CDF G satisfies

983133

V

vdG(v) =

983133

V

vdF (v) and983133 v

v

G(s)ds le983133 v

v

F (s)ds forallv isin V

20

That is D(micro) contains all distributions that are mean-preserving-contractions (MPC) of microIt is well known that D(micro) characterizes the set of distributions of Buyer posterior meansthat can be generated by any (uninformed-Seller) information structure

Definition 2 G is an incentive compatible distribution (ICD) if

983133

sgep

(pminus c(s))G(ds) = constant ge 0 forallp isin Supp(G)

That is when Buyerrsquos (mean) valuation is distributed according to an ICD G Seller isindifferent among all prices in the support of G assuming Buyer buys when indifferent Wefocus on a special family of ICDs whose supports are intervals [vlowast v

lowast] Given Condition 1such ICDs have analytical an expression

G(v) =

983099983105983105983103

983105983105983101

1minus983061

(1minus λ)v minus γ

(1minus λ)vlowast minus γ

983062 1λminus1

if λ ∕= 1

1minus evminusvlowast

γ if λ = 1

(3)

Given any vlowast and the above G(v) an upper bound vlowast is pinned down by the conditionEG[v] = EF [v] So a family of ICDs is parametrized by a single parameter vlowast and hence anICD is denoted by CDF 983144Gvlowast(v) with density 983144gvlowast(v) It is not difficult to verify that 983144Gvlowast(v) ismonotonic in vlowast for v in the support Consequently all ICDs are ordered according to theMPC order For vlowast isin [vE[v]] when vlowast increases the ICD increases in the MPC order

Proposition 2 Suppose Condition 1 is satisfied and let plowast equiv min983153vlowast983055983055 983144Gvlowast isin D(micro)

983154 It holds that

πuss (Γ) = plowast minus Emicro[c(v)]

Proposition 2 states that an uninformed Sellerrsquos minimum payoff (with price-independentbeliefs) is characterized by the ICDs that are MPCs of the prior distribution micro By defini-tion of ICDs Sellerrsquos profit when facing ICD 983144Gvlowast is vlowast minus E[v] Sellerrsquos minimum payoff isimplemented by the most dispersed ICD that is a MPC of the prior distribution

In proving Proposition 2 the key step is to show that given the prior micro garbling Buyerrsquosinformation such that the posterior mean distribution is an ICD makes Seller weakly worseoff It is then without loss to only consider ICDs to implement πus

s (Γ) We can thus workwith a tractable one-dimensional problem Consider the set D(micro) Suppose we find G(v)

such that (i) G isin D(micro) (ii) G is an ICD and (iii) existp such that983125 p

vG(s)ds =

983125 p

vF (s)ds Such

a G is the most-dispersed ICD that is a mean preserving contraction of F Consider the

21

following two manipulations

λ

983133 v

p

F (s)ds =

983133 v

p

(pminus c(s))dF (s)minus (1minus F (v)) (pminus c(v)) + λ(v minus p) (4)

λ

983133 v

p

G(s)ds =

983133 v

p

(pminus c(s))dG(s)minus (1minusG(v)) (pminus c(v)) + λ(v minus p) (5)

First by property (iii) above the LHS of Equation 4 equals the LHS of Equation 5 Secondsince

983125 v

p(F (s) minus G(s))ds is minimized at v this implies F (p) le G(p) Therefore the first

term on RHS of Equation 4 must be larger than that of Equation 5 Notice that the firstterm on RHS is Sellerrsquos profit when offering price p This implies that the profit fromoffering p given Buyer mean-valuation distribution G(v) is lower than that given F (v)which is lower than Sellerrsquos maximum profit given F (v) On the other hand by the ICDproperty p is already Sellerrsquos optimal price given G(v) The optimal profit given valuationdistribution F must be no lower than that of G Therefore we have proved that to minimizeSellerrsquos profit it is without loss to consider only ICDs within the set D(micro) which is a onedimensional subspace The detailed proof that such ICDs exist is relegated to the appendix

The monopoly-pricing environment with c(middot) = v is covered by Condition 1 with λ = 0

and γ = v The distribution G in (3) then reduces to that identified by Roesler and Szentes(2017)

If the prior micro has binary support then the cost function c(v) is linear In this special casewe can solve for πus

s (Γ) explicitly

Corollary 1 Suppose micro has binary support V = v1 v2 Let λ = c(v2)minusc(v1)v2minusv1

and let p be theunique solution to

(pminus c(p))1

λminus1 (pminus E[c(v)]) = (v2 minus c(v2))λ

λminus1 (6)

Then πuss (Γ) = max p v2minus E[c(v)]

References

AKERLOF G A (1970) ldquoThe Market for rdquoLemonsrdquo Quality Uncertainty and the MarketMechanismrdquo Quarterly Journal of Economics 84 488ndash500

BAGWELL K AND M H RIORDAN (1991) ldquoHigh and Declining Prices Signal ProductQualityrdquo American Economic Review 81 224ndash239

BAR-ISAAC H I JEWITT AND C LEAVER (2018) ldquoAdverse Selection Efficiency and theStructure of Informationrdquo Unpublished

22

BERGEMANN D B BROOKS AND S MORRIS (2015) ldquoThe Limits of Price Discrimina-tionrdquo American Economic Review 105 921ndash57

BERGEMANN D AND S MORRIS (2016) ldquoBayes Correlated Equilibrium and the Compar-ison of Information Structures in Gamesrdquo Theoretical Economics 11 487ndash522

BLACKWELL D (1953) ldquoEquivalent Comparisons of Experimentsrdquo Annals of MathematicalStatistics 24 265ndash272

CHO I-K AND D KREPS (1987) ldquoSignaling Games and Stable Equilibriardquo Quarterly Jour-nal of Economics 102 179ndash221

DU S (2018) ldquoRobust Mechanisms Under Common Valuationrdquo Econometrica 86 1569ndash1588

DURRETT R (1995) Probability Theory and Examples Duxbury Press 2nd ed

EINAV L AND A FINKELSTEIN (2011) ldquoSelection in Insurance Markets Theory and Em-pirics in Picturesrdquo Journal of Economic Perspectives 25 115ndash38

FUDENBERG D AND J TIROLE (1991) ldquoPerfect Bayesian Equilibrium and Sequential Equi-libriumrdquo Journal of Economic Theory 53 236ndash260

GARCIA D R TEPER AND M TSUR (2018) ldquoInformation Design in Insurance MarketsSelling Peaches in a Market for Lemonsrdquo Unpublished

GREENWALD B C (1986) ldquoAdverse Selection in the Labour Marketrdquo Review of EconomicStudies 53 325ndash347

JOVANOVIC B (1982) ldquoFavorable Selection with Asymmetric Informationrdquo QuarterlyJournal of Economics 97 535ndash539

KESSLER A S (2001) ldquoRevisiting the Lemons Marketrdquo International Economic Review 4225ndash41

KREPS D AND R WILSON (1982) ldquoSequential Equilibriardquo Econometrica 50 863ndash894

LEHMANN E L (1988) ldquoComparing Location Experimentsrdquo The Annals of Statistics 16521ndash533

LEVIN J (2001) ldquoInformation and the Market for Lemonsrdquo RAND Journal of Economics32 657ndash666

23

LIBGOBER J AND X MU (2019) ldquoInformational Robustness in Intertemporal PricingrdquoWorking paper

MAKRIS M AND L RENOU (2018) ldquoInformation Design in Multi-Stage Gamesrdquo Workingpaper

MILGROM P R AND R J WEBER (1985) ldquoDistributional Strategies for Games with In-complete Informationrdquo Mathematics of Operations Research 10 619ndash632

MUSSA M AND S ROSEN (1978) ldquoMonopoly and Product Qualityrdquo Journal of EconomicTheory 18 301ndash317

RAVID D A-K ROESLER AND B SZENTES (2019) ldquoLearning Before Trading On theInefficiency of Ignoring Free Informationrdquo Working paper

ROESLER A-K AND B SZENTES (2017) ldquoBuyer-Optimal Learning and Monopoly Pric-ingrdquo American Economic Review 107 2072ndash80

TERSTIEGE S AND C WASSER (2019) ldquoBuyer-optimal Extensionproof InformationrdquoWorking paper

Appendices omitted from this version but are available on request

24

the other Levin (2001) identifies conditions under which the volume of trade decreaseswhen one party is kept uninformed and the otherrsquos information become more effective inthe sense of Lehmann (1988) see also Kessler (2001) Assuming a linear payoff structureBar-Isaac Jewitt and Leaver (2018) consider how certain changes in Gaussian informationaffect the volume of trade surplus and a certain quantification of adverse selection

Finally Garcia Teper and Tsur (2018) solve for socially optimal information provisionmdashrather than characterizing the range of market outcomesmdashin an insurance setting with ad-verse selection owing to a cross-subsidization motive full information disclosure is typi-cally not optimal

The rest of the paper proceeds as follows We introduce our model equilibrium con-cept(s) and certain classes of information structures in Section 2 Section 3 presents themain results implementable payoffs when the information structure is arbitrary or varieswithin canonical classes Section 4 contains discussion and extensions Proofs and someadditional material are contained in the Appendices

2 Model

21 Primitives

Seller may sell an indivisible good to Buyer Buyerrsquos value for the good is v isin V sub Rwhere V is a compact (finite or infinite) set with v equiv minV lt maxV equiv v The valuev is drawn from a probability measure micro with support V Sellerrsquos cost of production isgiven by a function c(v)8 We assume c V rarr R is continuous v minus c(v) ge 0 for all vand E[v minus c(v)] gt 0 So trading surplus is nonnegative for all v and positive for a positivemeasure of v (Throughout expectations are with respect to the prior measure micro unlessindicated otherwise lsquopositiversquo means lsquostrictly positiversquo and similarly elsewhere) Note thatfunction c(v) need not be monotonic We call Γ = (c micro) an environment We refer to anenvironment with a constant c(middot) function as that of monopoly pricing

An information structure consists of signal spaces for each party and a joint signal dis-tribution (We abuse notation and sometimes write lsquodistributionrsquo even though lsquomeasurersquowould be more precise) Formally there is a probability space (ΩF P ) complete and sep-arable metric spaces Ts and Tb (with measurable structure given by their Borel sigma alge-bras) and an integrable function X Ω rarr TstimesTbtimesV We hereafter suppress the probabilityspace and define with an abuse of notation P (D) = P (Xminus1(D)) for any D isin Ts times Tb times V

measurable Each realization of random variable X is a triplet (tb ts v) where tb isin Tb

8 Subsection 41 discusses Sellerrsquos cost being stochastic even conditional on v

6

is Buyerrsquos signal and ts isin Ts is Sellerrsquos signal For i isin s b v let Pi denote the corre-sponding marginal distribution of P on dimension of Ti with the convention Tv equiv V Werequire Pv = micro this is the iterated expectation or Bayes plausibility requirement Denotean information structure by τ

The environment Γ and information structure τ define the following game

1 The random variables (tb ts v) are realized Signal tb is privately observed by Buyerand signal ts privately observed by Seller Neither party observes v

2 Seller posts a price p isin R

3 Buyer accepts or rejects the price If Buyer accepts his von-Neumann Morgensternpayoff is v minus p and Sellerrsquos is p minus c(v) If Buyer rejects both partiesrsquo payoffs arenormalized to 0

Note that because the signal spaces Tb and Ts are abstract and the two partiesrsquo signalscan be arbitrarily correlated conditional on v there is no loss of generality in assuming thateach party privately observes their own signal For example public information can becaptured by perfectly correlating (components of) tb and ts

22 Strategies and Equilibria

In the game defined by (Γ τ) denote Sellerrsquos strategy by σ and Buyerrsquos by α FollowingMilgrom and Weber (1985) we define σ as a distributional strategy σ is a joint distributionon R times Ts whose marginal distribution on Ts must be the Sellerrsquos signal distribution Soσ(middot|ts) is Sellerrsquos price distribution given her signal ts9 Buyerrsquos strategy α R times Tb rarr[0 1] maps each price-signal pair (p tb) into a trading probability A strategy profile (σα)

induces expected utilities for Buyer and Seller (πb πs) in the natural way

πb =

983133(v minus p)α(tb p)σ(dp|ts)P (dts dtb dv)

πs =

983133(pminus c(v))α(tb p)σ(dp|ts)P (dts dtb dv)

Our baseline equilibrium concept is weak Perfect Bayesian equilibrium Since Sellerrsquosaction is not preceded by Buyerrsquos we can dispense with specifying beliefs for Seller ForBuyer it suffices to focus on his belief about the value v given his signal and the price wedenote this distribution by ν(v|p tb)

9 Here σ(middot|ts) is the regular conditional distribution which exists and is unique almost everywhere becauseTs is a standard Borel space (Durrett 1995 pp 229ndash230) Similarly for subsequent such notation we dropldquoalmost everywhererdquo qualifiers unless essential

7

Definition 1 A strategy profile (σα) and a belief ν(v|p tb) is a weak perfect Bayesian equilib-rium (wPBE) of game (Γ τ) if

1 Buyer plays optimally at every information set given his belief

α(p tb) =

983099983103

9831011 if Eν(v|ptb)[v] gt p

0 if Eν(v|ptb)[v] lt p

2 Seller plays optimally

σ isin argmaxσ

983133(pminus c(v))α(p tb)σ(dp|ts)P (dts dtb dv)

3 Beliefs satisfy Bayes rule on path for every measurable D sube Rtimes Ts times Tb times V

983133

D

ν(dv|p tb)σ(dp|ts)P (dts dtb V ) =

983133

D

σ(dp|ts)P (dts dtb dv)

Notice that we have formulated Sellerrsquos optimality requirement ex ante but Buyerrsquos ateach information set The latter is needed to capture sequential rationality The formeris for (notational) convenience this choice is inconsequential because Seller moves beforeBuyer

Hereafter ldquoequilibriumrdquo without qualification refers to a wPBE As is well understoodwPBE permits significant latitude in beliefs off the equilibrium path We will discuss re-finements when appropriate

23 Implementable Payoffs and Canonical Information Structures

We now define the set of implementable equilibrium outcomesmdashthat is the payoffs thatobtain in some equilibrium under some information structuremdashand some canonical classesof information structures

For a game (Γ τ) let the equilibrium payoff set be

Π(Γ τ) equiv (πb πs) exist wPBE of (Γ τ) with payoffs (πb πs)

Denote the class of all information structures by T and define

Π(Γ) equiv983134

τisinT

Π(Γ τ)

8

That is for environment Γ Π(Γ) is the set of all equilibrium payoff pairs that obtain undersome information structure

Uninformed Seller An information structure has uninformed Seller if Ts is a singletonSellerrsquos own signal contains no information about Buyerrsquos value v and hence neither aboutSellerrsquos cost c(v) When discussing such information structures we write the associateddistribution as just P (tb v) and Sellerrsquos strategy as just σ(p) omitting the argument ts inboth cases The class of all uninformed-Seller information structure is denoted Tus

Fully-informed Buyer An information structure has fully-informed Buyer if Buyerrsquos signalfully reveals his value v Formally this holds if Tb = V and the conditional distribution onV P (middot|tb) satisfies P (tb|tb) = 1 We denote the class of fully-informed-Buyer informationstructures Tfb

More-informed Buyer An information structure has more-informed Buyer if Buyer hasmore information than Seller Formally this holds when v and ts are independent condi-tional on tb ie for any measurable Ds sub Ts and Dv sub V P (DscapDv|tb) = P (Ds|tb)P (Dv|tb)10

Another way to interpret this requirement is that random variable tb must be statisticallysufficient for ts with respect to v ie tb is more informative than ts about v in the sense ofBlackwell (1953) We denote the class of more-informed-Buyer information structures byTmb Naturally information structures with uninformed Seller or fully-informed Buyer arecases of more-informed Buyer both Tus and Tfb are subclasses of Tmb

No updating from price For more-informed-Buyer information structures it is desirableto impose further requirements on Buyerrsquos equilibrium belief Since Sellerrsquos price can onlydepend on her own signal and this signal contains no additional information about v givenBuyerrsquos signal the price is statistically uninformative about v given Buyerrsquos signal Con-sequently Buyerrsquos posterior belief should be price independent once his signal has beenconditioned upon Formally regardless of the price p the equilibrium belief ν(middot|p tb) mustsatisfy 983133

ν(dv|p tb)P (dtb Ts V ) =

983133P (dtb Ts dv)

We refer to this condition as price-independent belief Note that the condition is meaningfulregardless of whether Buyer is more informed than Seller In a more-informed-Buyer in-formation structure price-independent belief would be implied by the ldquono signaling what

10 We write sub to mean ldquoweak subsetrdquo

9

you donrsquot knowrdquo requirement (Fudenberg and Tirole 1991) frequently imposed in versionsof perfect Bayesian equilibrium and the concept of sequential equilibrium (Kreps and Wil-son 1982) in finite versions of our setting11

Some notation will be helpful Define

Πlowast(Γ τ) equiv (πb πs) exist wPBE of (Γ τ) with price-independent belief and payoffs(πb πs)

Πlowast(Γ) equiv983134

τisinT

Πlowast(Γ τ)

Πlowasti (Γ) equiv

983134

τisinTi

Πlowast(Γ τ) for i = us fbmb

So Πlowast and Πlowast are analogous to the implementable payoff sets Π and Π defined earlier butrestricted to equilibria with price-independent beliefs Πlowast

us Πlowastfb and Πlowast

mb are the imple-mentable payoff sets when further restricted to uninformed-Seller fully-informed-Buyerand more-informed-Buyer information structures Plainly for any environment Γ

Πlowastus(Γ) cupΠlowast

fb(Γ) sub Πlowastmb(Γ) sub Πlowast(Γ) sub Π(Γ)

3 Main Results

Our goal is to characterize equilibrium payoff pairs across information structures in anarbitrary environment Γ In particular we seek to characterize the five sets Π(Γ) Πlowast(Γ)Πlowast

mb(Γ) Πlowastus(Γ) and Πlowast

fb(Γ) Let

S(Γ) equiv E[v minus c(v)]

be the (expected) surplus from trade in environment Γ This quantity will play an impor-tant role

31 All Information Structures

Define Sellerrsquos payoff guarantee as

πs(Γ) equiv max v minus E [c(v)] 0

11 An example clarifying our terminology may be helpful If both Seller and Buyer are fully informed of vthen the natural equilibriummdashthe unique sequential equilibrium in a finite version of the gamemdashhas Sellerpricing at p = v and Buyerrsquos belief being degenerate on v regardless of Sellerrsquos price This equilibrium hasprice-independent belief even though ex-ante Sellerrsquos price are Buyerrsquos belief are perfectly correlated Thepoint is that Buyerrsquos belief does not depend on price conditional on his signal

10

To interpret this observe that it is optimal for Buyer to accept the price v no matter hisbelief Therefore Seller can guarantee herself the (expected) profit v minus E[c(v)] no matterwhat the information structure is More precisely she can guarantee v minus E[c(v)] minus ε forany ε gt 0 since sequential rationality requires Buyer to accept any price v minus ε SimilarlySeller can also guarantee zero profit offering a price p gt v Hence Sellerrsquos payoff in anyequilibrium of any information structure must be at least the payoff guarantee πs(Γ) above

On the other hand Buyer can guarantee himself the payoff πb = 0 by rejecting all pricesIt follows that the implementable set Π(Γ) must satisfy three simple constraints 1) Sellerrsquosldquoindividual rationalityrdquo constraint πs ge πs(Γ) 2) Buyerrsquos ldquoindividual rationalityrdquo con-straint πb ge 0 and 3) the feasibility constraint πb + πs le S(Γ)

Our first result is that these individual rationality and feasibility constraints are alsosufficient for a payoff pair to be implementable

Theorem 1 Consider all information structures and equilibria

Π(Γ) =

983099983105983103

983105983101

πb ge 0

(πb πs) πs ge πs(Γ)

πb + πs le S(Γ)

983100983105983104

983105983102

Theorem 1 says that the set Π(Γ) corresponds to the triangle AFG in Figure 1 In par-ticular Buyer can receive the entire surplus beyond Sellerrsquos payoff guarantee This is sur-prising as Seller has substantial bargaining power Note that when v le E[c(v)] an entirelyreasonable condition Sellerrsquos payoff guarantee is zero12

The proof of Theorem 1 is in fact straightforward Suppose for expositional simplicityv ge E[c(v)] Fix the trivial information structure in which neither player receives any infor-mation and consider the following family of strategy profiles Seller randomizes betweentwo prices some pl isin [vE[v]] and ph = E[v] Buyer accepts pl with probability one andaccepts ph with probability α(ph) where α(ph) isin [0 1] is specified to make Seller indifferentbetween the two prices That is α(ph)(ph minus E[c(v)]) = pl minus E[c(v)] The expected payoffsfrom this strategy profile are

πb = σ(pl)(E[v]minus pl) and πs = pl minus E[c(v)]

As pl traverses the interval [vE[v]] Sellerrsquos payoff πs traverses [πs(Γ) S(Γ)] Given any

12 In monopoly pricing with c(middot) = v Sellerrsquos payoff guarantee of zero is lower than the revenue guaranteeidentified in Du (2018 Section 5) which is typically positive Dursquos notion is different from ours

11

pl Buyerrsquos payoff πb traverses [0 S(Γ)minusπs] as σ(pl) traverses [0 1] Therefore the proposedstrategy profiles induce all the payoff pairs stated in Theorem 1

We are left to specify beliefs ν for Buyer After prices pl and ph Buyer holds the priorbelief micro After any other (necessarily off-path) price Buyerrsquos belief is that v = v and soBuyer rejects all prices p isin [vinfin)pl ph It is straightforward to confirm that the specified(σα ν) constitute a wPBE

To get more insight into this construction consider its implication for the monopolypricing with c(middot) = v The equilibrium with pl = v (hence α(ph) = 0) and σ(pl) = 1

corresponds to the monopolist deterministically pricing at v and Buyer purchasing Giventhat both sides of the market receive no information why doesnrsquot the monopolist deviate toany price in (vE[v]) The reason is that in this equilibrium the consumer will then not buybecause he updates to v = v Such belief is compatible with wPBE because the equilibriumconcept places no restrictions on off-path beliefs This may seem like a game-theoreticmisdirection Buyerrsquos belief is not consistent with ldquono signaling what you donrsquot knowrdquoPut differently since we have a (weakly) more-informed Buyer information structure weought to impose the price-independent belief condition described in Subsection 23 thatwould imply Buyer must purchase at any price p lt E[v]

But the message of Theorem 1 does not rely on the permissiveness of wPBE To illus-trate continue with the above monopoly-pricing environment and suppose v has positiveprior probability Consider Buyer remaining uninformed but Seller learning whether v = v

or v gt v Now Buyerrsquos off-path belief that v = v is consistent with ldquono signaling whatyou donrsquot knowrdquo More generally using richer information structures we can prove thatany payoff pair identified in Theorem 1 can be approximately implemented as a sequentialequilibrium (Kreps and Wilson 1982) in a suitably discretized game

Theorem 1lowast Fix any ε gt 0 There is a finite information structure and a finite price grid inducinga game with a set of sequential equilibrium payoffs that is an ε-net of Π(Γ)13

In fact the proof of Theorem 1lowast establishes even more the sequential equilibria in thediscretized games satisfy a natural version of the D1 refinement (Cho and Kreps 1987) Werelegate the logic to the Appendix but mention here that we use imperfectly-correlatedsignals for Buyer and Seller14

13 An information structure is finite if the signal spaces Tb and Ts are finite Sequential equilibrium isdefined in the obvious way for the ldquoinducedrdquo finite game where Nature directly draws (tb ts) rather thanfirst drawing v and playersrsquo payoffs are defined directly using (tb ts) For X sub R2 and ε gt 0 the set A sub Xis an ε-net of X if for each x isin X there is a isin A such that 983042xminus a983042 lt ε where 983042middot983042 is the Euclidean distance

14 The idea behind D1 is to ask for any off-path price whether one type of Seller would deviate for any

12

Even when our environment is specialized to monopoly pricing it is worth highlightingtwo contrasts between Theorem 11lowast and results of Bergemann et al (2015) and Roeslerand Szentes (2017) First we find that by not restricting the monopolist to be uninformedthe implementable payoff set typically expands rather dramatically trade can be efficientwith the monopolist securing none of the surplus beyond her payoff guarantee (πs) whichmay be zero (Roesler and Szentes establish implicitly that the implementable set withan uninformed monopolist is a superset of Bergemann et alrsquos which has a fully-informedconsumer) We will see in Subsection 32 that what is crucial to this expansion is price-dependent beliefs In particular the proof of Theorem 1lowast uses an information structure inwhich Buyer is not better informed than Seller mdash if he were then sequential equilibriumwould imply price-independent beliefs Second Theorem 1lowast establishes that for a givenε gt 0 a single information structure (and price grid) can be used to approximate the entirepayoff set Π(Γ) analogously to the construction described after Theorem 1 that used asingle information structure15 Bergemann et al (2015) and Roesler and Szentes (2017) onthe other hand vary information structures to span their payoff sets

32 More-informed Buyer and Price-independent Beliefs

In some economic settings it is plausible that Buyer is more informed than Seller Howdoes a restriction to such information structures ie τ isin Tmb affect the implementablepayoff set It turns out that what is in fact crucial is price-independent beliefs We have ex-plained earlier why it is desirable to impose this condition when Buyer is more informedbut that the condition is well defined even otherwise If Buyer is not more informed thanSeller price-independent beliefs ought to be viewed as an equilibrium restriction in gen-eral To reduce repetition readers should be bear in mind for the rest of this subsectionthe qualifier ldquowith price-independent beliefsrdquo applies unless stated explicitly otherwise

It is useful to define

πuss (Γ) equiv inf πs exist(πb πs) isin Πlowast

us(Γ) (1)

as the lowest payoff that an uninformed Seller can obtain no matter Buyerrsquos information(among equilibria with price-independent beliefs) Plainly πus

s (Γ) ge πs(Γ) In monopoly

Buyer mixed response that another type would Our construction has multiple Buyer types that are imper-fectly correlated with Seller types So different types of Seller have different beliefs about Buyer types Thisblunts the power of dominance considerations to the point where D1 does not exclude any Seller type fromthe support of Buyerrsquos off-path belief

15 In fact if one lets the price grid vary with ε then a single information structure implements exactlyrather than approximately in sequential equilibrium all payoffs ε-away from the boundary of Π(Γ) See in the Appendix for a formal statement

13

pricing with V = [v v] and c(middot) = v Roesler and Szentesrsquos (2017) characterization of theconsumer-optimal information structure identifies πus

s establishing that πuss gt πs If there

is no trade due to adverse selection when Seller is uniformed and Buyer has some informa-tion then πus

s = πs = 0 We do not have a general explicit formula for πuss Subsection 43

provides it for linear c(middot) Nonetheless we establish next that (i) the only additional restric-tion on equilibrium payoffs imposed by price-independent beliefs is a lower bound of πus

s

for Seller and (ii) uninformed-Seller information structures implement all such payoffs

Theorem 2 Consider equilibria with price-independent beliefs

1 Πlowast(Γ) = Πlowastmb(Γ) = Πlowast

us(Γ)

2 Πlowastus(Γ) = (πb πs) isin Π(Γ) πs ge πus

s (Γ)

3 For any (πb πs) isin Πlowastus(Γ) with πs gt πus

s (Γ) there is τ isin Tus with Π(Γ τ) = (πb πs)

Remark 1 We believe the substance of Theorem 2 would hold using discretizations and se-quential equilibria analogous to Theorem 1lowast As previously noted sequential equilibriumimplies price-independent beliefs when Buyer is more informed than Seller

To digest Theorem 2 note that Πlowast(Γ) sup Πlowastmb(Γ) sup Πlowast

us(Γ) is trivial So part 1 of thetheorem amounts to establishing the reverse inclusions The intuition for thismdashgiven part2rsquos characterization of Πlowast

usmdashis fairly straightforward with price-independent beliefs ad-ditional information cannot harm Seller even though it could alter the set of equilibria SoSellerrsquos lowest payoff obtains when she is uninformed

The characterization in part 2 of payoffs with an uninformed Seller corresponds to thetriangle ADE in Figure 1 Part 3 of the theorem assures ldquounique implementationrdquo of allimplementable payoffs satisfying πs gt πus

s (Γ) That is for any such payoff pair there is anuninformed-Seller information structure such that all equilibria (with price-independentbeliefs) induce exactly that payoff pair Unique implementation is appealing for multiplereasons one of which is that it obviates concerns about which among multiple payoff-distinct equilibria is more reasonable For example Ravid Roesler and Szentes (2019)bring to the fore the issue of equilibrium multiplicity in Roesler and Szentesrsquos (2017) infor-mation structure

Let us describe how we obtain the characterization of Πlowastus(Γ) and unique implementa-

tion There are two steps The first ensures that there is some information structure callit τ lowast isin Tus that implements Sellerrsquos payoff πus

s (Γ) That is we ensure that the infimum

14

in (1) is in fact a minimum16 While this argument is technical knowing τ lowast exists is usefulin what follows The second and economically insightful step is to construct informa-tion structures that implement every point in the triangle Πlowast

us(Γ) by suitably garbling theinformation structure τ lowast The construction is illustrated in Figure 2 Consider the distri-bution of Buyerrsquos posterior mean of his valuation v in information structure τ lowast (Givenprice-independent beliefs Buyerrsquos posterior mean is sufficient for his decision) For sim-plicity suppose this posterior-mean distribution has a density as depicted by the red curvein Figure 2 Fix any (πb πs) isin Πlowast

us(Γ)

First there is some number zlowast such that πb + πs is the total surplus from trading onlywhen Buyerrsquos posterior mean is greater than zlowast Next there is some price plowast ge zlowast suchthat Sellerrsquos payoff is πs if all these trades were to occur at price plowast17 Note that plowast must beno larger than the expected Buyer posterior mean conditional on that being above zlowast forotherwise πb lt 0 We claim that the information structure τ lowast can be garbled so that plowast isan equilibrium price and trade occurs only when Buyerrsquos posterior mean is greater thanzlowast The garbling is illustrated in Figure 2 by the blue distribution There is one signal that

pzv

ν(v)

Figure 2 Construction of information structure in Theorem 2

Buyer receives when the original posterior mean is between zlowast and plowast and also receiveswith some probability when the original posterior mean is above plowast The probability ischosen to make the posterior mean from this signal exactly plowast Apart from this one newsignal Buyer receives the original signal in τ lowast Plainly this is a garbling of τ lowast and hence isfeasible

16 The difficulty is in establishing suitable continuity Uninformed-Seller information structures can beviewed as probability measures over Buyerrsquos beliefs with convergence in the sense of the weak topologyThis topology ensures continuity with respect to probability measures of expectations of continuous orat least Lipschitz (and bounded) functions However Sellerrsquos expected payoff is not the expectation of aLipschitz function as Sellerrsquos profit is truncated at the price she charges

17 That plowast ge zlowast follows from πs ge πuss (Γ) as πs(Γ) itself is weakly larger than Sellerrsquos payoff from posting

price zlowast (and thus trading with the same set of Buyer posterior means) under information structure τlowast

15

Figure 2 makes clear why the new information structure has an equilibrium with priceplowast and Buyer breaking indifference in favor of trading (i) Sellerrsquos profit from posting anyprice below zlowast is the same as under τ lowast and hence no larger than πus

s (Γ) (ii) similarly Sellerrsquosprofit from posting any price above plowast is no higher than some fraction of πus

s (Γ) and (iii)any price between zlowast and plowast is worse that price plowast Moreover since Sellerrsquos profit fromoffering any price other than plowast is no more than πus

s (Γ) it follows that when πs gt πuss (Γ)

Buyer must break indifference as specified for Seller to have an optimal price and theequilibrium payoffs are unique

Remark 2 The above logic establishes that given any τ isin Tus that implements some (πb πs)τ can be garbled to uniquely implement any (πprime

b πprimes) isin Π(Γ) such that πprime

s gt πs That isan uninformed Sellerrsquos payoff can always be strictly raised and Buyerrsquos payoff reduced(strictly so long as it was not already zero) by garbling Buyerrsquos information

Remark 3 According to Theorem 11lowast and Theorem 2 uninformed-Seller information struc-tures cannot implement all implementable payoff pairs in an environment Γ if and only ifπuss (Γ) gt πs(Γ) This inequality fails in environment Γ if πus

s (Γ) = 0 since that impliesπuss (Γ) = πs(Γ) = 0 An example is when there is no trade due to adverse selection when

Seller is uniformed and Buyer has some information On the other hand it is sufficient forthe inequality that

v le E[c(v)] and forallv isin V c(v) lt v (2)

To see why (2) implies πuss (Γ) gt πs(Γ) notice that in any uninformed-Seller information

structure Seller can price at slightly less than Buyerrsquos highest posterior mean valuationand guarantee trade with only (a neighborhood of) that Buyer type If c(v) lt v for all v thisgives Seller a positive expected payoff and hence πus

s (Γ) gt 018 But the first inequality in (2)is equivalent to πs(Γ) = 0 We observe that Condition (2) is compatible with severe adverseselection resulting in very little trade when Seller is uninformed and Buyer (partially orfully) informed

33 Fully-Informed Buyer

We now turn to the third canonical class of information structures Buyer is fully in-formed of his value v As this is a special case of more-informed Buyer we maintain price-independent beliefs throughout this subsection

18 More precisely as V is compact c(v) lt v for all v implies there exists ε gt 0 such that v minus c(v) gt εGiven any uninformed-Seller information structure let mv be the highest posterior mean valuation in thesupport of the posterior means induced by Buyerrsquos signals So there is positive probability of Buyer signalswith posterior mean valuations at least mv minus ε2 By pricing at mv minus ε2 Sellerrsquos expected cost conditionalon trade is bounded above by mv minus ε and hence Sellerrsquos profit conditional on trade is at least ε2 gt 0 Itfollows that πus

s gt 0

16

Faced with a fully informed Buyer and any sequentially rational Buyer strategy an un-informed Seller can guarantee a profit level

πfbs (Γ) equiv sup

p

983133 v

p

(pminus c(v))micro(dv)

regardless of her information Plainly πfbs (Γ) ge πus

s (Γ) In monopoly pricing with c(middot) = vRoesler and Szentes (2017) have shown that πfb

s gt πuss if there is no trade due to adverse

selection when Buyer is fully informed and Seller is uniformed then πfbs = πus

s = 0 Belowwe establish that when Buyer is fully informed πfb

s is the only additional constraint onequilibrium payoffs Let Bε(πb πs) denote an ε-ball around (πb πs)

Theorem 3 Consider a fully informed Buyer and price-independent beliefs

1 Πlowastfb(Γ) = (πb πs) isin Π(Γ) πs ge πfb

s (Γ)

2 For any (πb πs) isin Πlowastfb(Γ) and any ε gt 0 there is τ isin Tfb with Π(Γ τ) sub Bε(πb πs)

The payoff set characterized in Theorem 3 corresponds to the triangle ABE in Figure 1Part 2 of the theorem establishes ldquoapproximately unique implementationrdquo for any point inthe triangle there is an information structure under which all equilibria yield payoff pairsin a neighborhood of that point

Here is the idea behind part 1 of Theorem 3 When Buyer is fully informed an informa-tion structure can be viewed as dividing vrsquos prior distribution micro into a set of microi that aver-age to micro with Seller informed of which microi she faces Theorem 3 is proven by establishingthat we can divide micro suitably so that against each microi Seller is indifferent between pricingat all prices in the support of microi including the price corresponding to πfb

s in that environ-ment We call such a microi a (Seller-price) incentive compatible distribution or ICD (Bergemannet al (2015) call their analogous construct an extreme market) The Appendix provides aldquogreedyrdquo algorithm to compute ICDs19

We can sketch how the algorithm works and construct a set of ICDs that average to theprior Suppose V = v1 v2 vK with vi lt vi+1 for i isin 1 K minus 1 and c(v) lt v forall v Given any small-enough mass of vK there is a unique mass of type vKminus1 that makesSeller indifferent between charging price vK and vKminus1 (If the mass is too low Seller prefersvK if it is too high she prefers vKminus1) Iterating down to keep Seller indifferent betweenall prices pins down an ICD Choose the maximum mass of type vK for which this worksRemove that ICD and repeat

19 The algorithm is defined for finite V and we take limits to handle the infinite case

17

Crucially whenever an ICD is removed the price corresponding to πfbs (Γ) remains op-

timal in the remaining ldquomarketrdquo this follows from the ICDrsquos defining property of Sellerindifference and an accounting identity Therefore Sellerrsquos profit in this segmentation ofICDs remains πfb

s (Γ) Moreover it is also optimal for Seller to always (ie for each microi) priceso that there is full trade or no trade Hence Buyerrsquos expected payoff can be either 0 or theentire surplus less πfb

s (Γ) It follows that the fully-informed Buyer information structuredefined by this set of ICDs implements point B and C in Figure 1 The entire triangle ABC

can then be implemented by convexification randomizing over this information structure(and two equilibria) and full information (where Seller obtains all the surplus)

The above logic of incentive compatible distributions reduces to that of Bergemann et al(2015) in the context of monopoly pricing Even in the monopoly pricing case Theorem 3part 2 strengthens the conclusions of Bergemann et al (2015) by ensuring approximatelyunique implementation

Remark 4 Theorems 1ndash3 imply that fully-informed-Buyer information structures imple-ment all implementable payoff pairs if and only if πfb

s (Γ) = πs(Γ) In that case trianglesAFG and ABC coincide in Figure 1 It follows that πfb

s (Γ) = πs(Γ) only when a fully-informed Buyer and uninformed Seller can result in full trade (v ge E[c(v)] and Sellerprices at v) or no trade (v le E[c(v)] and Seller prices at some p ge v) Interestinglywhen πfb

s (Γ) gt πs(Γ) fully-informed-Buyer information structures cannot even implementall payoff pairs implementable by uninformed-Seller information structures ie trianglesABC and ADE in Figure 1 are distinct That is when πfb

s (Γ) gt πs(Γ) there is an uninformed-Seller information structure that implements some πs lt πfb

s (Γ)20 Theorems 2ndash3 further im-ply in this case that Buyer can obtain a strictly higher payoff when he is not fully informed(and Seller is uninformed)

4 Discussion

41 Multidimensionality

Suppose Buyer and Sellerrsquos cost and valuation pair (c v) is a two-dimensional randomvariable distributed according to joint distribution micro with a compact support The ex-tension of our maintained assumption of commonly known gains from trade is for all

20 Pick any pprime gt v such that pprimeminusE[c(v)] lt πfbs (Γ) Following the construction described after Theorem 2 we

can mix all valuations v le pprime with a fraction λ gt 0 of valuations v gt pprime so that the mixture has posterior meanexactly pprime The remaining fraction 1 minus λ of valuations above pprime are revealed to Buyer With this uninformed-Seller information structure consider any equilibrium in which Buyer purchases when indifferent (Suchan equilibrium with price-independent belief exists) Sellerrsquos profit is at most (1 minus λ)πfb

s (Γ) from any pricep gt pprime and pprime minus E[c(v)] from price p = pprime Hence Sellerrsquos profit is strictly less than πfb

s (Γ)

18

(c v) isin Supp(micro) v ge c and E[v minus c] gt 0 An information structure is now a joint distribu-tion P (tb ts v c) whose marginal distribution on (v c) is micro

We claim that the content of our key results Theorem 1Theorem 1lowast Theorem 2 andTheorem 3 still hold To see why let v be the lowest valuation in the support of micro Sellerrsquosindividual rationality constraint is now maxv minus E[c] 0 as she can guarantee this by set-ting either a sufficiently high price or a price (arbitrarily close to) v regardless of her signalWe can define a cost function c(vprime) equiv Emicro[c|v = vprime] ge vprime This results in an environment sat-isfying all the maintained assumptions of our baseline model except that c(middot) may not becontinuous Recall that such continuity is not needed for proving Theorem 1 Theorem 1lowastBoth Theorem 2 and Theorem 3 use continuity of c(middot) to guarantee that Eν [c(v)] is a contin-uous function of ν isin ∆(V ) for a key convergence result However in the two-dimensionaltype environment Eν [c] is still a continuous function of ν isin ∆(V times C) Note that Theo-rem 3 uses upper semi-continuity of Sellerrsquos profit in price boundedness of c and c le v issufficient for such upper semi-continuity

42 Negative Trading Surplus

Returning to our baseline model we next discuss cases in which trade sometimes gen-erates negative surplus That is we drop the assumption that c(v) le v we do not requireE[v minus c(v)] gt 0 either Define Sλ(Γ) for λ isin [1infin) as

Sλ(Γ) equiv983133 v

v

[v minus c(v) + λ(v minus v)]+ micro(dv)

where [middot]+ equiv max middot 0 It is readily verified that S1(Γ) = E [[v minus c(v)]+] and limλrarr0

Sλ(Γ)λ =

E[v] minus v Allowing negative trading surplus does not affect our definition of wPBE Sothe notation Π(Γ) and πs(Γ) still have the same meanings as before The next propositionshows that Π(Γ) is now characterized by three constraints as before the two individualrationality constraints πs ge πs(Γ) and πb ge 0 and different now a Pareto frontier definedby all Sλ(Γ)

Proposition 1 Consider all information structures and equilibria when trade can generate negativesurplus

Π(Γ) =

983099983105983103

983105983101

πb ge 0

(πb πs) πs ge πs(Γ)

λπb + πs le Sλ(Γ) forallλ ge 1

983100983105983104

983105983102

Proposition 1 shows that when the trade can be inefficient it may be impossible to lower

19

Sellerrsquos profit without losing total surplus (ie πb + πs) This result is intuitive at the ex-tremes Suppose for simplicity v gt E[c(v)] If πs = πs(Γ) then since πs(Γ) = v minus E[c(v)]Seller must find it optimal to price at v and trade with probability one including whentrade generates negative surplus So Buyerrsquos payoff is E[v]minus v = limλrarrinfin Sλ(Γ)λ and totalsurplus is E[v minus c(v)] which includes the maximum amount of inefficient trade On theother hand an information structure that publicly reveals (only) whether trade is efficient(ie whether v ge c(v)) can implement efficient trade with all the surplus accruing to SellerIn this case Seller obtains S1(Γ) while Buyer obtains zero The proposition establishes thatin between S1(Γ) and πs(Γ) information structures and equilibria that lower Sellerrsquos profitmust reduce total surplus as the Pareto frontier Sλ(Γ) is a concave function with maximalslope minus1

The function capturing the Pareto frontier Sλ(Γ) is the maximal weighted sum of Buyerand Seller payoff assuming that Seller will trade at the low price p = v whenever it createspositive weighted total payoff Since we place higher weight on buyer (λ ge 1) S1(Γ) isthe upper bound for the weighted total payoff In the proof of Proposition 1 we showthat these bounds are tight mdash any payoff pair on the frontier can be implemented by someinformation structure The key observation is that whenever v minus c(v) + λ(v minus v) lt 0 itfollows that vminus c(v) lt (1minusλ)(vminus v) le 0 That is it is incentive compatible for Seller to notsell when it is common knowledge that v creates negative weighted total payoff Thereforewe can simply release a public signal indicating the sign of weighted total payoff and ifpositive induce Seller to sell at v which implies all bounds Sλ(Γ)

43 Linear Cost

When Sellerrsquos cost function c(v) is linear we can explicitly characterize an informationstructure that implements Sellerrsquos minimum implementable payoff πus

s (Γ) when Seller isuninformed (or per Theorem 2 when Buyer is better informed) subject to price-independentbeliefs We note that given the discussion in Subsection 41 a linear c(v) subsumes richerenvironments in which conditional expectations are linear including Gaussian environ-ments (cf Bar-Isaac et al 2018)

Condition 1 c(v) = λv + γ for some λ isin R

Let F (v) be the cumulative distribution function (CDF) corresponding to the prior mea-sure micro Let D(micro) be the set of all distributions whose CDF G satisfies

983133

V

vdG(v) =

983133

V

vdF (v) and983133 v

v

G(s)ds le983133 v

v

F (s)ds forallv isin V

20

That is D(micro) contains all distributions that are mean-preserving-contractions (MPC) of microIt is well known that D(micro) characterizes the set of distributions of Buyer posterior meansthat can be generated by any (uninformed-Seller) information structure

Definition 2 G is an incentive compatible distribution (ICD) if

983133

sgep

(pminus c(s))G(ds) = constant ge 0 forallp isin Supp(G)

That is when Buyerrsquos (mean) valuation is distributed according to an ICD G Seller isindifferent among all prices in the support of G assuming Buyer buys when indifferent Wefocus on a special family of ICDs whose supports are intervals [vlowast v

lowast] Given Condition 1such ICDs have analytical an expression

G(v) =

983099983105983105983103

983105983105983101

1minus983061

(1minus λ)v minus γ

(1minus λ)vlowast minus γ

983062 1λminus1

if λ ∕= 1

1minus evminusvlowast

γ if λ = 1

(3)

Given any vlowast and the above G(v) an upper bound vlowast is pinned down by the conditionEG[v] = EF [v] So a family of ICDs is parametrized by a single parameter vlowast and hence anICD is denoted by CDF 983144Gvlowast(v) with density 983144gvlowast(v) It is not difficult to verify that 983144Gvlowast(v) ismonotonic in vlowast for v in the support Consequently all ICDs are ordered according to theMPC order For vlowast isin [vE[v]] when vlowast increases the ICD increases in the MPC order

Proposition 2 Suppose Condition 1 is satisfied and let plowast equiv min983153vlowast983055983055 983144Gvlowast isin D(micro)

983154 It holds that

πuss (Γ) = plowast minus Emicro[c(v)]

Proposition 2 states that an uninformed Sellerrsquos minimum payoff (with price-independentbeliefs) is characterized by the ICDs that are MPCs of the prior distribution micro By defini-tion of ICDs Sellerrsquos profit when facing ICD 983144Gvlowast is vlowast minus E[v] Sellerrsquos minimum payoff isimplemented by the most dispersed ICD that is a MPC of the prior distribution

In proving Proposition 2 the key step is to show that given the prior micro garbling Buyerrsquosinformation such that the posterior mean distribution is an ICD makes Seller weakly worseoff It is then without loss to only consider ICDs to implement πus

s (Γ) We can thus workwith a tractable one-dimensional problem Consider the set D(micro) Suppose we find G(v)

such that (i) G isin D(micro) (ii) G is an ICD and (iii) existp such that983125 p

vG(s)ds =

983125 p

vF (s)ds Such

a G is the most-dispersed ICD that is a mean preserving contraction of F Consider the

21

following two manipulations

λ

983133 v

p

F (s)ds =

983133 v

p

(pminus c(s))dF (s)minus (1minus F (v)) (pminus c(v)) + λ(v minus p) (4)

λ

983133 v

p

G(s)ds =

983133 v

p

(pminus c(s))dG(s)minus (1minusG(v)) (pminus c(v)) + λ(v minus p) (5)

First by property (iii) above the LHS of Equation 4 equals the LHS of Equation 5 Secondsince

983125 v

p(F (s) minus G(s))ds is minimized at v this implies F (p) le G(p) Therefore the first

term on RHS of Equation 4 must be larger than that of Equation 5 Notice that the firstterm on RHS is Sellerrsquos profit when offering price p This implies that the profit fromoffering p given Buyer mean-valuation distribution G(v) is lower than that given F (v)which is lower than Sellerrsquos maximum profit given F (v) On the other hand by the ICDproperty p is already Sellerrsquos optimal price given G(v) The optimal profit given valuationdistribution F must be no lower than that of G Therefore we have proved that to minimizeSellerrsquos profit it is without loss to consider only ICDs within the set D(micro) which is a onedimensional subspace The detailed proof that such ICDs exist is relegated to the appendix

The monopoly-pricing environment with c(middot) = v is covered by Condition 1 with λ = 0

and γ = v The distribution G in (3) then reduces to that identified by Roesler and Szentes(2017)

If the prior micro has binary support then the cost function c(v) is linear In this special casewe can solve for πus

s (Γ) explicitly

Corollary 1 Suppose micro has binary support V = v1 v2 Let λ = c(v2)minusc(v1)v2minusv1

and let p be theunique solution to

(pminus c(p))1

λminus1 (pminus E[c(v)]) = (v2 minus c(v2))λ

λminus1 (6)

Then πuss (Γ) = max p v2minus E[c(v)]

References

AKERLOF G A (1970) ldquoThe Market for rdquoLemonsrdquo Quality Uncertainty and the MarketMechanismrdquo Quarterly Journal of Economics 84 488ndash500

BAGWELL K AND M H RIORDAN (1991) ldquoHigh and Declining Prices Signal ProductQualityrdquo American Economic Review 81 224ndash239

BAR-ISAAC H I JEWITT AND C LEAVER (2018) ldquoAdverse Selection Efficiency and theStructure of Informationrdquo Unpublished

22

BERGEMANN D B BROOKS AND S MORRIS (2015) ldquoThe Limits of Price Discrimina-tionrdquo American Economic Review 105 921ndash57

BERGEMANN D AND S MORRIS (2016) ldquoBayes Correlated Equilibrium and the Compar-ison of Information Structures in Gamesrdquo Theoretical Economics 11 487ndash522

BLACKWELL D (1953) ldquoEquivalent Comparisons of Experimentsrdquo Annals of MathematicalStatistics 24 265ndash272

CHO I-K AND D KREPS (1987) ldquoSignaling Games and Stable Equilibriardquo Quarterly Jour-nal of Economics 102 179ndash221

DU S (2018) ldquoRobust Mechanisms Under Common Valuationrdquo Econometrica 86 1569ndash1588

DURRETT R (1995) Probability Theory and Examples Duxbury Press 2nd ed

EINAV L AND A FINKELSTEIN (2011) ldquoSelection in Insurance Markets Theory and Em-pirics in Picturesrdquo Journal of Economic Perspectives 25 115ndash38

FUDENBERG D AND J TIROLE (1991) ldquoPerfect Bayesian Equilibrium and Sequential Equi-libriumrdquo Journal of Economic Theory 53 236ndash260

GARCIA D R TEPER AND M TSUR (2018) ldquoInformation Design in Insurance MarketsSelling Peaches in a Market for Lemonsrdquo Unpublished

GREENWALD B C (1986) ldquoAdverse Selection in the Labour Marketrdquo Review of EconomicStudies 53 325ndash347

JOVANOVIC B (1982) ldquoFavorable Selection with Asymmetric Informationrdquo QuarterlyJournal of Economics 97 535ndash539

KESSLER A S (2001) ldquoRevisiting the Lemons Marketrdquo International Economic Review 4225ndash41

KREPS D AND R WILSON (1982) ldquoSequential Equilibriardquo Econometrica 50 863ndash894

LEHMANN E L (1988) ldquoComparing Location Experimentsrdquo The Annals of Statistics 16521ndash533

LEVIN J (2001) ldquoInformation and the Market for Lemonsrdquo RAND Journal of Economics32 657ndash666

23

LIBGOBER J AND X MU (2019) ldquoInformational Robustness in Intertemporal PricingrdquoWorking paper

MAKRIS M AND L RENOU (2018) ldquoInformation Design in Multi-Stage Gamesrdquo Workingpaper

MILGROM P R AND R J WEBER (1985) ldquoDistributional Strategies for Games with In-complete Informationrdquo Mathematics of Operations Research 10 619ndash632

MUSSA M AND S ROSEN (1978) ldquoMonopoly and Product Qualityrdquo Journal of EconomicTheory 18 301ndash317

RAVID D A-K ROESLER AND B SZENTES (2019) ldquoLearning Before Trading On theInefficiency of Ignoring Free Informationrdquo Working paper

ROESLER A-K AND B SZENTES (2017) ldquoBuyer-Optimal Learning and Monopoly Pric-ingrdquo American Economic Review 107 2072ndash80

TERSTIEGE S AND C WASSER (2019) ldquoBuyer-optimal Extensionproof InformationrdquoWorking paper

Appendices omitted from this version but are available on request

24

is Buyerrsquos signal and ts isin Ts is Sellerrsquos signal For i isin s b v let Pi denote the corre-sponding marginal distribution of P on dimension of Ti with the convention Tv equiv V Werequire Pv = micro this is the iterated expectation or Bayes plausibility requirement Denotean information structure by τ

The environment Γ and information structure τ define the following game

1 The random variables (tb ts v) are realized Signal tb is privately observed by Buyerand signal ts privately observed by Seller Neither party observes v

2 Seller posts a price p isin R

3 Buyer accepts or rejects the price If Buyer accepts his von-Neumann Morgensternpayoff is v minus p and Sellerrsquos is p minus c(v) If Buyer rejects both partiesrsquo payoffs arenormalized to 0

Note that because the signal spaces Tb and Ts are abstract and the two partiesrsquo signalscan be arbitrarily correlated conditional on v there is no loss of generality in assuming thateach party privately observes their own signal For example public information can becaptured by perfectly correlating (components of) tb and ts

22 Strategies and Equilibria

In the game defined by (Γ τ) denote Sellerrsquos strategy by σ and Buyerrsquos by α FollowingMilgrom and Weber (1985) we define σ as a distributional strategy σ is a joint distributionon R times Ts whose marginal distribution on Ts must be the Sellerrsquos signal distribution Soσ(middot|ts) is Sellerrsquos price distribution given her signal ts9 Buyerrsquos strategy α R times Tb rarr[0 1] maps each price-signal pair (p tb) into a trading probability A strategy profile (σα)

induces expected utilities for Buyer and Seller (πb πs) in the natural way

πb =

983133(v minus p)α(tb p)σ(dp|ts)P (dts dtb dv)

πs =

983133(pminus c(v))α(tb p)σ(dp|ts)P (dts dtb dv)

Our baseline equilibrium concept is weak Perfect Bayesian equilibrium Since Sellerrsquosaction is not preceded by Buyerrsquos we can dispense with specifying beliefs for Seller ForBuyer it suffices to focus on his belief about the value v given his signal and the price wedenote this distribution by ν(v|p tb)

9 Here σ(middot|ts) is the regular conditional distribution which exists and is unique almost everywhere becauseTs is a standard Borel space (Durrett 1995 pp 229ndash230) Similarly for subsequent such notation we dropldquoalmost everywhererdquo qualifiers unless essential

7

Definition 1 A strategy profile (σα) and a belief ν(v|p tb) is a weak perfect Bayesian equilib-rium (wPBE) of game (Γ τ) if

1 Buyer plays optimally at every information set given his belief

α(p tb) =

983099983103

9831011 if Eν(v|ptb)[v] gt p

0 if Eν(v|ptb)[v] lt p

2 Seller plays optimally

σ isin argmaxσ

983133(pminus c(v))α(p tb)σ(dp|ts)P (dts dtb dv)

3 Beliefs satisfy Bayes rule on path for every measurable D sube Rtimes Ts times Tb times V

983133

D

ν(dv|p tb)σ(dp|ts)P (dts dtb V ) =

983133

D

σ(dp|ts)P (dts dtb dv)

Notice that we have formulated Sellerrsquos optimality requirement ex ante but Buyerrsquos ateach information set The latter is needed to capture sequential rationality The formeris for (notational) convenience this choice is inconsequential because Seller moves beforeBuyer

Hereafter ldquoequilibriumrdquo without qualification refers to a wPBE As is well understoodwPBE permits significant latitude in beliefs off the equilibrium path We will discuss re-finements when appropriate

23 Implementable Payoffs and Canonical Information Structures

We now define the set of implementable equilibrium outcomesmdashthat is the payoffs thatobtain in some equilibrium under some information structuremdashand some canonical classesof information structures

For a game (Γ τ) let the equilibrium payoff set be

Π(Γ τ) equiv (πb πs) exist wPBE of (Γ τ) with payoffs (πb πs)

Denote the class of all information structures by T and define

Π(Γ) equiv983134

τisinT

Π(Γ τ)

8

That is for environment Γ Π(Γ) is the set of all equilibrium payoff pairs that obtain undersome information structure

Uninformed Seller An information structure has uninformed Seller if Ts is a singletonSellerrsquos own signal contains no information about Buyerrsquos value v and hence neither aboutSellerrsquos cost c(v) When discussing such information structures we write the associateddistribution as just P (tb v) and Sellerrsquos strategy as just σ(p) omitting the argument ts inboth cases The class of all uninformed-Seller information structure is denoted Tus

Fully-informed Buyer An information structure has fully-informed Buyer if Buyerrsquos signalfully reveals his value v Formally this holds if Tb = V and the conditional distribution onV P (middot|tb) satisfies P (tb|tb) = 1 We denote the class of fully-informed-Buyer informationstructures Tfb

More-informed Buyer An information structure has more-informed Buyer if Buyer hasmore information than Seller Formally this holds when v and ts are independent condi-tional on tb ie for any measurable Ds sub Ts and Dv sub V P (DscapDv|tb) = P (Ds|tb)P (Dv|tb)10

Another way to interpret this requirement is that random variable tb must be statisticallysufficient for ts with respect to v ie tb is more informative than ts about v in the sense ofBlackwell (1953) We denote the class of more-informed-Buyer information structures byTmb Naturally information structures with uninformed Seller or fully-informed Buyer arecases of more-informed Buyer both Tus and Tfb are subclasses of Tmb

No updating from price For more-informed-Buyer information structures it is desirableto impose further requirements on Buyerrsquos equilibrium belief Since Sellerrsquos price can onlydepend on her own signal and this signal contains no additional information about v givenBuyerrsquos signal the price is statistically uninformative about v given Buyerrsquos signal Con-sequently Buyerrsquos posterior belief should be price independent once his signal has beenconditioned upon Formally regardless of the price p the equilibrium belief ν(middot|p tb) mustsatisfy 983133

ν(dv|p tb)P (dtb Ts V ) =

983133P (dtb Ts dv)

We refer to this condition as price-independent belief Note that the condition is meaningfulregardless of whether Buyer is more informed than Seller In a more-informed-Buyer in-formation structure price-independent belief would be implied by the ldquono signaling what

10 We write sub to mean ldquoweak subsetrdquo

9

you donrsquot knowrdquo requirement (Fudenberg and Tirole 1991) frequently imposed in versionsof perfect Bayesian equilibrium and the concept of sequential equilibrium (Kreps and Wil-son 1982) in finite versions of our setting11

Some notation will be helpful Define

Πlowast(Γ τ) equiv (πb πs) exist wPBE of (Γ τ) with price-independent belief and payoffs(πb πs)

Πlowast(Γ) equiv983134

τisinT

Πlowast(Γ τ)

Πlowasti (Γ) equiv

983134

τisinTi

Πlowast(Γ τ) for i = us fbmb

So Πlowast and Πlowast are analogous to the implementable payoff sets Π and Π defined earlier butrestricted to equilibria with price-independent beliefs Πlowast

us Πlowastfb and Πlowast

mb are the imple-mentable payoff sets when further restricted to uninformed-Seller fully-informed-Buyerand more-informed-Buyer information structures Plainly for any environment Γ

Πlowastus(Γ) cupΠlowast

fb(Γ) sub Πlowastmb(Γ) sub Πlowast(Γ) sub Π(Γ)

3 Main Results

Our goal is to characterize equilibrium payoff pairs across information structures in anarbitrary environment Γ In particular we seek to characterize the five sets Π(Γ) Πlowast(Γ)Πlowast

mb(Γ) Πlowastus(Γ) and Πlowast

fb(Γ) Let

S(Γ) equiv E[v minus c(v)]

be the (expected) surplus from trade in environment Γ This quantity will play an impor-tant role

31 All Information Structures

Define Sellerrsquos payoff guarantee as

πs(Γ) equiv max v minus E [c(v)] 0

11 An example clarifying our terminology may be helpful If both Seller and Buyer are fully informed of vthen the natural equilibriummdashthe unique sequential equilibrium in a finite version of the gamemdashhas Sellerpricing at p = v and Buyerrsquos belief being degenerate on v regardless of Sellerrsquos price This equilibrium hasprice-independent belief even though ex-ante Sellerrsquos price are Buyerrsquos belief are perfectly correlated Thepoint is that Buyerrsquos belief does not depend on price conditional on his signal

10

To interpret this observe that it is optimal for Buyer to accept the price v no matter hisbelief Therefore Seller can guarantee herself the (expected) profit v minus E[c(v)] no matterwhat the information structure is More precisely she can guarantee v minus E[c(v)] minus ε forany ε gt 0 since sequential rationality requires Buyer to accept any price v minus ε SimilarlySeller can also guarantee zero profit offering a price p gt v Hence Sellerrsquos payoff in anyequilibrium of any information structure must be at least the payoff guarantee πs(Γ) above

On the other hand Buyer can guarantee himself the payoff πb = 0 by rejecting all pricesIt follows that the implementable set Π(Γ) must satisfy three simple constraints 1) Sellerrsquosldquoindividual rationalityrdquo constraint πs ge πs(Γ) 2) Buyerrsquos ldquoindividual rationalityrdquo con-straint πb ge 0 and 3) the feasibility constraint πb + πs le S(Γ)

Our first result is that these individual rationality and feasibility constraints are alsosufficient for a payoff pair to be implementable

Theorem 1 Consider all information structures and equilibria

Π(Γ) =

983099983105983103

983105983101

πb ge 0

(πb πs) πs ge πs(Γ)

πb + πs le S(Γ)

983100983105983104

983105983102

Theorem 1 says that the set Π(Γ) corresponds to the triangle AFG in Figure 1 In par-ticular Buyer can receive the entire surplus beyond Sellerrsquos payoff guarantee This is sur-prising as Seller has substantial bargaining power Note that when v le E[c(v)] an entirelyreasonable condition Sellerrsquos payoff guarantee is zero12

The proof of Theorem 1 is in fact straightforward Suppose for expositional simplicityv ge E[c(v)] Fix the trivial information structure in which neither player receives any infor-mation and consider the following family of strategy profiles Seller randomizes betweentwo prices some pl isin [vE[v]] and ph = E[v] Buyer accepts pl with probability one andaccepts ph with probability α(ph) where α(ph) isin [0 1] is specified to make Seller indifferentbetween the two prices That is α(ph)(ph minus E[c(v)]) = pl minus E[c(v)] The expected payoffsfrom this strategy profile are

πb = σ(pl)(E[v]minus pl) and πs = pl minus E[c(v)]

As pl traverses the interval [vE[v]] Sellerrsquos payoff πs traverses [πs(Γ) S(Γ)] Given any

12 In monopoly pricing with c(middot) = v Sellerrsquos payoff guarantee of zero is lower than the revenue guaranteeidentified in Du (2018 Section 5) which is typically positive Dursquos notion is different from ours

11

pl Buyerrsquos payoff πb traverses [0 S(Γ)minusπs] as σ(pl) traverses [0 1] Therefore the proposedstrategy profiles induce all the payoff pairs stated in Theorem 1

We are left to specify beliefs ν for Buyer After prices pl and ph Buyer holds the priorbelief micro After any other (necessarily off-path) price Buyerrsquos belief is that v = v and soBuyer rejects all prices p isin [vinfin)pl ph It is straightforward to confirm that the specified(σα ν) constitute a wPBE

To get more insight into this construction consider its implication for the monopolypricing with c(middot) = v The equilibrium with pl = v (hence α(ph) = 0) and σ(pl) = 1

corresponds to the monopolist deterministically pricing at v and Buyer purchasing Giventhat both sides of the market receive no information why doesnrsquot the monopolist deviate toany price in (vE[v]) The reason is that in this equilibrium the consumer will then not buybecause he updates to v = v Such belief is compatible with wPBE because the equilibriumconcept places no restrictions on off-path beliefs This may seem like a game-theoreticmisdirection Buyerrsquos belief is not consistent with ldquono signaling what you donrsquot knowrdquoPut differently since we have a (weakly) more-informed Buyer information structure weought to impose the price-independent belief condition described in Subsection 23 thatwould imply Buyer must purchase at any price p lt E[v]

But the message of Theorem 1 does not rely on the permissiveness of wPBE To illus-trate continue with the above monopoly-pricing environment and suppose v has positiveprior probability Consider Buyer remaining uninformed but Seller learning whether v = v

or v gt v Now Buyerrsquos off-path belief that v = v is consistent with ldquono signaling whatyou donrsquot knowrdquo More generally using richer information structures we can prove thatany payoff pair identified in Theorem 1 can be approximately implemented as a sequentialequilibrium (Kreps and Wilson 1982) in a suitably discretized game

Theorem 1lowast Fix any ε gt 0 There is a finite information structure and a finite price grid inducinga game with a set of sequential equilibrium payoffs that is an ε-net of Π(Γ)13

In fact the proof of Theorem 1lowast establishes even more the sequential equilibria in thediscretized games satisfy a natural version of the D1 refinement (Cho and Kreps 1987) Werelegate the logic to the Appendix but mention here that we use imperfectly-correlatedsignals for Buyer and Seller14

13 An information structure is finite if the signal spaces Tb and Ts are finite Sequential equilibrium isdefined in the obvious way for the ldquoinducedrdquo finite game where Nature directly draws (tb ts) rather thanfirst drawing v and playersrsquo payoffs are defined directly using (tb ts) For X sub R2 and ε gt 0 the set A sub Xis an ε-net of X if for each x isin X there is a isin A such that 983042xminus a983042 lt ε where 983042middot983042 is the Euclidean distance

14 The idea behind D1 is to ask for any off-path price whether one type of Seller would deviate for any

12

Even when our environment is specialized to monopoly pricing it is worth highlightingtwo contrasts between Theorem 11lowast and results of Bergemann et al (2015) and Roeslerand Szentes (2017) First we find that by not restricting the monopolist to be uninformedthe implementable payoff set typically expands rather dramatically trade can be efficientwith the monopolist securing none of the surplus beyond her payoff guarantee (πs) whichmay be zero (Roesler and Szentes establish implicitly that the implementable set withan uninformed monopolist is a superset of Bergemann et alrsquos which has a fully-informedconsumer) We will see in Subsection 32 that what is crucial to this expansion is price-dependent beliefs In particular the proof of Theorem 1lowast uses an information structure inwhich Buyer is not better informed than Seller mdash if he were then sequential equilibriumwould imply price-independent beliefs Second Theorem 1lowast establishes that for a givenε gt 0 a single information structure (and price grid) can be used to approximate the entirepayoff set Π(Γ) analogously to the construction described after Theorem 1 that used asingle information structure15 Bergemann et al (2015) and Roesler and Szentes (2017) onthe other hand vary information structures to span their payoff sets

32 More-informed Buyer and Price-independent Beliefs

In some economic settings it is plausible that Buyer is more informed than Seller Howdoes a restriction to such information structures ie τ isin Tmb affect the implementablepayoff set It turns out that what is in fact crucial is price-independent beliefs We have ex-plained earlier why it is desirable to impose this condition when Buyer is more informedbut that the condition is well defined even otherwise If Buyer is not more informed thanSeller price-independent beliefs ought to be viewed as an equilibrium restriction in gen-eral To reduce repetition readers should be bear in mind for the rest of this subsectionthe qualifier ldquowith price-independent beliefsrdquo applies unless stated explicitly otherwise

It is useful to define

πuss (Γ) equiv inf πs exist(πb πs) isin Πlowast

us(Γ) (1)

as the lowest payoff that an uninformed Seller can obtain no matter Buyerrsquos information(among equilibria with price-independent beliefs) Plainly πus

s (Γ) ge πs(Γ) In monopoly

Buyer mixed response that another type would Our construction has multiple Buyer types that are imper-fectly correlated with Seller types So different types of Seller have different beliefs about Buyer types Thisblunts the power of dominance considerations to the point where D1 does not exclude any Seller type fromthe support of Buyerrsquos off-path belief

15 In fact if one lets the price grid vary with ε then a single information structure implements exactlyrather than approximately in sequential equilibrium all payoffs ε-away from the boundary of Π(Γ) See in the Appendix for a formal statement

13

pricing with V = [v v] and c(middot) = v Roesler and Szentesrsquos (2017) characterization of theconsumer-optimal information structure identifies πus

s establishing that πuss gt πs If there

is no trade due to adverse selection when Seller is uniformed and Buyer has some informa-tion then πus

s = πs = 0 We do not have a general explicit formula for πuss Subsection 43

provides it for linear c(middot) Nonetheless we establish next that (i) the only additional restric-tion on equilibrium payoffs imposed by price-independent beliefs is a lower bound of πus

s

for Seller and (ii) uninformed-Seller information structures implement all such payoffs

Theorem 2 Consider equilibria with price-independent beliefs

1 Πlowast(Γ) = Πlowastmb(Γ) = Πlowast

us(Γ)

2 Πlowastus(Γ) = (πb πs) isin Π(Γ) πs ge πus

s (Γ)

3 For any (πb πs) isin Πlowastus(Γ) with πs gt πus

s (Γ) there is τ isin Tus with Π(Γ τ) = (πb πs)

Remark 1 We believe the substance of Theorem 2 would hold using discretizations and se-quential equilibria analogous to Theorem 1lowast As previously noted sequential equilibriumimplies price-independent beliefs when Buyer is more informed than Seller

To digest Theorem 2 note that Πlowast(Γ) sup Πlowastmb(Γ) sup Πlowast

us(Γ) is trivial So part 1 of thetheorem amounts to establishing the reverse inclusions The intuition for thismdashgiven part2rsquos characterization of Πlowast

usmdashis fairly straightforward with price-independent beliefs ad-ditional information cannot harm Seller even though it could alter the set of equilibria SoSellerrsquos lowest payoff obtains when she is uninformed

The characterization in part 2 of payoffs with an uninformed Seller corresponds to thetriangle ADE in Figure 1 Part 3 of the theorem assures ldquounique implementationrdquo of allimplementable payoffs satisfying πs gt πus

s (Γ) That is for any such payoff pair there is anuninformed-Seller information structure such that all equilibria (with price-independentbeliefs) induce exactly that payoff pair Unique implementation is appealing for multiplereasons one of which is that it obviates concerns about which among multiple payoff-distinct equilibria is more reasonable For example Ravid Roesler and Szentes (2019)bring to the fore the issue of equilibrium multiplicity in Roesler and Szentesrsquos (2017) infor-mation structure

Let us describe how we obtain the characterization of Πlowastus(Γ) and unique implementa-

tion There are two steps The first ensures that there is some information structure callit τ lowast isin Tus that implements Sellerrsquos payoff πus

s (Γ) That is we ensure that the infimum

14

in (1) is in fact a minimum16 While this argument is technical knowing τ lowast exists is usefulin what follows The second and economically insightful step is to construct informa-tion structures that implement every point in the triangle Πlowast

us(Γ) by suitably garbling theinformation structure τ lowast The construction is illustrated in Figure 2 Consider the distri-bution of Buyerrsquos posterior mean of his valuation v in information structure τ lowast (Givenprice-independent beliefs Buyerrsquos posterior mean is sufficient for his decision) For sim-plicity suppose this posterior-mean distribution has a density as depicted by the red curvein Figure 2 Fix any (πb πs) isin Πlowast

us(Γ)

First there is some number zlowast such that πb + πs is the total surplus from trading onlywhen Buyerrsquos posterior mean is greater than zlowast Next there is some price plowast ge zlowast suchthat Sellerrsquos payoff is πs if all these trades were to occur at price plowast17 Note that plowast must beno larger than the expected Buyer posterior mean conditional on that being above zlowast forotherwise πb lt 0 We claim that the information structure τ lowast can be garbled so that plowast isan equilibrium price and trade occurs only when Buyerrsquos posterior mean is greater thanzlowast The garbling is illustrated in Figure 2 by the blue distribution There is one signal that

pzv

ν(v)

Figure 2 Construction of information structure in Theorem 2

Buyer receives when the original posterior mean is between zlowast and plowast and also receiveswith some probability when the original posterior mean is above plowast The probability ischosen to make the posterior mean from this signal exactly plowast Apart from this one newsignal Buyer receives the original signal in τ lowast Plainly this is a garbling of τ lowast and hence isfeasible

16 The difficulty is in establishing suitable continuity Uninformed-Seller information structures can beviewed as probability measures over Buyerrsquos beliefs with convergence in the sense of the weak topologyThis topology ensures continuity with respect to probability measures of expectations of continuous orat least Lipschitz (and bounded) functions However Sellerrsquos expected payoff is not the expectation of aLipschitz function as Sellerrsquos profit is truncated at the price she charges

17 That plowast ge zlowast follows from πs ge πuss (Γ) as πs(Γ) itself is weakly larger than Sellerrsquos payoff from posting

price zlowast (and thus trading with the same set of Buyer posterior means) under information structure τlowast

15

Figure 2 makes clear why the new information structure has an equilibrium with priceplowast and Buyer breaking indifference in favor of trading (i) Sellerrsquos profit from posting anyprice below zlowast is the same as under τ lowast and hence no larger than πus

s (Γ) (ii) similarly Sellerrsquosprofit from posting any price above plowast is no higher than some fraction of πus

s (Γ) and (iii)any price between zlowast and plowast is worse that price plowast Moreover since Sellerrsquos profit fromoffering any price other than plowast is no more than πus

s (Γ) it follows that when πs gt πuss (Γ)

Buyer must break indifference as specified for Seller to have an optimal price and theequilibrium payoffs are unique

Remark 2 The above logic establishes that given any τ isin Tus that implements some (πb πs)τ can be garbled to uniquely implement any (πprime

b πprimes) isin Π(Γ) such that πprime

s gt πs That isan uninformed Sellerrsquos payoff can always be strictly raised and Buyerrsquos payoff reduced(strictly so long as it was not already zero) by garbling Buyerrsquos information

Remark 3 According to Theorem 11lowast and Theorem 2 uninformed-Seller information struc-tures cannot implement all implementable payoff pairs in an environment Γ if and only ifπuss (Γ) gt πs(Γ) This inequality fails in environment Γ if πus

s (Γ) = 0 since that impliesπuss (Γ) = πs(Γ) = 0 An example is when there is no trade due to adverse selection when

Seller is uniformed and Buyer has some information On the other hand it is sufficient forthe inequality that

v le E[c(v)] and forallv isin V c(v) lt v (2)

To see why (2) implies πuss (Γ) gt πs(Γ) notice that in any uninformed-Seller information

structure Seller can price at slightly less than Buyerrsquos highest posterior mean valuationand guarantee trade with only (a neighborhood of) that Buyer type If c(v) lt v for all v thisgives Seller a positive expected payoff and hence πus

s (Γ) gt 018 But the first inequality in (2)is equivalent to πs(Γ) = 0 We observe that Condition (2) is compatible with severe adverseselection resulting in very little trade when Seller is uninformed and Buyer (partially orfully) informed

33 Fully-Informed Buyer

We now turn to the third canonical class of information structures Buyer is fully in-formed of his value v As this is a special case of more-informed Buyer we maintain price-independent beliefs throughout this subsection

18 More precisely as V is compact c(v) lt v for all v implies there exists ε gt 0 such that v minus c(v) gt εGiven any uninformed-Seller information structure let mv be the highest posterior mean valuation in thesupport of the posterior means induced by Buyerrsquos signals So there is positive probability of Buyer signalswith posterior mean valuations at least mv minus ε2 By pricing at mv minus ε2 Sellerrsquos expected cost conditionalon trade is bounded above by mv minus ε and hence Sellerrsquos profit conditional on trade is at least ε2 gt 0 Itfollows that πus

s gt 0

16

Faced with a fully informed Buyer and any sequentially rational Buyer strategy an un-informed Seller can guarantee a profit level

πfbs (Γ) equiv sup

p

983133 v

p

(pminus c(v))micro(dv)

regardless of her information Plainly πfbs (Γ) ge πus

s (Γ) In monopoly pricing with c(middot) = vRoesler and Szentes (2017) have shown that πfb

s gt πuss if there is no trade due to adverse

selection when Buyer is fully informed and Seller is uniformed then πfbs = πus

s = 0 Belowwe establish that when Buyer is fully informed πfb

s is the only additional constraint onequilibrium payoffs Let Bε(πb πs) denote an ε-ball around (πb πs)

Theorem 3 Consider a fully informed Buyer and price-independent beliefs

1 Πlowastfb(Γ) = (πb πs) isin Π(Γ) πs ge πfb

s (Γ)

2 For any (πb πs) isin Πlowastfb(Γ) and any ε gt 0 there is τ isin Tfb with Π(Γ τ) sub Bε(πb πs)

The payoff set characterized in Theorem 3 corresponds to the triangle ABE in Figure 1Part 2 of the theorem establishes ldquoapproximately unique implementationrdquo for any point inthe triangle there is an information structure under which all equilibria yield payoff pairsin a neighborhood of that point

Here is the idea behind part 1 of Theorem 3 When Buyer is fully informed an informa-tion structure can be viewed as dividing vrsquos prior distribution micro into a set of microi that aver-age to micro with Seller informed of which microi she faces Theorem 3 is proven by establishingthat we can divide micro suitably so that against each microi Seller is indifferent between pricingat all prices in the support of microi including the price corresponding to πfb

s in that environ-ment We call such a microi a (Seller-price) incentive compatible distribution or ICD (Bergemannet al (2015) call their analogous construct an extreme market) The Appendix provides aldquogreedyrdquo algorithm to compute ICDs19

We can sketch how the algorithm works and construct a set of ICDs that average to theprior Suppose V = v1 v2 vK with vi lt vi+1 for i isin 1 K minus 1 and c(v) lt v forall v Given any small-enough mass of vK there is a unique mass of type vKminus1 that makesSeller indifferent between charging price vK and vKminus1 (If the mass is too low Seller prefersvK if it is too high she prefers vKminus1) Iterating down to keep Seller indifferent betweenall prices pins down an ICD Choose the maximum mass of type vK for which this worksRemove that ICD and repeat

19 The algorithm is defined for finite V and we take limits to handle the infinite case

17

Crucially whenever an ICD is removed the price corresponding to πfbs (Γ) remains op-

timal in the remaining ldquomarketrdquo this follows from the ICDrsquos defining property of Sellerindifference and an accounting identity Therefore Sellerrsquos profit in this segmentation ofICDs remains πfb

s (Γ) Moreover it is also optimal for Seller to always (ie for each microi) priceso that there is full trade or no trade Hence Buyerrsquos expected payoff can be either 0 or theentire surplus less πfb

s (Γ) It follows that the fully-informed Buyer information structuredefined by this set of ICDs implements point B and C in Figure 1 The entire triangle ABC

can then be implemented by convexification randomizing over this information structure(and two equilibria) and full information (where Seller obtains all the surplus)

The above logic of incentive compatible distributions reduces to that of Bergemann et al(2015) in the context of monopoly pricing Even in the monopoly pricing case Theorem 3part 2 strengthens the conclusions of Bergemann et al (2015) by ensuring approximatelyunique implementation

Remark 4 Theorems 1ndash3 imply that fully-informed-Buyer information structures imple-ment all implementable payoff pairs if and only if πfb

s (Γ) = πs(Γ) In that case trianglesAFG and ABC coincide in Figure 1 It follows that πfb

s (Γ) = πs(Γ) only when a fully-informed Buyer and uninformed Seller can result in full trade (v ge E[c(v)] and Sellerprices at v) or no trade (v le E[c(v)] and Seller prices at some p ge v) Interestinglywhen πfb

s (Γ) gt πs(Γ) fully-informed-Buyer information structures cannot even implementall payoff pairs implementable by uninformed-Seller information structures ie trianglesABC and ADE in Figure 1 are distinct That is when πfb

s (Γ) gt πs(Γ) there is an uninformed-Seller information structure that implements some πs lt πfb

s (Γ)20 Theorems 2ndash3 further im-ply in this case that Buyer can obtain a strictly higher payoff when he is not fully informed(and Seller is uninformed)

4 Discussion

41 Multidimensionality

Suppose Buyer and Sellerrsquos cost and valuation pair (c v) is a two-dimensional randomvariable distributed according to joint distribution micro with a compact support The ex-tension of our maintained assumption of commonly known gains from trade is for all

20 Pick any pprime gt v such that pprimeminusE[c(v)] lt πfbs (Γ) Following the construction described after Theorem 2 we

can mix all valuations v le pprime with a fraction λ gt 0 of valuations v gt pprime so that the mixture has posterior meanexactly pprime The remaining fraction 1 minus λ of valuations above pprime are revealed to Buyer With this uninformed-Seller information structure consider any equilibrium in which Buyer purchases when indifferent (Suchan equilibrium with price-independent belief exists) Sellerrsquos profit is at most (1 minus λ)πfb

s (Γ) from any pricep gt pprime and pprime minus E[c(v)] from price p = pprime Hence Sellerrsquos profit is strictly less than πfb

s (Γ)

18

(c v) isin Supp(micro) v ge c and E[v minus c] gt 0 An information structure is now a joint distribu-tion P (tb ts v c) whose marginal distribution on (v c) is micro

We claim that the content of our key results Theorem 1Theorem 1lowast Theorem 2 andTheorem 3 still hold To see why let v be the lowest valuation in the support of micro Sellerrsquosindividual rationality constraint is now maxv minus E[c] 0 as she can guarantee this by set-ting either a sufficiently high price or a price (arbitrarily close to) v regardless of her signalWe can define a cost function c(vprime) equiv Emicro[c|v = vprime] ge vprime This results in an environment sat-isfying all the maintained assumptions of our baseline model except that c(middot) may not becontinuous Recall that such continuity is not needed for proving Theorem 1 Theorem 1lowastBoth Theorem 2 and Theorem 3 use continuity of c(middot) to guarantee that Eν [c(v)] is a contin-uous function of ν isin ∆(V ) for a key convergence result However in the two-dimensionaltype environment Eν [c] is still a continuous function of ν isin ∆(V times C) Note that Theo-rem 3 uses upper semi-continuity of Sellerrsquos profit in price boundedness of c and c le v issufficient for such upper semi-continuity

42 Negative Trading Surplus

Returning to our baseline model we next discuss cases in which trade sometimes gen-erates negative surplus That is we drop the assumption that c(v) le v we do not requireE[v minus c(v)] gt 0 either Define Sλ(Γ) for λ isin [1infin) as

Sλ(Γ) equiv983133 v

v

[v minus c(v) + λ(v minus v)]+ micro(dv)

where [middot]+ equiv max middot 0 It is readily verified that S1(Γ) = E [[v minus c(v)]+] and limλrarr0

Sλ(Γ)λ =

E[v] minus v Allowing negative trading surplus does not affect our definition of wPBE Sothe notation Π(Γ) and πs(Γ) still have the same meanings as before The next propositionshows that Π(Γ) is now characterized by three constraints as before the two individualrationality constraints πs ge πs(Γ) and πb ge 0 and different now a Pareto frontier definedby all Sλ(Γ)

Proposition 1 Consider all information structures and equilibria when trade can generate negativesurplus

Π(Γ) =

983099983105983103

983105983101

πb ge 0

(πb πs) πs ge πs(Γ)

λπb + πs le Sλ(Γ) forallλ ge 1

983100983105983104

983105983102

Proposition 1 shows that when the trade can be inefficient it may be impossible to lower

19

Sellerrsquos profit without losing total surplus (ie πb + πs) This result is intuitive at the ex-tremes Suppose for simplicity v gt E[c(v)] If πs = πs(Γ) then since πs(Γ) = v minus E[c(v)]Seller must find it optimal to price at v and trade with probability one including whentrade generates negative surplus So Buyerrsquos payoff is E[v]minus v = limλrarrinfin Sλ(Γ)λ and totalsurplus is E[v minus c(v)] which includes the maximum amount of inefficient trade On theother hand an information structure that publicly reveals (only) whether trade is efficient(ie whether v ge c(v)) can implement efficient trade with all the surplus accruing to SellerIn this case Seller obtains S1(Γ) while Buyer obtains zero The proposition establishes thatin between S1(Γ) and πs(Γ) information structures and equilibria that lower Sellerrsquos profitmust reduce total surplus as the Pareto frontier Sλ(Γ) is a concave function with maximalslope minus1

The function capturing the Pareto frontier Sλ(Γ) is the maximal weighted sum of Buyerand Seller payoff assuming that Seller will trade at the low price p = v whenever it createspositive weighted total payoff Since we place higher weight on buyer (λ ge 1) S1(Γ) isthe upper bound for the weighted total payoff In the proof of Proposition 1 we showthat these bounds are tight mdash any payoff pair on the frontier can be implemented by someinformation structure The key observation is that whenever v minus c(v) + λ(v minus v) lt 0 itfollows that vminus c(v) lt (1minusλ)(vminus v) le 0 That is it is incentive compatible for Seller to notsell when it is common knowledge that v creates negative weighted total payoff Thereforewe can simply release a public signal indicating the sign of weighted total payoff and ifpositive induce Seller to sell at v which implies all bounds Sλ(Γ)

43 Linear Cost

When Sellerrsquos cost function c(v) is linear we can explicitly characterize an informationstructure that implements Sellerrsquos minimum implementable payoff πus

s (Γ) when Seller isuninformed (or per Theorem 2 when Buyer is better informed) subject to price-independentbeliefs We note that given the discussion in Subsection 41 a linear c(v) subsumes richerenvironments in which conditional expectations are linear including Gaussian environ-ments (cf Bar-Isaac et al 2018)

Condition 1 c(v) = λv + γ for some λ isin R

Let F (v) be the cumulative distribution function (CDF) corresponding to the prior mea-sure micro Let D(micro) be the set of all distributions whose CDF G satisfies

983133

V

vdG(v) =

983133

V

vdF (v) and983133 v

v

G(s)ds le983133 v

v

F (s)ds forallv isin V

20

That is D(micro) contains all distributions that are mean-preserving-contractions (MPC) of microIt is well known that D(micro) characterizes the set of distributions of Buyer posterior meansthat can be generated by any (uninformed-Seller) information structure

Definition 2 G is an incentive compatible distribution (ICD) if

983133

sgep

(pminus c(s))G(ds) = constant ge 0 forallp isin Supp(G)

That is when Buyerrsquos (mean) valuation is distributed according to an ICD G Seller isindifferent among all prices in the support of G assuming Buyer buys when indifferent Wefocus on a special family of ICDs whose supports are intervals [vlowast v

lowast] Given Condition 1such ICDs have analytical an expression

G(v) =

983099983105983105983103

983105983105983101

1minus983061

(1minus λ)v minus γ

(1minus λ)vlowast minus γ

983062 1λminus1

if λ ∕= 1

1minus evminusvlowast

γ if λ = 1

(3)

Given any vlowast and the above G(v) an upper bound vlowast is pinned down by the conditionEG[v] = EF [v] So a family of ICDs is parametrized by a single parameter vlowast and hence anICD is denoted by CDF 983144Gvlowast(v) with density 983144gvlowast(v) It is not difficult to verify that 983144Gvlowast(v) ismonotonic in vlowast for v in the support Consequently all ICDs are ordered according to theMPC order For vlowast isin [vE[v]] when vlowast increases the ICD increases in the MPC order

Proposition 2 Suppose Condition 1 is satisfied and let plowast equiv min983153vlowast983055983055 983144Gvlowast isin D(micro)

983154 It holds that

πuss (Γ) = plowast minus Emicro[c(v)]

Proposition 2 states that an uninformed Sellerrsquos minimum payoff (with price-independentbeliefs) is characterized by the ICDs that are MPCs of the prior distribution micro By defini-tion of ICDs Sellerrsquos profit when facing ICD 983144Gvlowast is vlowast minus E[v] Sellerrsquos minimum payoff isimplemented by the most dispersed ICD that is a MPC of the prior distribution

In proving Proposition 2 the key step is to show that given the prior micro garbling Buyerrsquosinformation such that the posterior mean distribution is an ICD makes Seller weakly worseoff It is then without loss to only consider ICDs to implement πus

s (Γ) We can thus workwith a tractable one-dimensional problem Consider the set D(micro) Suppose we find G(v)

such that (i) G isin D(micro) (ii) G is an ICD and (iii) existp such that983125 p

vG(s)ds =

983125 p

vF (s)ds Such

a G is the most-dispersed ICD that is a mean preserving contraction of F Consider the

21

following two manipulations

λ

983133 v

p

F (s)ds =

983133 v

p

(pminus c(s))dF (s)minus (1minus F (v)) (pminus c(v)) + λ(v minus p) (4)

λ

983133 v

p

G(s)ds =

983133 v

p

(pminus c(s))dG(s)minus (1minusG(v)) (pminus c(v)) + λ(v minus p) (5)

First by property (iii) above the LHS of Equation 4 equals the LHS of Equation 5 Secondsince

983125 v

p(F (s) minus G(s))ds is minimized at v this implies F (p) le G(p) Therefore the first

term on RHS of Equation 4 must be larger than that of Equation 5 Notice that the firstterm on RHS is Sellerrsquos profit when offering price p This implies that the profit fromoffering p given Buyer mean-valuation distribution G(v) is lower than that given F (v)which is lower than Sellerrsquos maximum profit given F (v) On the other hand by the ICDproperty p is already Sellerrsquos optimal price given G(v) The optimal profit given valuationdistribution F must be no lower than that of G Therefore we have proved that to minimizeSellerrsquos profit it is without loss to consider only ICDs within the set D(micro) which is a onedimensional subspace The detailed proof that such ICDs exist is relegated to the appendix

The monopoly-pricing environment with c(middot) = v is covered by Condition 1 with λ = 0

and γ = v The distribution G in (3) then reduces to that identified by Roesler and Szentes(2017)

If the prior micro has binary support then the cost function c(v) is linear In this special casewe can solve for πus

s (Γ) explicitly

Corollary 1 Suppose micro has binary support V = v1 v2 Let λ = c(v2)minusc(v1)v2minusv1

and let p be theunique solution to

(pminus c(p))1

λminus1 (pminus E[c(v)]) = (v2 minus c(v2))λ

λminus1 (6)

Then πuss (Γ) = max p v2minus E[c(v)]

References

AKERLOF G A (1970) ldquoThe Market for rdquoLemonsrdquo Quality Uncertainty and the MarketMechanismrdquo Quarterly Journal of Economics 84 488ndash500

BAGWELL K AND M H RIORDAN (1991) ldquoHigh and Declining Prices Signal ProductQualityrdquo American Economic Review 81 224ndash239

BAR-ISAAC H I JEWITT AND C LEAVER (2018) ldquoAdverse Selection Efficiency and theStructure of Informationrdquo Unpublished

22

BERGEMANN D B BROOKS AND S MORRIS (2015) ldquoThe Limits of Price Discrimina-tionrdquo American Economic Review 105 921ndash57

BERGEMANN D AND S MORRIS (2016) ldquoBayes Correlated Equilibrium and the Compar-ison of Information Structures in Gamesrdquo Theoretical Economics 11 487ndash522

BLACKWELL D (1953) ldquoEquivalent Comparisons of Experimentsrdquo Annals of MathematicalStatistics 24 265ndash272

CHO I-K AND D KREPS (1987) ldquoSignaling Games and Stable Equilibriardquo Quarterly Jour-nal of Economics 102 179ndash221

DU S (2018) ldquoRobust Mechanisms Under Common Valuationrdquo Econometrica 86 1569ndash1588

DURRETT R (1995) Probability Theory and Examples Duxbury Press 2nd ed

EINAV L AND A FINKELSTEIN (2011) ldquoSelection in Insurance Markets Theory and Em-pirics in Picturesrdquo Journal of Economic Perspectives 25 115ndash38

FUDENBERG D AND J TIROLE (1991) ldquoPerfect Bayesian Equilibrium and Sequential Equi-libriumrdquo Journal of Economic Theory 53 236ndash260

GARCIA D R TEPER AND M TSUR (2018) ldquoInformation Design in Insurance MarketsSelling Peaches in a Market for Lemonsrdquo Unpublished

GREENWALD B C (1986) ldquoAdverse Selection in the Labour Marketrdquo Review of EconomicStudies 53 325ndash347

JOVANOVIC B (1982) ldquoFavorable Selection with Asymmetric Informationrdquo QuarterlyJournal of Economics 97 535ndash539

KESSLER A S (2001) ldquoRevisiting the Lemons Marketrdquo International Economic Review 4225ndash41

KREPS D AND R WILSON (1982) ldquoSequential Equilibriardquo Econometrica 50 863ndash894

LEHMANN E L (1988) ldquoComparing Location Experimentsrdquo The Annals of Statistics 16521ndash533

LEVIN J (2001) ldquoInformation and the Market for Lemonsrdquo RAND Journal of Economics32 657ndash666

23

LIBGOBER J AND X MU (2019) ldquoInformational Robustness in Intertemporal PricingrdquoWorking paper

MAKRIS M AND L RENOU (2018) ldquoInformation Design in Multi-Stage Gamesrdquo Workingpaper

MILGROM P R AND R J WEBER (1985) ldquoDistributional Strategies for Games with In-complete Informationrdquo Mathematics of Operations Research 10 619ndash632

MUSSA M AND S ROSEN (1978) ldquoMonopoly and Product Qualityrdquo Journal of EconomicTheory 18 301ndash317

RAVID D A-K ROESLER AND B SZENTES (2019) ldquoLearning Before Trading On theInefficiency of Ignoring Free Informationrdquo Working paper

ROESLER A-K AND B SZENTES (2017) ldquoBuyer-Optimal Learning and Monopoly Pric-ingrdquo American Economic Review 107 2072ndash80

TERSTIEGE S AND C WASSER (2019) ldquoBuyer-optimal Extensionproof InformationrdquoWorking paper

Appendices omitted from this version but are available on request

24

Definition 1 A strategy profile (σα) and a belief ν(v|p tb) is a weak perfect Bayesian equilib-rium (wPBE) of game (Γ τ) if

1 Buyer plays optimally at every information set given his belief

α(p tb) =

983099983103

9831011 if Eν(v|ptb)[v] gt p

0 if Eν(v|ptb)[v] lt p

2 Seller plays optimally

σ isin argmaxσ

983133(pminus c(v))α(p tb)σ(dp|ts)P (dts dtb dv)

3 Beliefs satisfy Bayes rule on path for every measurable D sube Rtimes Ts times Tb times V

983133

D

ν(dv|p tb)σ(dp|ts)P (dts dtb V ) =

983133

D

σ(dp|ts)P (dts dtb dv)

Notice that we have formulated Sellerrsquos optimality requirement ex ante but Buyerrsquos ateach information set The latter is needed to capture sequential rationality The formeris for (notational) convenience this choice is inconsequential because Seller moves beforeBuyer

Hereafter ldquoequilibriumrdquo without qualification refers to a wPBE As is well understoodwPBE permits significant latitude in beliefs off the equilibrium path We will discuss re-finements when appropriate

23 Implementable Payoffs and Canonical Information Structures

We now define the set of implementable equilibrium outcomesmdashthat is the payoffs thatobtain in some equilibrium under some information structuremdashand some canonical classesof information structures

For a game (Γ τ) let the equilibrium payoff set be

Π(Γ τ) equiv (πb πs) exist wPBE of (Γ τ) with payoffs (πb πs)

Denote the class of all information structures by T and define

Π(Γ) equiv983134

τisinT

Π(Γ τ)

8

That is for environment Γ Π(Γ) is the set of all equilibrium payoff pairs that obtain undersome information structure

Uninformed Seller An information structure has uninformed Seller if Ts is a singletonSellerrsquos own signal contains no information about Buyerrsquos value v and hence neither aboutSellerrsquos cost c(v) When discussing such information structures we write the associateddistribution as just P (tb v) and Sellerrsquos strategy as just σ(p) omitting the argument ts inboth cases The class of all uninformed-Seller information structure is denoted Tus

Fully-informed Buyer An information structure has fully-informed Buyer if Buyerrsquos signalfully reveals his value v Formally this holds if Tb = V and the conditional distribution onV P (middot|tb) satisfies P (tb|tb) = 1 We denote the class of fully-informed-Buyer informationstructures Tfb

More-informed Buyer An information structure has more-informed Buyer if Buyer hasmore information than Seller Formally this holds when v and ts are independent condi-tional on tb ie for any measurable Ds sub Ts and Dv sub V P (DscapDv|tb) = P (Ds|tb)P (Dv|tb)10

Another way to interpret this requirement is that random variable tb must be statisticallysufficient for ts with respect to v ie tb is more informative than ts about v in the sense ofBlackwell (1953) We denote the class of more-informed-Buyer information structures byTmb Naturally information structures with uninformed Seller or fully-informed Buyer arecases of more-informed Buyer both Tus and Tfb are subclasses of Tmb

No updating from price For more-informed-Buyer information structures it is desirableto impose further requirements on Buyerrsquos equilibrium belief Since Sellerrsquos price can onlydepend on her own signal and this signal contains no additional information about v givenBuyerrsquos signal the price is statistically uninformative about v given Buyerrsquos signal Con-sequently Buyerrsquos posterior belief should be price independent once his signal has beenconditioned upon Formally regardless of the price p the equilibrium belief ν(middot|p tb) mustsatisfy 983133

ν(dv|p tb)P (dtb Ts V ) =

983133P (dtb Ts dv)

We refer to this condition as price-independent belief Note that the condition is meaningfulregardless of whether Buyer is more informed than Seller In a more-informed-Buyer in-formation structure price-independent belief would be implied by the ldquono signaling what

10 We write sub to mean ldquoweak subsetrdquo

9

you donrsquot knowrdquo requirement (Fudenberg and Tirole 1991) frequently imposed in versionsof perfect Bayesian equilibrium and the concept of sequential equilibrium (Kreps and Wil-son 1982) in finite versions of our setting11

Some notation will be helpful Define

Πlowast(Γ τ) equiv (πb πs) exist wPBE of (Γ τ) with price-independent belief and payoffs(πb πs)

Πlowast(Γ) equiv983134

τisinT

Πlowast(Γ τ)

Πlowasti (Γ) equiv

983134

τisinTi

Πlowast(Γ τ) for i = us fbmb

So Πlowast and Πlowast are analogous to the implementable payoff sets Π and Π defined earlier butrestricted to equilibria with price-independent beliefs Πlowast

us Πlowastfb and Πlowast

mb are the imple-mentable payoff sets when further restricted to uninformed-Seller fully-informed-Buyerand more-informed-Buyer information structures Plainly for any environment Γ

Πlowastus(Γ) cupΠlowast

fb(Γ) sub Πlowastmb(Γ) sub Πlowast(Γ) sub Π(Γ)

3 Main Results

Our goal is to characterize equilibrium payoff pairs across information structures in anarbitrary environment Γ In particular we seek to characterize the five sets Π(Γ) Πlowast(Γ)Πlowast

mb(Γ) Πlowastus(Γ) and Πlowast

fb(Γ) Let

S(Γ) equiv E[v minus c(v)]

be the (expected) surplus from trade in environment Γ This quantity will play an impor-tant role

31 All Information Structures

Define Sellerrsquos payoff guarantee as

πs(Γ) equiv max v minus E [c(v)] 0

11 An example clarifying our terminology may be helpful If both Seller and Buyer are fully informed of vthen the natural equilibriummdashthe unique sequential equilibrium in a finite version of the gamemdashhas Sellerpricing at p = v and Buyerrsquos belief being degenerate on v regardless of Sellerrsquos price This equilibrium hasprice-independent belief even though ex-ante Sellerrsquos price are Buyerrsquos belief are perfectly correlated Thepoint is that Buyerrsquos belief does not depend on price conditional on his signal

10

To interpret this observe that it is optimal for Buyer to accept the price v no matter hisbelief Therefore Seller can guarantee herself the (expected) profit v minus E[c(v)] no matterwhat the information structure is More precisely she can guarantee v minus E[c(v)] minus ε forany ε gt 0 since sequential rationality requires Buyer to accept any price v minus ε SimilarlySeller can also guarantee zero profit offering a price p gt v Hence Sellerrsquos payoff in anyequilibrium of any information structure must be at least the payoff guarantee πs(Γ) above

On the other hand Buyer can guarantee himself the payoff πb = 0 by rejecting all pricesIt follows that the implementable set Π(Γ) must satisfy three simple constraints 1) Sellerrsquosldquoindividual rationalityrdquo constraint πs ge πs(Γ) 2) Buyerrsquos ldquoindividual rationalityrdquo con-straint πb ge 0 and 3) the feasibility constraint πb + πs le S(Γ)

Our first result is that these individual rationality and feasibility constraints are alsosufficient for a payoff pair to be implementable

Theorem 1 Consider all information structures and equilibria

Π(Γ) =

983099983105983103

983105983101

πb ge 0

(πb πs) πs ge πs(Γ)

πb + πs le S(Γ)

983100983105983104

983105983102

Theorem 1 says that the set Π(Γ) corresponds to the triangle AFG in Figure 1 In par-ticular Buyer can receive the entire surplus beyond Sellerrsquos payoff guarantee This is sur-prising as Seller has substantial bargaining power Note that when v le E[c(v)] an entirelyreasonable condition Sellerrsquos payoff guarantee is zero12

The proof of Theorem 1 is in fact straightforward Suppose for expositional simplicityv ge E[c(v)] Fix the trivial information structure in which neither player receives any infor-mation and consider the following family of strategy profiles Seller randomizes betweentwo prices some pl isin [vE[v]] and ph = E[v] Buyer accepts pl with probability one andaccepts ph with probability α(ph) where α(ph) isin [0 1] is specified to make Seller indifferentbetween the two prices That is α(ph)(ph minus E[c(v)]) = pl minus E[c(v)] The expected payoffsfrom this strategy profile are

πb = σ(pl)(E[v]minus pl) and πs = pl minus E[c(v)]

As pl traverses the interval [vE[v]] Sellerrsquos payoff πs traverses [πs(Γ) S(Γ)] Given any

12 In monopoly pricing with c(middot) = v Sellerrsquos payoff guarantee of zero is lower than the revenue guaranteeidentified in Du (2018 Section 5) which is typically positive Dursquos notion is different from ours

11

pl Buyerrsquos payoff πb traverses [0 S(Γ)minusπs] as σ(pl) traverses [0 1] Therefore the proposedstrategy profiles induce all the payoff pairs stated in Theorem 1

We are left to specify beliefs ν for Buyer After prices pl and ph Buyer holds the priorbelief micro After any other (necessarily off-path) price Buyerrsquos belief is that v = v and soBuyer rejects all prices p isin [vinfin)pl ph It is straightforward to confirm that the specified(σα ν) constitute a wPBE

To get more insight into this construction consider its implication for the monopolypricing with c(middot) = v The equilibrium with pl = v (hence α(ph) = 0) and σ(pl) = 1

corresponds to the monopolist deterministically pricing at v and Buyer purchasing Giventhat both sides of the market receive no information why doesnrsquot the monopolist deviate toany price in (vE[v]) The reason is that in this equilibrium the consumer will then not buybecause he updates to v = v Such belief is compatible with wPBE because the equilibriumconcept places no restrictions on off-path beliefs This may seem like a game-theoreticmisdirection Buyerrsquos belief is not consistent with ldquono signaling what you donrsquot knowrdquoPut differently since we have a (weakly) more-informed Buyer information structure weought to impose the price-independent belief condition described in Subsection 23 thatwould imply Buyer must purchase at any price p lt E[v]

But the message of Theorem 1 does not rely on the permissiveness of wPBE To illus-trate continue with the above monopoly-pricing environment and suppose v has positiveprior probability Consider Buyer remaining uninformed but Seller learning whether v = v

or v gt v Now Buyerrsquos off-path belief that v = v is consistent with ldquono signaling whatyou donrsquot knowrdquo More generally using richer information structures we can prove thatany payoff pair identified in Theorem 1 can be approximately implemented as a sequentialequilibrium (Kreps and Wilson 1982) in a suitably discretized game

Theorem 1lowast Fix any ε gt 0 There is a finite information structure and a finite price grid inducinga game with a set of sequential equilibrium payoffs that is an ε-net of Π(Γ)13

In fact the proof of Theorem 1lowast establishes even more the sequential equilibria in thediscretized games satisfy a natural version of the D1 refinement (Cho and Kreps 1987) Werelegate the logic to the Appendix but mention here that we use imperfectly-correlatedsignals for Buyer and Seller14

13 An information structure is finite if the signal spaces Tb and Ts are finite Sequential equilibrium isdefined in the obvious way for the ldquoinducedrdquo finite game where Nature directly draws (tb ts) rather thanfirst drawing v and playersrsquo payoffs are defined directly using (tb ts) For X sub R2 and ε gt 0 the set A sub Xis an ε-net of X if for each x isin X there is a isin A such that 983042xminus a983042 lt ε where 983042middot983042 is the Euclidean distance

14 The idea behind D1 is to ask for any off-path price whether one type of Seller would deviate for any

12

Even when our environment is specialized to monopoly pricing it is worth highlightingtwo contrasts between Theorem 11lowast and results of Bergemann et al (2015) and Roeslerand Szentes (2017) First we find that by not restricting the monopolist to be uninformedthe implementable payoff set typically expands rather dramatically trade can be efficientwith the monopolist securing none of the surplus beyond her payoff guarantee (πs) whichmay be zero (Roesler and Szentes establish implicitly that the implementable set withan uninformed monopolist is a superset of Bergemann et alrsquos which has a fully-informedconsumer) We will see in Subsection 32 that what is crucial to this expansion is price-dependent beliefs In particular the proof of Theorem 1lowast uses an information structure inwhich Buyer is not better informed than Seller mdash if he were then sequential equilibriumwould imply price-independent beliefs Second Theorem 1lowast establishes that for a givenε gt 0 a single information structure (and price grid) can be used to approximate the entirepayoff set Π(Γ) analogously to the construction described after Theorem 1 that used asingle information structure15 Bergemann et al (2015) and Roesler and Szentes (2017) onthe other hand vary information structures to span their payoff sets

32 More-informed Buyer and Price-independent Beliefs

In some economic settings it is plausible that Buyer is more informed than Seller Howdoes a restriction to such information structures ie τ isin Tmb affect the implementablepayoff set It turns out that what is in fact crucial is price-independent beliefs We have ex-plained earlier why it is desirable to impose this condition when Buyer is more informedbut that the condition is well defined even otherwise If Buyer is not more informed thanSeller price-independent beliefs ought to be viewed as an equilibrium restriction in gen-eral To reduce repetition readers should be bear in mind for the rest of this subsectionthe qualifier ldquowith price-independent beliefsrdquo applies unless stated explicitly otherwise

It is useful to define

πuss (Γ) equiv inf πs exist(πb πs) isin Πlowast

us(Γ) (1)

as the lowest payoff that an uninformed Seller can obtain no matter Buyerrsquos information(among equilibria with price-independent beliefs) Plainly πus

s (Γ) ge πs(Γ) In monopoly

Buyer mixed response that another type would Our construction has multiple Buyer types that are imper-fectly correlated with Seller types So different types of Seller have different beliefs about Buyer types Thisblunts the power of dominance considerations to the point where D1 does not exclude any Seller type fromthe support of Buyerrsquos off-path belief

15 In fact if one lets the price grid vary with ε then a single information structure implements exactlyrather than approximately in sequential equilibrium all payoffs ε-away from the boundary of Π(Γ) See in the Appendix for a formal statement

13

pricing with V = [v v] and c(middot) = v Roesler and Szentesrsquos (2017) characterization of theconsumer-optimal information structure identifies πus

s establishing that πuss gt πs If there

is no trade due to adverse selection when Seller is uniformed and Buyer has some informa-tion then πus

s = πs = 0 We do not have a general explicit formula for πuss Subsection 43

provides it for linear c(middot) Nonetheless we establish next that (i) the only additional restric-tion on equilibrium payoffs imposed by price-independent beliefs is a lower bound of πus

s

for Seller and (ii) uninformed-Seller information structures implement all such payoffs

Theorem 2 Consider equilibria with price-independent beliefs

1 Πlowast(Γ) = Πlowastmb(Γ) = Πlowast

us(Γ)

2 Πlowastus(Γ) = (πb πs) isin Π(Γ) πs ge πus

s (Γ)

3 For any (πb πs) isin Πlowastus(Γ) with πs gt πus

s (Γ) there is τ isin Tus with Π(Γ τ) = (πb πs)

Remark 1 We believe the substance of Theorem 2 would hold using discretizations and se-quential equilibria analogous to Theorem 1lowast As previously noted sequential equilibriumimplies price-independent beliefs when Buyer is more informed than Seller

To digest Theorem 2 note that Πlowast(Γ) sup Πlowastmb(Γ) sup Πlowast

us(Γ) is trivial So part 1 of thetheorem amounts to establishing the reverse inclusions The intuition for thismdashgiven part2rsquos characterization of Πlowast

usmdashis fairly straightforward with price-independent beliefs ad-ditional information cannot harm Seller even though it could alter the set of equilibria SoSellerrsquos lowest payoff obtains when she is uninformed

The characterization in part 2 of payoffs with an uninformed Seller corresponds to thetriangle ADE in Figure 1 Part 3 of the theorem assures ldquounique implementationrdquo of allimplementable payoffs satisfying πs gt πus

s (Γ) That is for any such payoff pair there is anuninformed-Seller information structure such that all equilibria (with price-independentbeliefs) induce exactly that payoff pair Unique implementation is appealing for multiplereasons one of which is that it obviates concerns about which among multiple payoff-distinct equilibria is more reasonable For example Ravid Roesler and Szentes (2019)bring to the fore the issue of equilibrium multiplicity in Roesler and Szentesrsquos (2017) infor-mation structure

Let us describe how we obtain the characterization of Πlowastus(Γ) and unique implementa-

tion There are two steps The first ensures that there is some information structure callit τ lowast isin Tus that implements Sellerrsquos payoff πus

s (Γ) That is we ensure that the infimum

14

in (1) is in fact a minimum16 While this argument is technical knowing τ lowast exists is usefulin what follows The second and economically insightful step is to construct informa-tion structures that implement every point in the triangle Πlowast

us(Γ) by suitably garbling theinformation structure τ lowast The construction is illustrated in Figure 2 Consider the distri-bution of Buyerrsquos posterior mean of his valuation v in information structure τ lowast (Givenprice-independent beliefs Buyerrsquos posterior mean is sufficient for his decision) For sim-plicity suppose this posterior-mean distribution has a density as depicted by the red curvein Figure 2 Fix any (πb πs) isin Πlowast

us(Γ)

First there is some number zlowast such that πb + πs is the total surplus from trading onlywhen Buyerrsquos posterior mean is greater than zlowast Next there is some price plowast ge zlowast suchthat Sellerrsquos payoff is πs if all these trades were to occur at price plowast17 Note that plowast must beno larger than the expected Buyer posterior mean conditional on that being above zlowast forotherwise πb lt 0 We claim that the information structure τ lowast can be garbled so that plowast isan equilibrium price and trade occurs only when Buyerrsquos posterior mean is greater thanzlowast The garbling is illustrated in Figure 2 by the blue distribution There is one signal that

pzv

ν(v)

Figure 2 Construction of information structure in Theorem 2

Buyer receives when the original posterior mean is between zlowast and plowast and also receiveswith some probability when the original posterior mean is above plowast The probability ischosen to make the posterior mean from this signal exactly plowast Apart from this one newsignal Buyer receives the original signal in τ lowast Plainly this is a garbling of τ lowast and hence isfeasible

16 The difficulty is in establishing suitable continuity Uninformed-Seller information structures can beviewed as probability measures over Buyerrsquos beliefs with convergence in the sense of the weak topologyThis topology ensures continuity with respect to probability measures of expectations of continuous orat least Lipschitz (and bounded) functions However Sellerrsquos expected payoff is not the expectation of aLipschitz function as Sellerrsquos profit is truncated at the price she charges

17 That plowast ge zlowast follows from πs ge πuss (Γ) as πs(Γ) itself is weakly larger than Sellerrsquos payoff from posting

price zlowast (and thus trading with the same set of Buyer posterior means) under information structure τlowast

15

Figure 2 makes clear why the new information structure has an equilibrium with priceplowast and Buyer breaking indifference in favor of trading (i) Sellerrsquos profit from posting anyprice below zlowast is the same as under τ lowast and hence no larger than πus

s (Γ) (ii) similarly Sellerrsquosprofit from posting any price above plowast is no higher than some fraction of πus

s (Γ) and (iii)any price between zlowast and plowast is worse that price plowast Moreover since Sellerrsquos profit fromoffering any price other than plowast is no more than πus

s (Γ) it follows that when πs gt πuss (Γ)

Buyer must break indifference as specified for Seller to have an optimal price and theequilibrium payoffs are unique

Remark 2 The above logic establishes that given any τ isin Tus that implements some (πb πs)τ can be garbled to uniquely implement any (πprime

b πprimes) isin Π(Γ) such that πprime

s gt πs That isan uninformed Sellerrsquos payoff can always be strictly raised and Buyerrsquos payoff reduced(strictly so long as it was not already zero) by garbling Buyerrsquos information

Remark 3 According to Theorem 11lowast and Theorem 2 uninformed-Seller information struc-tures cannot implement all implementable payoff pairs in an environment Γ if and only ifπuss (Γ) gt πs(Γ) This inequality fails in environment Γ if πus

s (Γ) = 0 since that impliesπuss (Γ) = πs(Γ) = 0 An example is when there is no trade due to adverse selection when

Seller is uniformed and Buyer has some information On the other hand it is sufficient forthe inequality that

v le E[c(v)] and forallv isin V c(v) lt v (2)

To see why (2) implies πuss (Γ) gt πs(Γ) notice that in any uninformed-Seller information

structure Seller can price at slightly less than Buyerrsquos highest posterior mean valuationand guarantee trade with only (a neighborhood of) that Buyer type If c(v) lt v for all v thisgives Seller a positive expected payoff and hence πus

s (Γ) gt 018 But the first inequality in (2)is equivalent to πs(Γ) = 0 We observe that Condition (2) is compatible with severe adverseselection resulting in very little trade when Seller is uninformed and Buyer (partially orfully) informed

33 Fully-Informed Buyer

We now turn to the third canonical class of information structures Buyer is fully in-formed of his value v As this is a special case of more-informed Buyer we maintain price-independent beliefs throughout this subsection

18 More precisely as V is compact c(v) lt v for all v implies there exists ε gt 0 such that v minus c(v) gt εGiven any uninformed-Seller information structure let mv be the highest posterior mean valuation in thesupport of the posterior means induced by Buyerrsquos signals So there is positive probability of Buyer signalswith posterior mean valuations at least mv minus ε2 By pricing at mv minus ε2 Sellerrsquos expected cost conditionalon trade is bounded above by mv minus ε and hence Sellerrsquos profit conditional on trade is at least ε2 gt 0 Itfollows that πus

s gt 0

16

Faced with a fully informed Buyer and any sequentially rational Buyer strategy an un-informed Seller can guarantee a profit level

πfbs (Γ) equiv sup

p

983133 v

p

(pminus c(v))micro(dv)

regardless of her information Plainly πfbs (Γ) ge πus

s (Γ) In monopoly pricing with c(middot) = vRoesler and Szentes (2017) have shown that πfb

s gt πuss if there is no trade due to adverse

selection when Buyer is fully informed and Seller is uniformed then πfbs = πus

s = 0 Belowwe establish that when Buyer is fully informed πfb

s is the only additional constraint onequilibrium payoffs Let Bε(πb πs) denote an ε-ball around (πb πs)

Theorem 3 Consider a fully informed Buyer and price-independent beliefs

1 Πlowastfb(Γ) = (πb πs) isin Π(Γ) πs ge πfb

s (Γ)

2 For any (πb πs) isin Πlowastfb(Γ) and any ε gt 0 there is τ isin Tfb with Π(Γ τ) sub Bε(πb πs)

The payoff set characterized in Theorem 3 corresponds to the triangle ABE in Figure 1Part 2 of the theorem establishes ldquoapproximately unique implementationrdquo for any point inthe triangle there is an information structure under which all equilibria yield payoff pairsin a neighborhood of that point

Here is the idea behind part 1 of Theorem 3 When Buyer is fully informed an informa-tion structure can be viewed as dividing vrsquos prior distribution micro into a set of microi that aver-age to micro with Seller informed of which microi she faces Theorem 3 is proven by establishingthat we can divide micro suitably so that against each microi Seller is indifferent between pricingat all prices in the support of microi including the price corresponding to πfb

s in that environ-ment We call such a microi a (Seller-price) incentive compatible distribution or ICD (Bergemannet al (2015) call their analogous construct an extreme market) The Appendix provides aldquogreedyrdquo algorithm to compute ICDs19

We can sketch how the algorithm works and construct a set of ICDs that average to theprior Suppose V = v1 v2 vK with vi lt vi+1 for i isin 1 K minus 1 and c(v) lt v forall v Given any small-enough mass of vK there is a unique mass of type vKminus1 that makesSeller indifferent between charging price vK and vKminus1 (If the mass is too low Seller prefersvK if it is too high she prefers vKminus1) Iterating down to keep Seller indifferent betweenall prices pins down an ICD Choose the maximum mass of type vK for which this worksRemove that ICD and repeat

19 The algorithm is defined for finite V and we take limits to handle the infinite case

17

Crucially whenever an ICD is removed the price corresponding to πfbs (Γ) remains op-

timal in the remaining ldquomarketrdquo this follows from the ICDrsquos defining property of Sellerindifference and an accounting identity Therefore Sellerrsquos profit in this segmentation ofICDs remains πfb

s (Γ) Moreover it is also optimal for Seller to always (ie for each microi) priceso that there is full trade or no trade Hence Buyerrsquos expected payoff can be either 0 or theentire surplus less πfb

s (Γ) It follows that the fully-informed Buyer information structuredefined by this set of ICDs implements point B and C in Figure 1 The entire triangle ABC

can then be implemented by convexification randomizing over this information structure(and two equilibria) and full information (where Seller obtains all the surplus)

The above logic of incentive compatible distributions reduces to that of Bergemann et al(2015) in the context of monopoly pricing Even in the monopoly pricing case Theorem 3part 2 strengthens the conclusions of Bergemann et al (2015) by ensuring approximatelyunique implementation

Remark 4 Theorems 1ndash3 imply that fully-informed-Buyer information structures imple-ment all implementable payoff pairs if and only if πfb

s (Γ) = πs(Γ) In that case trianglesAFG and ABC coincide in Figure 1 It follows that πfb

s (Γ) = πs(Γ) only when a fully-informed Buyer and uninformed Seller can result in full trade (v ge E[c(v)] and Sellerprices at v) or no trade (v le E[c(v)] and Seller prices at some p ge v) Interestinglywhen πfb

s (Γ) gt πs(Γ) fully-informed-Buyer information structures cannot even implementall payoff pairs implementable by uninformed-Seller information structures ie trianglesABC and ADE in Figure 1 are distinct That is when πfb

s (Γ) gt πs(Γ) there is an uninformed-Seller information structure that implements some πs lt πfb

s (Γ)20 Theorems 2ndash3 further im-ply in this case that Buyer can obtain a strictly higher payoff when he is not fully informed(and Seller is uninformed)

4 Discussion

41 Multidimensionality

Suppose Buyer and Sellerrsquos cost and valuation pair (c v) is a two-dimensional randomvariable distributed according to joint distribution micro with a compact support The ex-tension of our maintained assumption of commonly known gains from trade is for all

20 Pick any pprime gt v such that pprimeminusE[c(v)] lt πfbs (Γ) Following the construction described after Theorem 2 we

can mix all valuations v le pprime with a fraction λ gt 0 of valuations v gt pprime so that the mixture has posterior meanexactly pprime The remaining fraction 1 minus λ of valuations above pprime are revealed to Buyer With this uninformed-Seller information structure consider any equilibrium in which Buyer purchases when indifferent (Suchan equilibrium with price-independent belief exists) Sellerrsquos profit is at most (1 minus λ)πfb

s (Γ) from any pricep gt pprime and pprime minus E[c(v)] from price p = pprime Hence Sellerrsquos profit is strictly less than πfb

s (Γ)

18

(c v) isin Supp(micro) v ge c and E[v minus c] gt 0 An information structure is now a joint distribu-tion P (tb ts v c) whose marginal distribution on (v c) is micro

We claim that the content of our key results Theorem 1Theorem 1lowast Theorem 2 andTheorem 3 still hold To see why let v be the lowest valuation in the support of micro Sellerrsquosindividual rationality constraint is now maxv minus E[c] 0 as she can guarantee this by set-ting either a sufficiently high price or a price (arbitrarily close to) v regardless of her signalWe can define a cost function c(vprime) equiv Emicro[c|v = vprime] ge vprime This results in an environment sat-isfying all the maintained assumptions of our baseline model except that c(middot) may not becontinuous Recall that such continuity is not needed for proving Theorem 1 Theorem 1lowastBoth Theorem 2 and Theorem 3 use continuity of c(middot) to guarantee that Eν [c(v)] is a contin-uous function of ν isin ∆(V ) for a key convergence result However in the two-dimensionaltype environment Eν [c] is still a continuous function of ν isin ∆(V times C) Note that Theo-rem 3 uses upper semi-continuity of Sellerrsquos profit in price boundedness of c and c le v issufficient for such upper semi-continuity

42 Negative Trading Surplus

Returning to our baseline model we next discuss cases in which trade sometimes gen-erates negative surplus That is we drop the assumption that c(v) le v we do not requireE[v minus c(v)] gt 0 either Define Sλ(Γ) for λ isin [1infin) as

Sλ(Γ) equiv983133 v

v

[v minus c(v) + λ(v minus v)]+ micro(dv)

where [middot]+ equiv max middot 0 It is readily verified that S1(Γ) = E [[v minus c(v)]+] and limλrarr0

Sλ(Γ)λ =

E[v] minus v Allowing negative trading surplus does not affect our definition of wPBE Sothe notation Π(Γ) and πs(Γ) still have the same meanings as before The next propositionshows that Π(Γ) is now characterized by three constraints as before the two individualrationality constraints πs ge πs(Γ) and πb ge 0 and different now a Pareto frontier definedby all Sλ(Γ)

Proposition 1 Consider all information structures and equilibria when trade can generate negativesurplus

Π(Γ) =

983099983105983103

983105983101

πb ge 0

(πb πs) πs ge πs(Γ)

λπb + πs le Sλ(Γ) forallλ ge 1

983100983105983104

983105983102

Proposition 1 shows that when the trade can be inefficient it may be impossible to lower

19

Sellerrsquos profit without losing total surplus (ie πb + πs) This result is intuitive at the ex-tremes Suppose for simplicity v gt E[c(v)] If πs = πs(Γ) then since πs(Γ) = v minus E[c(v)]Seller must find it optimal to price at v and trade with probability one including whentrade generates negative surplus So Buyerrsquos payoff is E[v]minus v = limλrarrinfin Sλ(Γ)λ and totalsurplus is E[v minus c(v)] which includes the maximum amount of inefficient trade On theother hand an information structure that publicly reveals (only) whether trade is efficient(ie whether v ge c(v)) can implement efficient trade with all the surplus accruing to SellerIn this case Seller obtains S1(Γ) while Buyer obtains zero The proposition establishes thatin between S1(Γ) and πs(Γ) information structures and equilibria that lower Sellerrsquos profitmust reduce total surplus as the Pareto frontier Sλ(Γ) is a concave function with maximalslope minus1

The function capturing the Pareto frontier Sλ(Γ) is the maximal weighted sum of Buyerand Seller payoff assuming that Seller will trade at the low price p = v whenever it createspositive weighted total payoff Since we place higher weight on buyer (λ ge 1) S1(Γ) isthe upper bound for the weighted total payoff In the proof of Proposition 1 we showthat these bounds are tight mdash any payoff pair on the frontier can be implemented by someinformation structure The key observation is that whenever v minus c(v) + λ(v minus v) lt 0 itfollows that vminus c(v) lt (1minusλ)(vminus v) le 0 That is it is incentive compatible for Seller to notsell when it is common knowledge that v creates negative weighted total payoff Thereforewe can simply release a public signal indicating the sign of weighted total payoff and ifpositive induce Seller to sell at v which implies all bounds Sλ(Γ)

43 Linear Cost

When Sellerrsquos cost function c(v) is linear we can explicitly characterize an informationstructure that implements Sellerrsquos minimum implementable payoff πus

s (Γ) when Seller isuninformed (or per Theorem 2 when Buyer is better informed) subject to price-independentbeliefs We note that given the discussion in Subsection 41 a linear c(v) subsumes richerenvironments in which conditional expectations are linear including Gaussian environ-ments (cf Bar-Isaac et al 2018)

Condition 1 c(v) = λv + γ for some λ isin R

Let F (v) be the cumulative distribution function (CDF) corresponding to the prior mea-sure micro Let D(micro) be the set of all distributions whose CDF G satisfies

983133

V

vdG(v) =

983133

V

vdF (v) and983133 v

v

G(s)ds le983133 v

v

F (s)ds forallv isin V

20

That is D(micro) contains all distributions that are mean-preserving-contractions (MPC) of microIt is well known that D(micro) characterizes the set of distributions of Buyer posterior meansthat can be generated by any (uninformed-Seller) information structure

Definition 2 G is an incentive compatible distribution (ICD) if

983133

sgep

(pminus c(s))G(ds) = constant ge 0 forallp isin Supp(G)

That is when Buyerrsquos (mean) valuation is distributed according to an ICD G Seller isindifferent among all prices in the support of G assuming Buyer buys when indifferent Wefocus on a special family of ICDs whose supports are intervals [vlowast v

lowast] Given Condition 1such ICDs have analytical an expression

G(v) =

983099983105983105983103

983105983105983101

1minus983061

(1minus λ)v minus γ

(1minus λ)vlowast minus γ

983062 1λminus1

if λ ∕= 1

1minus evminusvlowast

γ if λ = 1

(3)

Given any vlowast and the above G(v) an upper bound vlowast is pinned down by the conditionEG[v] = EF [v] So a family of ICDs is parametrized by a single parameter vlowast and hence anICD is denoted by CDF 983144Gvlowast(v) with density 983144gvlowast(v) It is not difficult to verify that 983144Gvlowast(v) ismonotonic in vlowast for v in the support Consequently all ICDs are ordered according to theMPC order For vlowast isin [vE[v]] when vlowast increases the ICD increases in the MPC order

Proposition 2 Suppose Condition 1 is satisfied and let plowast equiv min983153vlowast983055983055 983144Gvlowast isin D(micro)

983154 It holds that

πuss (Γ) = plowast minus Emicro[c(v)]

Proposition 2 states that an uninformed Sellerrsquos minimum payoff (with price-independentbeliefs) is characterized by the ICDs that are MPCs of the prior distribution micro By defini-tion of ICDs Sellerrsquos profit when facing ICD 983144Gvlowast is vlowast minus E[v] Sellerrsquos minimum payoff isimplemented by the most dispersed ICD that is a MPC of the prior distribution

In proving Proposition 2 the key step is to show that given the prior micro garbling Buyerrsquosinformation such that the posterior mean distribution is an ICD makes Seller weakly worseoff It is then without loss to only consider ICDs to implement πus

s (Γ) We can thus workwith a tractable one-dimensional problem Consider the set D(micro) Suppose we find G(v)

such that (i) G isin D(micro) (ii) G is an ICD and (iii) existp such that983125 p

vG(s)ds =

983125 p

vF (s)ds Such

a G is the most-dispersed ICD that is a mean preserving contraction of F Consider the

21

following two manipulations

λ

983133 v

p

F (s)ds =

983133 v

p

(pminus c(s))dF (s)minus (1minus F (v)) (pminus c(v)) + λ(v minus p) (4)

λ

983133 v

p

G(s)ds =

983133 v

p

(pminus c(s))dG(s)minus (1minusG(v)) (pminus c(v)) + λ(v minus p) (5)

First by property (iii) above the LHS of Equation 4 equals the LHS of Equation 5 Secondsince

983125 v

p(F (s) minus G(s))ds is minimized at v this implies F (p) le G(p) Therefore the first

term on RHS of Equation 4 must be larger than that of Equation 5 Notice that the firstterm on RHS is Sellerrsquos profit when offering price p This implies that the profit fromoffering p given Buyer mean-valuation distribution G(v) is lower than that given F (v)which is lower than Sellerrsquos maximum profit given F (v) On the other hand by the ICDproperty p is already Sellerrsquos optimal price given G(v) The optimal profit given valuationdistribution F must be no lower than that of G Therefore we have proved that to minimizeSellerrsquos profit it is without loss to consider only ICDs within the set D(micro) which is a onedimensional subspace The detailed proof that such ICDs exist is relegated to the appendix

The monopoly-pricing environment with c(middot) = v is covered by Condition 1 with λ = 0

and γ = v The distribution G in (3) then reduces to that identified by Roesler and Szentes(2017)

If the prior micro has binary support then the cost function c(v) is linear In this special casewe can solve for πus

s (Γ) explicitly

Corollary 1 Suppose micro has binary support V = v1 v2 Let λ = c(v2)minusc(v1)v2minusv1

and let p be theunique solution to

(pminus c(p))1

λminus1 (pminus E[c(v)]) = (v2 minus c(v2))λ

λminus1 (6)

Then πuss (Γ) = max p v2minus E[c(v)]

References

AKERLOF G A (1970) ldquoThe Market for rdquoLemonsrdquo Quality Uncertainty and the MarketMechanismrdquo Quarterly Journal of Economics 84 488ndash500

BAGWELL K AND M H RIORDAN (1991) ldquoHigh and Declining Prices Signal ProductQualityrdquo American Economic Review 81 224ndash239

BAR-ISAAC H I JEWITT AND C LEAVER (2018) ldquoAdverse Selection Efficiency and theStructure of Informationrdquo Unpublished

22

BERGEMANN D B BROOKS AND S MORRIS (2015) ldquoThe Limits of Price Discrimina-tionrdquo American Economic Review 105 921ndash57

BERGEMANN D AND S MORRIS (2016) ldquoBayes Correlated Equilibrium and the Compar-ison of Information Structures in Gamesrdquo Theoretical Economics 11 487ndash522

BLACKWELL D (1953) ldquoEquivalent Comparisons of Experimentsrdquo Annals of MathematicalStatistics 24 265ndash272

CHO I-K AND D KREPS (1987) ldquoSignaling Games and Stable Equilibriardquo Quarterly Jour-nal of Economics 102 179ndash221

DU S (2018) ldquoRobust Mechanisms Under Common Valuationrdquo Econometrica 86 1569ndash1588

DURRETT R (1995) Probability Theory and Examples Duxbury Press 2nd ed

EINAV L AND A FINKELSTEIN (2011) ldquoSelection in Insurance Markets Theory and Em-pirics in Picturesrdquo Journal of Economic Perspectives 25 115ndash38

FUDENBERG D AND J TIROLE (1991) ldquoPerfect Bayesian Equilibrium and Sequential Equi-libriumrdquo Journal of Economic Theory 53 236ndash260

GARCIA D R TEPER AND M TSUR (2018) ldquoInformation Design in Insurance MarketsSelling Peaches in a Market for Lemonsrdquo Unpublished

GREENWALD B C (1986) ldquoAdverse Selection in the Labour Marketrdquo Review of EconomicStudies 53 325ndash347

JOVANOVIC B (1982) ldquoFavorable Selection with Asymmetric Informationrdquo QuarterlyJournal of Economics 97 535ndash539

KESSLER A S (2001) ldquoRevisiting the Lemons Marketrdquo International Economic Review 4225ndash41

KREPS D AND R WILSON (1982) ldquoSequential Equilibriardquo Econometrica 50 863ndash894

LEHMANN E L (1988) ldquoComparing Location Experimentsrdquo The Annals of Statistics 16521ndash533

LEVIN J (2001) ldquoInformation and the Market for Lemonsrdquo RAND Journal of Economics32 657ndash666

23

LIBGOBER J AND X MU (2019) ldquoInformational Robustness in Intertemporal PricingrdquoWorking paper

MAKRIS M AND L RENOU (2018) ldquoInformation Design in Multi-Stage Gamesrdquo Workingpaper

MILGROM P R AND R J WEBER (1985) ldquoDistributional Strategies for Games with In-complete Informationrdquo Mathematics of Operations Research 10 619ndash632

MUSSA M AND S ROSEN (1978) ldquoMonopoly and Product Qualityrdquo Journal of EconomicTheory 18 301ndash317

RAVID D A-K ROESLER AND B SZENTES (2019) ldquoLearning Before Trading On theInefficiency of Ignoring Free Informationrdquo Working paper

ROESLER A-K AND B SZENTES (2017) ldquoBuyer-Optimal Learning and Monopoly Pric-ingrdquo American Economic Review 107 2072ndash80

TERSTIEGE S AND C WASSER (2019) ldquoBuyer-optimal Extensionproof InformationrdquoWorking paper

Appendices omitted from this version but are available on request

24

That is for environment Γ Π(Γ) is the set of all equilibrium payoff pairs that obtain undersome information structure

Uninformed Seller An information structure has uninformed Seller if Ts is a singletonSellerrsquos own signal contains no information about Buyerrsquos value v and hence neither aboutSellerrsquos cost c(v) When discussing such information structures we write the associateddistribution as just P (tb v) and Sellerrsquos strategy as just σ(p) omitting the argument ts inboth cases The class of all uninformed-Seller information structure is denoted Tus

Fully-informed Buyer An information structure has fully-informed Buyer if Buyerrsquos signalfully reveals his value v Formally this holds if Tb = V and the conditional distribution onV P (middot|tb) satisfies P (tb|tb) = 1 We denote the class of fully-informed-Buyer informationstructures Tfb

More-informed Buyer An information structure has more-informed Buyer if Buyer hasmore information than Seller Formally this holds when v and ts are independent condi-tional on tb ie for any measurable Ds sub Ts and Dv sub V P (DscapDv|tb) = P (Ds|tb)P (Dv|tb)10

Another way to interpret this requirement is that random variable tb must be statisticallysufficient for ts with respect to v ie tb is more informative than ts about v in the sense ofBlackwell (1953) We denote the class of more-informed-Buyer information structures byTmb Naturally information structures with uninformed Seller or fully-informed Buyer arecases of more-informed Buyer both Tus and Tfb are subclasses of Tmb

No updating from price For more-informed-Buyer information structures it is desirableto impose further requirements on Buyerrsquos equilibrium belief Since Sellerrsquos price can onlydepend on her own signal and this signal contains no additional information about v givenBuyerrsquos signal the price is statistically uninformative about v given Buyerrsquos signal Con-sequently Buyerrsquos posterior belief should be price independent once his signal has beenconditioned upon Formally regardless of the price p the equilibrium belief ν(middot|p tb) mustsatisfy 983133

ν(dv|p tb)P (dtb Ts V ) =

983133P (dtb Ts dv)

We refer to this condition as price-independent belief Note that the condition is meaningfulregardless of whether Buyer is more informed than Seller In a more-informed-Buyer in-formation structure price-independent belief would be implied by the ldquono signaling what

10 We write sub to mean ldquoweak subsetrdquo

9

you donrsquot knowrdquo requirement (Fudenberg and Tirole 1991) frequently imposed in versionsof perfect Bayesian equilibrium and the concept of sequential equilibrium (Kreps and Wil-son 1982) in finite versions of our setting11

Some notation will be helpful Define

Πlowast(Γ τ) equiv (πb πs) exist wPBE of (Γ τ) with price-independent belief and payoffs(πb πs)

Πlowast(Γ) equiv983134

τisinT

Πlowast(Γ τ)

Πlowasti (Γ) equiv

983134

τisinTi

Πlowast(Γ τ) for i = us fbmb

So Πlowast and Πlowast are analogous to the implementable payoff sets Π and Π defined earlier butrestricted to equilibria with price-independent beliefs Πlowast

us Πlowastfb and Πlowast

mb are the imple-mentable payoff sets when further restricted to uninformed-Seller fully-informed-Buyerand more-informed-Buyer information structures Plainly for any environment Γ

Πlowastus(Γ) cupΠlowast

fb(Γ) sub Πlowastmb(Γ) sub Πlowast(Γ) sub Π(Γ)

3 Main Results

Our goal is to characterize equilibrium payoff pairs across information structures in anarbitrary environment Γ In particular we seek to characterize the five sets Π(Γ) Πlowast(Γ)Πlowast

mb(Γ) Πlowastus(Γ) and Πlowast

fb(Γ) Let

S(Γ) equiv E[v minus c(v)]

be the (expected) surplus from trade in environment Γ This quantity will play an impor-tant role

31 All Information Structures

Define Sellerrsquos payoff guarantee as

πs(Γ) equiv max v minus E [c(v)] 0

11 An example clarifying our terminology may be helpful If both Seller and Buyer are fully informed of vthen the natural equilibriummdashthe unique sequential equilibrium in a finite version of the gamemdashhas Sellerpricing at p = v and Buyerrsquos belief being degenerate on v regardless of Sellerrsquos price This equilibrium hasprice-independent belief even though ex-ante Sellerrsquos price are Buyerrsquos belief are perfectly correlated Thepoint is that Buyerrsquos belief does not depend on price conditional on his signal

10

To interpret this observe that it is optimal for Buyer to accept the price v no matter hisbelief Therefore Seller can guarantee herself the (expected) profit v minus E[c(v)] no matterwhat the information structure is More precisely she can guarantee v minus E[c(v)] minus ε forany ε gt 0 since sequential rationality requires Buyer to accept any price v minus ε SimilarlySeller can also guarantee zero profit offering a price p gt v Hence Sellerrsquos payoff in anyequilibrium of any information structure must be at least the payoff guarantee πs(Γ) above

On the other hand Buyer can guarantee himself the payoff πb = 0 by rejecting all pricesIt follows that the implementable set Π(Γ) must satisfy three simple constraints 1) Sellerrsquosldquoindividual rationalityrdquo constraint πs ge πs(Γ) 2) Buyerrsquos ldquoindividual rationalityrdquo con-straint πb ge 0 and 3) the feasibility constraint πb + πs le S(Γ)

Our first result is that these individual rationality and feasibility constraints are alsosufficient for a payoff pair to be implementable

Theorem 1 Consider all information structures and equilibria

Π(Γ) =

983099983105983103

983105983101

πb ge 0

(πb πs) πs ge πs(Γ)

πb + πs le S(Γ)

983100983105983104

983105983102

Theorem 1 says that the set Π(Γ) corresponds to the triangle AFG in Figure 1 In par-ticular Buyer can receive the entire surplus beyond Sellerrsquos payoff guarantee This is sur-prising as Seller has substantial bargaining power Note that when v le E[c(v)] an entirelyreasonable condition Sellerrsquos payoff guarantee is zero12

The proof of Theorem 1 is in fact straightforward Suppose for expositional simplicityv ge E[c(v)] Fix the trivial information structure in which neither player receives any infor-mation and consider the following family of strategy profiles Seller randomizes betweentwo prices some pl isin [vE[v]] and ph = E[v] Buyer accepts pl with probability one andaccepts ph with probability α(ph) where α(ph) isin [0 1] is specified to make Seller indifferentbetween the two prices That is α(ph)(ph minus E[c(v)]) = pl minus E[c(v)] The expected payoffsfrom this strategy profile are

πb = σ(pl)(E[v]minus pl) and πs = pl minus E[c(v)]

As pl traverses the interval [vE[v]] Sellerrsquos payoff πs traverses [πs(Γ) S(Γ)] Given any

12 In monopoly pricing with c(middot) = v Sellerrsquos payoff guarantee of zero is lower than the revenue guaranteeidentified in Du (2018 Section 5) which is typically positive Dursquos notion is different from ours

11

pl Buyerrsquos payoff πb traverses [0 S(Γ)minusπs] as σ(pl) traverses [0 1] Therefore the proposedstrategy profiles induce all the payoff pairs stated in Theorem 1

We are left to specify beliefs ν for Buyer After prices pl and ph Buyer holds the priorbelief micro After any other (necessarily off-path) price Buyerrsquos belief is that v = v and soBuyer rejects all prices p isin [vinfin)pl ph It is straightforward to confirm that the specified(σα ν) constitute a wPBE

To get more insight into this construction consider its implication for the monopolypricing with c(middot) = v The equilibrium with pl = v (hence α(ph) = 0) and σ(pl) = 1

corresponds to the monopolist deterministically pricing at v and Buyer purchasing Giventhat both sides of the market receive no information why doesnrsquot the monopolist deviate toany price in (vE[v]) The reason is that in this equilibrium the consumer will then not buybecause he updates to v = v Such belief is compatible with wPBE because the equilibriumconcept places no restrictions on off-path beliefs This may seem like a game-theoreticmisdirection Buyerrsquos belief is not consistent with ldquono signaling what you donrsquot knowrdquoPut differently since we have a (weakly) more-informed Buyer information structure weought to impose the price-independent belief condition described in Subsection 23 thatwould imply Buyer must purchase at any price p lt E[v]

But the message of Theorem 1 does not rely on the permissiveness of wPBE To illus-trate continue with the above monopoly-pricing environment and suppose v has positiveprior probability Consider Buyer remaining uninformed but Seller learning whether v = v

or v gt v Now Buyerrsquos off-path belief that v = v is consistent with ldquono signaling whatyou donrsquot knowrdquo More generally using richer information structures we can prove thatany payoff pair identified in Theorem 1 can be approximately implemented as a sequentialequilibrium (Kreps and Wilson 1982) in a suitably discretized game

Theorem 1lowast Fix any ε gt 0 There is a finite information structure and a finite price grid inducinga game with a set of sequential equilibrium payoffs that is an ε-net of Π(Γ)13

In fact the proof of Theorem 1lowast establishes even more the sequential equilibria in thediscretized games satisfy a natural version of the D1 refinement (Cho and Kreps 1987) Werelegate the logic to the Appendix but mention here that we use imperfectly-correlatedsignals for Buyer and Seller14

13 An information structure is finite if the signal spaces Tb and Ts are finite Sequential equilibrium isdefined in the obvious way for the ldquoinducedrdquo finite game where Nature directly draws (tb ts) rather thanfirst drawing v and playersrsquo payoffs are defined directly using (tb ts) For X sub R2 and ε gt 0 the set A sub Xis an ε-net of X if for each x isin X there is a isin A such that 983042xminus a983042 lt ε where 983042middot983042 is the Euclidean distance

14 The idea behind D1 is to ask for any off-path price whether one type of Seller would deviate for any

12

Even when our environment is specialized to monopoly pricing it is worth highlightingtwo contrasts between Theorem 11lowast and results of Bergemann et al (2015) and Roeslerand Szentes (2017) First we find that by not restricting the monopolist to be uninformedthe implementable payoff set typically expands rather dramatically trade can be efficientwith the monopolist securing none of the surplus beyond her payoff guarantee (πs) whichmay be zero (Roesler and Szentes establish implicitly that the implementable set withan uninformed monopolist is a superset of Bergemann et alrsquos which has a fully-informedconsumer) We will see in Subsection 32 that what is crucial to this expansion is price-dependent beliefs In particular the proof of Theorem 1lowast uses an information structure inwhich Buyer is not better informed than Seller mdash if he were then sequential equilibriumwould imply price-independent beliefs Second Theorem 1lowast establishes that for a givenε gt 0 a single information structure (and price grid) can be used to approximate the entirepayoff set Π(Γ) analogously to the construction described after Theorem 1 that used asingle information structure15 Bergemann et al (2015) and Roesler and Szentes (2017) onthe other hand vary information structures to span their payoff sets

32 More-informed Buyer and Price-independent Beliefs

In some economic settings it is plausible that Buyer is more informed than Seller Howdoes a restriction to such information structures ie τ isin Tmb affect the implementablepayoff set It turns out that what is in fact crucial is price-independent beliefs We have ex-plained earlier why it is desirable to impose this condition when Buyer is more informedbut that the condition is well defined even otherwise If Buyer is not more informed thanSeller price-independent beliefs ought to be viewed as an equilibrium restriction in gen-eral To reduce repetition readers should be bear in mind for the rest of this subsectionthe qualifier ldquowith price-independent beliefsrdquo applies unless stated explicitly otherwise

It is useful to define

πuss (Γ) equiv inf πs exist(πb πs) isin Πlowast

us(Γ) (1)

as the lowest payoff that an uninformed Seller can obtain no matter Buyerrsquos information(among equilibria with price-independent beliefs) Plainly πus

s (Γ) ge πs(Γ) In monopoly

Buyer mixed response that another type would Our construction has multiple Buyer types that are imper-fectly correlated with Seller types So different types of Seller have different beliefs about Buyer types Thisblunts the power of dominance considerations to the point where D1 does not exclude any Seller type fromthe support of Buyerrsquos off-path belief

15 In fact if one lets the price grid vary with ε then a single information structure implements exactlyrather than approximately in sequential equilibrium all payoffs ε-away from the boundary of Π(Γ) See in the Appendix for a formal statement

13

pricing with V = [v v] and c(middot) = v Roesler and Szentesrsquos (2017) characterization of theconsumer-optimal information structure identifies πus

s establishing that πuss gt πs If there

is no trade due to adverse selection when Seller is uniformed and Buyer has some informa-tion then πus

s = πs = 0 We do not have a general explicit formula for πuss Subsection 43

provides it for linear c(middot) Nonetheless we establish next that (i) the only additional restric-tion on equilibrium payoffs imposed by price-independent beliefs is a lower bound of πus

s

for Seller and (ii) uninformed-Seller information structures implement all such payoffs

Theorem 2 Consider equilibria with price-independent beliefs

1 Πlowast(Γ) = Πlowastmb(Γ) = Πlowast

us(Γ)

2 Πlowastus(Γ) = (πb πs) isin Π(Γ) πs ge πus

s (Γ)

3 For any (πb πs) isin Πlowastus(Γ) with πs gt πus

s (Γ) there is τ isin Tus with Π(Γ τ) = (πb πs)

Remark 1 We believe the substance of Theorem 2 would hold using discretizations and se-quential equilibria analogous to Theorem 1lowast As previously noted sequential equilibriumimplies price-independent beliefs when Buyer is more informed than Seller

To digest Theorem 2 note that Πlowast(Γ) sup Πlowastmb(Γ) sup Πlowast

us(Γ) is trivial So part 1 of thetheorem amounts to establishing the reverse inclusions The intuition for thismdashgiven part2rsquos characterization of Πlowast

usmdashis fairly straightforward with price-independent beliefs ad-ditional information cannot harm Seller even though it could alter the set of equilibria SoSellerrsquos lowest payoff obtains when she is uninformed

The characterization in part 2 of payoffs with an uninformed Seller corresponds to thetriangle ADE in Figure 1 Part 3 of the theorem assures ldquounique implementationrdquo of allimplementable payoffs satisfying πs gt πus

s (Γ) That is for any such payoff pair there is anuninformed-Seller information structure such that all equilibria (with price-independentbeliefs) induce exactly that payoff pair Unique implementation is appealing for multiplereasons one of which is that it obviates concerns about which among multiple payoff-distinct equilibria is more reasonable For example Ravid Roesler and Szentes (2019)bring to the fore the issue of equilibrium multiplicity in Roesler and Szentesrsquos (2017) infor-mation structure

Let us describe how we obtain the characterization of Πlowastus(Γ) and unique implementa-

tion There are two steps The first ensures that there is some information structure callit τ lowast isin Tus that implements Sellerrsquos payoff πus

s (Γ) That is we ensure that the infimum

14

in (1) is in fact a minimum16 While this argument is technical knowing τ lowast exists is usefulin what follows The second and economically insightful step is to construct informa-tion structures that implement every point in the triangle Πlowast

us(Γ) by suitably garbling theinformation structure τ lowast The construction is illustrated in Figure 2 Consider the distri-bution of Buyerrsquos posterior mean of his valuation v in information structure τ lowast (Givenprice-independent beliefs Buyerrsquos posterior mean is sufficient for his decision) For sim-plicity suppose this posterior-mean distribution has a density as depicted by the red curvein Figure 2 Fix any (πb πs) isin Πlowast

us(Γ)

First there is some number zlowast such that πb + πs is the total surplus from trading onlywhen Buyerrsquos posterior mean is greater than zlowast Next there is some price plowast ge zlowast suchthat Sellerrsquos payoff is πs if all these trades were to occur at price plowast17 Note that plowast must beno larger than the expected Buyer posterior mean conditional on that being above zlowast forotherwise πb lt 0 We claim that the information structure τ lowast can be garbled so that plowast isan equilibrium price and trade occurs only when Buyerrsquos posterior mean is greater thanzlowast The garbling is illustrated in Figure 2 by the blue distribution There is one signal that

pzv

ν(v)

Figure 2 Construction of information structure in Theorem 2

Buyer receives when the original posterior mean is between zlowast and plowast and also receiveswith some probability when the original posterior mean is above plowast The probability ischosen to make the posterior mean from this signal exactly plowast Apart from this one newsignal Buyer receives the original signal in τ lowast Plainly this is a garbling of τ lowast and hence isfeasible

16 The difficulty is in establishing suitable continuity Uninformed-Seller information structures can beviewed as probability measures over Buyerrsquos beliefs with convergence in the sense of the weak topologyThis topology ensures continuity with respect to probability measures of expectations of continuous orat least Lipschitz (and bounded) functions However Sellerrsquos expected payoff is not the expectation of aLipschitz function as Sellerrsquos profit is truncated at the price she charges

17 That plowast ge zlowast follows from πs ge πuss (Γ) as πs(Γ) itself is weakly larger than Sellerrsquos payoff from posting

price zlowast (and thus trading with the same set of Buyer posterior means) under information structure τlowast

15

Figure 2 makes clear why the new information structure has an equilibrium with priceplowast and Buyer breaking indifference in favor of trading (i) Sellerrsquos profit from posting anyprice below zlowast is the same as under τ lowast and hence no larger than πus

s (Γ) (ii) similarly Sellerrsquosprofit from posting any price above plowast is no higher than some fraction of πus

s (Γ) and (iii)any price between zlowast and plowast is worse that price plowast Moreover since Sellerrsquos profit fromoffering any price other than plowast is no more than πus

s (Γ) it follows that when πs gt πuss (Γ)

Buyer must break indifference as specified for Seller to have an optimal price and theequilibrium payoffs are unique

Remark 2 The above logic establishes that given any τ isin Tus that implements some (πb πs)τ can be garbled to uniquely implement any (πprime

b πprimes) isin Π(Γ) such that πprime

s gt πs That isan uninformed Sellerrsquos payoff can always be strictly raised and Buyerrsquos payoff reduced(strictly so long as it was not already zero) by garbling Buyerrsquos information

Remark 3 According to Theorem 11lowast and Theorem 2 uninformed-Seller information struc-tures cannot implement all implementable payoff pairs in an environment Γ if and only ifπuss (Γ) gt πs(Γ) This inequality fails in environment Γ if πus

s (Γ) = 0 since that impliesπuss (Γ) = πs(Γ) = 0 An example is when there is no trade due to adverse selection when

Seller is uniformed and Buyer has some information On the other hand it is sufficient forthe inequality that

v le E[c(v)] and forallv isin V c(v) lt v (2)

To see why (2) implies πuss (Γ) gt πs(Γ) notice that in any uninformed-Seller information

structure Seller can price at slightly less than Buyerrsquos highest posterior mean valuationand guarantee trade with only (a neighborhood of) that Buyer type If c(v) lt v for all v thisgives Seller a positive expected payoff and hence πus

s (Γ) gt 018 But the first inequality in (2)is equivalent to πs(Γ) = 0 We observe that Condition (2) is compatible with severe adverseselection resulting in very little trade when Seller is uninformed and Buyer (partially orfully) informed

33 Fully-Informed Buyer

We now turn to the third canonical class of information structures Buyer is fully in-formed of his value v As this is a special case of more-informed Buyer we maintain price-independent beliefs throughout this subsection

18 More precisely as V is compact c(v) lt v for all v implies there exists ε gt 0 such that v minus c(v) gt εGiven any uninformed-Seller information structure let mv be the highest posterior mean valuation in thesupport of the posterior means induced by Buyerrsquos signals So there is positive probability of Buyer signalswith posterior mean valuations at least mv minus ε2 By pricing at mv minus ε2 Sellerrsquos expected cost conditionalon trade is bounded above by mv minus ε and hence Sellerrsquos profit conditional on trade is at least ε2 gt 0 Itfollows that πus

s gt 0

16

Faced with a fully informed Buyer and any sequentially rational Buyer strategy an un-informed Seller can guarantee a profit level

πfbs (Γ) equiv sup

p

983133 v

p

(pminus c(v))micro(dv)

regardless of her information Plainly πfbs (Γ) ge πus

s (Γ) In monopoly pricing with c(middot) = vRoesler and Szentes (2017) have shown that πfb

s gt πuss if there is no trade due to adverse

selection when Buyer is fully informed and Seller is uniformed then πfbs = πus

s = 0 Belowwe establish that when Buyer is fully informed πfb

s is the only additional constraint onequilibrium payoffs Let Bε(πb πs) denote an ε-ball around (πb πs)

Theorem 3 Consider a fully informed Buyer and price-independent beliefs

1 Πlowastfb(Γ) = (πb πs) isin Π(Γ) πs ge πfb

s (Γ)

2 For any (πb πs) isin Πlowastfb(Γ) and any ε gt 0 there is τ isin Tfb with Π(Γ τ) sub Bε(πb πs)

The payoff set characterized in Theorem 3 corresponds to the triangle ABE in Figure 1Part 2 of the theorem establishes ldquoapproximately unique implementationrdquo for any point inthe triangle there is an information structure under which all equilibria yield payoff pairsin a neighborhood of that point

Here is the idea behind part 1 of Theorem 3 When Buyer is fully informed an informa-tion structure can be viewed as dividing vrsquos prior distribution micro into a set of microi that aver-age to micro with Seller informed of which microi she faces Theorem 3 is proven by establishingthat we can divide micro suitably so that against each microi Seller is indifferent between pricingat all prices in the support of microi including the price corresponding to πfb

s in that environ-ment We call such a microi a (Seller-price) incentive compatible distribution or ICD (Bergemannet al (2015) call their analogous construct an extreme market) The Appendix provides aldquogreedyrdquo algorithm to compute ICDs19

We can sketch how the algorithm works and construct a set of ICDs that average to theprior Suppose V = v1 v2 vK with vi lt vi+1 for i isin 1 K minus 1 and c(v) lt v forall v Given any small-enough mass of vK there is a unique mass of type vKminus1 that makesSeller indifferent between charging price vK and vKminus1 (If the mass is too low Seller prefersvK if it is too high she prefers vKminus1) Iterating down to keep Seller indifferent betweenall prices pins down an ICD Choose the maximum mass of type vK for which this worksRemove that ICD and repeat

19 The algorithm is defined for finite V and we take limits to handle the infinite case

17

Crucially whenever an ICD is removed the price corresponding to πfbs (Γ) remains op-

timal in the remaining ldquomarketrdquo this follows from the ICDrsquos defining property of Sellerindifference and an accounting identity Therefore Sellerrsquos profit in this segmentation ofICDs remains πfb

s (Γ) Moreover it is also optimal for Seller to always (ie for each microi) priceso that there is full trade or no trade Hence Buyerrsquos expected payoff can be either 0 or theentire surplus less πfb

s (Γ) It follows that the fully-informed Buyer information structuredefined by this set of ICDs implements point B and C in Figure 1 The entire triangle ABC

can then be implemented by convexification randomizing over this information structure(and two equilibria) and full information (where Seller obtains all the surplus)

The above logic of incentive compatible distributions reduces to that of Bergemann et al(2015) in the context of monopoly pricing Even in the monopoly pricing case Theorem 3part 2 strengthens the conclusions of Bergemann et al (2015) by ensuring approximatelyunique implementation

Remark 4 Theorems 1ndash3 imply that fully-informed-Buyer information structures imple-ment all implementable payoff pairs if and only if πfb

s (Γ) = πs(Γ) In that case trianglesAFG and ABC coincide in Figure 1 It follows that πfb

s (Γ) = πs(Γ) only when a fully-informed Buyer and uninformed Seller can result in full trade (v ge E[c(v)] and Sellerprices at v) or no trade (v le E[c(v)] and Seller prices at some p ge v) Interestinglywhen πfb

s (Γ) gt πs(Γ) fully-informed-Buyer information structures cannot even implementall payoff pairs implementable by uninformed-Seller information structures ie trianglesABC and ADE in Figure 1 are distinct That is when πfb

s (Γ) gt πs(Γ) there is an uninformed-Seller information structure that implements some πs lt πfb

s (Γ)20 Theorems 2ndash3 further im-ply in this case that Buyer can obtain a strictly higher payoff when he is not fully informed(and Seller is uninformed)

4 Discussion

41 Multidimensionality

Suppose Buyer and Sellerrsquos cost and valuation pair (c v) is a two-dimensional randomvariable distributed according to joint distribution micro with a compact support The ex-tension of our maintained assumption of commonly known gains from trade is for all

20 Pick any pprime gt v such that pprimeminusE[c(v)] lt πfbs (Γ) Following the construction described after Theorem 2 we

can mix all valuations v le pprime with a fraction λ gt 0 of valuations v gt pprime so that the mixture has posterior meanexactly pprime The remaining fraction 1 minus λ of valuations above pprime are revealed to Buyer With this uninformed-Seller information structure consider any equilibrium in which Buyer purchases when indifferent (Suchan equilibrium with price-independent belief exists) Sellerrsquos profit is at most (1 minus λ)πfb

s (Γ) from any pricep gt pprime and pprime minus E[c(v)] from price p = pprime Hence Sellerrsquos profit is strictly less than πfb

s (Γ)

18

(c v) isin Supp(micro) v ge c and E[v minus c] gt 0 An information structure is now a joint distribu-tion P (tb ts v c) whose marginal distribution on (v c) is micro

We claim that the content of our key results Theorem 1Theorem 1lowast Theorem 2 andTheorem 3 still hold To see why let v be the lowest valuation in the support of micro Sellerrsquosindividual rationality constraint is now maxv minus E[c] 0 as she can guarantee this by set-ting either a sufficiently high price or a price (arbitrarily close to) v regardless of her signalWe can define a cost function c(vprime) equiv Emicro[c|v = vprime] ge vprime This results in an environment sat-isfying all the maintained assumptions of our baseline model except that c(middot) may not becontinuous Recall that such continuity is not needed for proving Theorem 1 Theorem 1lowastBoth Theorem 2 and Theorem 3 use continuity of c(middot) to guarantee that Eν [c(v)] is a contin-uous function of ν isin ∆(V ) for a key convergence result However in the two-dimensionaltype environment Eν [c] is still a continuous function of ν isin ∆(V times C) Note that Theo-rem 3 uses upper semi-continuity of Sellerrsquos profit in price boundedness of c and c le v issufficient for such upper semi-continuity

42 Negative Trading Surplus

Returning to our baseline model we next discuss cases in which trade sometimes gen-erates negative surplus That is we drop the assumption that c(v) le v we do not requireE[v minus c(v)] gt 0 either Define Sλ(Γ) for λ isin [1infin) as

Sλ(Γ) equiv983133 v

v

[v minus c(v) + λ(v minus v)]+ micro(dv)

where [middot]+ equiv max middot 0 It is readily verified that S1(Γ) = E [[v minus c(v)]+] and limλrarr0

Sλ(Γ)λ =

E[v] minus v Allowing negative trading surplus does not affect our definition of wPBE Sothe notation Π(Γ) and πs(Γ) still have the same meanings as before The next propositionshows that Π(Γ) is now characterized by three constraints as before the two individualrationality constraints πs ge πs(Γ) and πb ge 0 and different now a Pareto frontier definedby all Sλ(Γ)

Proposition 1 Consider all information structures and equilibria when trade can generate negativesurplus

Π(Γ) =

983099983105983103

983105983101

πb ge 0

(πb πs) πs ge πs(Γ)

λπb + πs le Sλ(Γ) forallλ ge 1

983100983105983104

983105983102

Proposition 1 shows that when the trade can be inefficient it may be impossible to lower

19

Sellerrsquos profit without losing total surplus (ie πb + πs) This result is intuitive at the ex-tremes Suppose for simplicity v gt E[c(v)] If πs = πs(Γ) then since πs(Γ) = v minus E[c(v)]Seller must find it optimal to price at v and trade with probability one including whentrade generates negative surplus So Buyerrsquos payoff is E[v]minus v = limλrarrinfin Sλ(Γ)λ and totalsurplus is E[v minus c(v)] which includes the maximum amount of inefficient trade On theother hand an information structure that publicly reveals (only) whether trade is efficient(ie whether v ge c(v)) can implement efficient trade with all the surplus accruing to SellerIn this case Seller obtains S1(Γ) while Buyer obtains zero The proposition establishes thatin between S1(Γ) and πs(Γ) information structures and equilibria that lower Sellerrsquos profitmust reduce total surplus as the Pareto frontier Sλ(Γ) is a concave function with maximalslope minus1

The function capturing the Pareto frontier Sλ(Γ) is the maximal weighted sum of Buyerand Seller payoff assuming that Seller will trade at the low price p = v whenever it createspositive weighted total payoff Since we place higher weight on buyer (λ ge 1) S1(Γ) isthe upper bound for the weighted total payoff In the proof of Proposition 1 we showthat these bounds are tight mdash any payoff pair on the frontier can be implemented by someinformation structure The key observation is that whenever v minus c(v) + λ(v minus v) lt 0 itfollows that vminus c(v) lt (1minusλ)(vminus v) le 0 That is it is incentive compatible for Seller to notsell when it is common knowledge that v creates negative weighted total payoff Thereforewe can simply release a public signal indicating the sign of weighted total payoff and ifpositive induce Seller to sell at v which implies all bounds Sλ(Γ)

43 Linear Cost

When Sellerrsquos cost function c(v) is linear we can explicitly characterize an informationstructure that implements Sellerrsquos minimum implementable payoff πus

s (Γ) when Seller isuninformed (or per Theorem 2 when Buyer is better informed) subject to price-independentbeliefs We note that given the discussion in Subsection 41 a linear c(v) subsumes richerenvironments in which conditional expectations are linear including Gaussian environ-ments (cf Bar-Isaac et al 2018)

Condition 1 c(v) = λv + γ for some λ isin R

Let F (v) be the cumulative distribution function (CDF) corresponding to the prior mea-sure micro Let D(micro) be the set of all distributions whose CDF G satisfies

983133

V

vdG(v) =

983133

V

vdF (v) and983133 v

v

G(s)ds le983133 v

v

F (s)ds forallv isin V

20

That is D(micro) contains all distributions that are mean-preserving-contractions (MPC) of microIt is well known that D(micro) characterizes the set of distributions of Buyer posterior meansthat can be generated by any (uninformed-Seller) information structure

Definition 2 G is an incentive compatible distribution (ICD) if

983133

sgep

(pminus c(s))G(ds) = constant ge 0 forallp isin Supp(G)

That is when Buyerrsquos (mean) valuation is distributed according to an ICD G Seller isindifferent among all prices in the support of G assuming Buyer buys when indifferent Wefocus on a special family of ICDs whose supports are intervals [vlowast v

lowast] Given Condition 1such ICDs have analytical an expression

G(v) =

983099983105983105983103

983105983105983101

1minus983061

(1minus λ)v minus γ

(1minus λ)vlowast minus γ

983062 1λminus1

if λ ∕= 1

1minus evminusvlowast

γ if λ = 1

(3)

Given any vlowast and the above G(v) an upper bound vlowast is pinned down by the conditionEG[v] = EF [v] So a family of ICDs is parametrized by a single parameter vlowast and hence anICD is denoted by CDF 983144Gvlowast(v) with density 983144gvlowast(v) It is not difficult to verify that 983144Gvlowast(v) ismonotonic in vlowast for v in the support Consequently all ICDs are ordered according to theMPC order For vlowast isin [vE[v]] when vlowast increases the ICD increases in the MPC order

Proposition 2 Suppose Condition 1 is satisfied and let plowast equiv min983153vlowast983055983055 983144Gvlowast isin D(micro)

983154 It holds that

πuss (Γ) = plowast minus Emicro[c(v)]

Proposition 2 states that an uninformed Sellerrsquos minimum payoff (with price-independentbeliefs) is characterized by the ICDs that are MPCs of the prior distribution micro By defini-tion of ICDs Sellerrsquos profit when facing ICD 983144Gvlowast is vlowast minus E[v] Sellerrsquos minimum payoff isimplemented by the most dispersed ICD that is a MPC of the prior distribution

In proving Proposition 2 the key step is to show that given the prior micro garbling Buyerrsquosinformation such that the posterior mean distribution is an ICD makes Seller weakly worseoff It is then without loss to only consider ICDs to implement πus

s (Γ) We can thus workwith a tractable one-dimensional problem Consider the set D(micro) Suppose we find G(v)

such that (i) G isin D(micro) (ii) G is an ICD and (iii) existp such that983125 p

vG(s)ds =

983125 p

vF (s)ds Such

a G is the most-dispersed ICD that is a mean preserving contraction of F Consider the

21

following two manipulations

λ

983133 v

p

F (s)ds =

983133 v

p

(pminus c(s))dF (s)minus (1minus F (v)) (pminus c(v)) + λ(v minus p) (4)

λ

983133 v

p

G(s)ds =

983133 v

p

(pminus c(s))dG(s)minus (1minusG(v)) (pminus c(v)) + λ(v minus p) (5)

First by property (iii) above the LHS of Equation 4 equals the LHS of Equation 5 Secondsince

983125 v

p(F (s) minus G(s))ds is minimized at v this implies F (p) le G(p) Therefore the first

term on RHS of Equation 4 must be larger than that of Equation 5 Notice that the firstterm on RHS is Sellerrsquos profit when offering price p This implies that the profit fromoffering p given Buyer mean-valuation distribution G(v) is lower than that given F (v)which is lower than Sellerrsquos maximum profit given F (v) On the other hand by the ICDproperty p is already Sellerrsquos optimal price given G(v) The optimal profit given valuationdistribution F must be no lower than that of G Therefore we have proved that to minimizeSellerrsquos profit it is without loss to consider only ICDs within the set D(micro) which is a onedimensional subspace The detailed proof that such ICDs exist is relegated to the appendix

The monopoly-pricing environment with c(middot) = v is covered by Condition 1 with λ = 0

and γ = v The distribution G in (3) then reduces to that identified by Roesler and Szentes(2017)

If the prior micro has binary support then the cost function c(v) is linear In this special casewe can solve for πus

s (Γ) explicitly

Corollary 1 Suppose micro has binary support V = v1 v2 Let λ = c(v2)minusc(v1)v2minusv1

and let p be theunique solution to

(pminus c(p))1

λminus1 (pminus E[c(v)]) = (v2 minus c(v2))λ

λminus1 (6)

Then πuss (Γ) = max p v2minus E[c(v)]

References

AKERLOF G A (1970) ldquoThe Market for rdquoLemonsrdquo Quality Uncertainty and the MarketMechanismrdquo Quarterly Journal of Economics 84 488ndash500

BAGWELL K AND M H RIORDAN (1991) ldquoHigh and Declining Prices Signal ProductQualityrdquo American Economic Review 81 224ndash239

BAR-ISAAC H I JEWITT AND C LEAVER (2018) ldquoAdverse Selection Efficiency and theStructure of Informationrdquo Unpublished

22

BERGEMANN D B BROOKS AND S MORRIS (2015) ldquoThe Limits of Price Discrimina-tionrdquo American Economic Review 105 921ndash57

BERGEMANN D AND S MORRIS (2016) ldquoBayes Correlated Equilibrium and the Compar-ison of Information Structures in Gamesrdquo Theoretical Economics 11 487ndash522

BLACKWELL D (1953) ldquoEquivalent Comparisons of Experimentsrdquo Annals of MathematicalStatistics 24 265ndash272

CHO I-K AND D KREPS (1987) ldquoSignaling Games and Stable Equilibriardquo Quarterly Jour-nal of Economics 102 179ndash221

DU S (2018) ldquoRobust Mechanisms Under Common Valuationrdquo Econometrica 86 1569ndash1588

DURRETT R (1995) Probability Theory and Examples Duxbury Press 2nd ed

EINAV L AND A FINKELSTEIN (2011) ldquoSelection in Insurance Markets Theory and Em-pirics in Picturesrdquo Journal of Economic Perspectives 25 115ndash38

FUDENBERG D AND J TIROLE (1991) ldquoPerfect Bayesian Equilibrium and Sequential Equi-libriumrdquo Journal of Economic Theory 53 236ndash260

GARCIA D R TEPER AND M TSUR (2018) ldquoInformation Design in Insurance MarketsSelling Peaches in a Market for Lemonsrdquo Unpublished

GREENWALD B C (1986) ldquoAdverse Selection in the Labour Marketrdquo Review of EconomicStudies 53 325ndash347

JOVANOVIC B (1982) ldquoFavorable Selection with Asymmetric Informationrdquo QuarterlyJournal of Economics 97 535ndash539

KESSLER A S (2001) ldquoRevisiting the Lemons Marketrdquo International Economic Review 4225ndash41

KREPS D AND R WILSON (1982) ldquoSequential Equilibriardquo Econometrica 50 863ndash894

LEHMANN E L (1988) ldquoComparing Location Experimentsrdquo The Annals of Statistics 16521ndash533

LEVIN J (2001) ldquoInformation and the Market for Lemonsrdquo RAND Journal of Economics32 657ndash666

23

LIBGOBER J AND X MU (2019) ldquoInformational Robustness in Intertemporal PricingrdquoWorking paper

MAKRIS M AND L RENOU (2018) ldquoInformation Design in Multi-Stage Gamesrdquo Workingpaper

MILGROM P R AND R J WEBER (1985) ldquoDistributional Strategies for Games with In-complete Informationrdquo Mathematics of Operations Research 10 619ndash632

MUSSA M AND S ROSEN (1978) ldquoMonopoly and Product Qualityrdquo Journal of EconomicTheory 18 301ndash317

RAVID D A-K ROESLER AND B SZENTES (2019) ldquoLearning Before Trading On theInefficiency of Ignoring Free Informationrdquo Working paper

ROESLER A-K AND B SZENTES (2017) ldquoBuyer-Optimal Learning and Monopoly Pric-ingrdquo American Economic Review 107 2072ndash80

TERSTIEGE S AND C WASSER (2019) ldquoBuyer-optimal Extensionproof InformationrdquoWorking paper

Appendices omitted from this version but are available on request

24

you donrsquot knowrdquo requirement (Fudenberg and Tirole 1991) frequently imposed in versionsof perfect Bayesian equilibrium and the concept of sequential equilibrium (Kreps and Wil-son 1982) in finite versions of our setting11

Some notation will be helpful Define

Πlowast(Γ τ) equiv (πb πs) exist wPBE of (Γ τ) with price-independent belief and payoffs(πb πs)

Πlowast(Γ) equiv983134

τisinT

Πlowast(Γ τ)

Πlowasti (Γ) equiv

983134

τisinTi

Πlowast(Γ τ) for i = us fbmb

So Πlowast and Πlowast are analogous to the implementable payoff sets Π and Π defined earlier butrestricted to equilibria with price-independent beliefs Πlowast

us Πlowastfb and Πlowast

mb are the imple-mentable payoff sets when further restricted to uninformed-Seller fully-informed-Buyerand more-informed-Buyer information structures Plainly for any environment Γ

Πlowastus(Γ) cupΠlowast

fb(Γ) sub Πlowastmb(Γ) sub Πlowast(Γ) sub Π(Γ)

3 Main Results

Our goal is to characterize equilibrium payoff pairs across information structures in anarbitrary environment Γ In particular we seek to characterize the five sets Π(Γ) Πlowast(Γ)Πlowast

mb(Γ) Πlowastus(Γ) and Πlowast

fb(Γ) Let

S(Γ) equiv E[v minus c(v)]

be the (expected) surplus from trade in environment Γ This quantity will play an impor-tant role

31 All Information Structures

Define Sellerrsquos payoff guarantee as

πs(Γ) equiv max v minus E [c(v)] 0

11 An example clarifying our terminology may be helpful If both Seller and Buyer are fully informed of vthen the natural equilibriummdashthe unique sequential equilibrium in a finite version of the gamemdashhas Sellerpricing at p = v and Buyerrsquos belief being degenerate on v regardless of Sellerrsquos price This equilibrium hasprice-independent belief even though ex-ante Sellerrsquos price are Buyerrsquos belief are perfectly correlated Thepoint is that Buyerrsquos belief does not depend on price conditional on his signal

10

To interpret this observe that it is optimal for Buyer to accept the price v no matter hisbelief Therefore Seller can guarantee herself the (expected) profit v minus E[c(v)] no matterwhat the information structure is More precisely she can guarantee v minus E[c(v)] minus ε forany ε gt 0 since sequential rationality requires Buyer to accept any price v minus ε SimilarlySeller can also guarantee zero profit offering a price p gt v Hence Sellerrsquos payoff in anyequilibrium of any information structure must be at least the payoff guarantee πs(Γ) above

On the other hand Buyer can guarantee himself the payoff πb = 0 by rejecting all pricesIt follows that the implementable set Π(Γ) must satisfy three simple constraints 1) Sellerrsquosldquoindividual rationalityrdquo constraint πs ge πs(Γ) 2) Buyerrsquos ldquoindividual rationalityrdquo con-straint πb ge 0 and 3) the feasibility constraint πb + πs le S(Γ)

Our first result is that these individual rationality and feasibility constraints are alsosufficient for a payoff pair to be implementable

Theorem 1 Consider all information structures and equilibria

Π(Γ) =

983099983105983103

983105983101

πb ge 0

(πb πs) πs ge πs(Γ)

πb + πs le S(Γ)

983100983105983104

983105983102

Theorem 1 says that the set Π(Γ) corresponds to the triangle AFG in Figure 1 In par-ticular Buyer can receive the entire surplus beyond Sellerrsquos payoff guarantee This is sur-prising as Seller has substantial bargaining power Note that when v le E[c(v)] an entirelyreasonable condition Sellerrsquos payoff guarantee is zero12

The proof of Theorem 1 is in fact straightforward Suppose for expositional simplicityv ge E[c(v)] Fix the trivial information structure in which neither player receives any infor-mation and consider the following family of strategy profiles Seller randomizes betweentwo prices some pl isin [vE[v]] and ph = E[v] Buyer accepts pl with probability one andaccepts ph with probability α(ph) where α(ph) isin [0 1] is specified to make Seller indifferentbetween the two prices That is α(ph)(ph minus E[c(v)]) = pl minus E[c(v)] The expected payoffsfrom this strategy profile are

πb = σ(pl)(E[v]minus pl) and πs = pl minus E[c(v)]

As pl traverses the interval [vE[v]] Sellerrsquos payoff πs traverses [πs(Γ) S(Γ)] Given any

12 In monopoly pricing with c(middot) = v Sellerrsquos payoff guarantee of zero is lower than the revenue guaranteeidentified in Du (2018 Section 5) which is typically positive Dursquos notion is different from ours

11

pl Buyerrsquos payoff πb traverses [0 S(Γ)minusπs] as σ(pl) traverses [0 1] Therefore the proposedstrategy profiles induce all the payoff pairs stated in Theorem 1

We are left to specify beliefs ν for Buyer After prices pl and ph Buyer holds the priorbelief micro After any other (necessarily off-path) price Buyerrsquos belief is that v = v and soBuyer rejects all prices p isin [vinfin)pl ph It is straightforward to confirm that the specified(σα ν) constitute a wPBE

To get more insight into this construction consider its implication for the monopolypricing with c(middot) = v The equilibrium with pl = v (hence α(ph) = 0) and σ(pl) = 1

corresponds to the monopolist deterministically pricing at v and Buyer purchasing Giventhat both sides of the market receive no information why doesnrsquot the monopolist deviate toany price in (vE[v]) The reason is that in this equilibrium the consumer will then not buybecause he updates to v = v Such belief is compatible with wPBE because the equilibriumconcept places no restrictions on off-path beliefs This may seem like a game-theoreticmisdirection Buyerrsquos belief is not consistent with ldquono signaling what you donrsquot knowrdquoPut differently since we have a (weakly) more-informed Buyer information structure weought to impose the price-independent belief condition described in Subsection 23 thatwould imply Buyer must purchase at any price p lt E[v]

But the message of Theorem 1 does not rely on the permissiveness of wPBE To illus-trate continue with the above monopoly-pricing environment and suppose v has positiveprior probability Consider Buyer remaining uninformed but Seller learning whether v = v

or v gt v Now Buyerrsquos off-path belief that v = v is consistent with ldquono signaling whatyou donrsquot knowrdquo More generally using richer information structures we can prove thatany payoff pair identified in Theorem 1 can be approximately implemented as a sequentialequilibrium (Kreps and Wilson 1982) in a suitably discretized game

Theorem 1lowast Fix any ε gt 0 There is a finite information structure and a finite price grid inducinga game with a set of sequential equilibrium payoffs that is an ε-net of Π(Γ)13

In fact the proof of Theorem 1lowast establishes even more the sequential equilibria in thediscretized games satisfy a natural version of the D1 refinement (Cho and Kreps 1987) Werelegate the logic to the Appendix but mention here that we use imperfectly-correlatedsignals for Buyer and Seller14

13 An information structure is finite if the signal spaces Tb and Ts are finite Sequential equilibrium isdefined in the obvious way for the ldquoinducedrdquo finite game where Nature directly draws (tb ts) rather thanfirst drawing v and playersrsquo payoffs are defined directly using (tb ts) For X sub R2 and ε gt 0 the set A sub Xis an ε-net of X if for each x isin X there is a isin A such that 983042xminus a983042 lt ε where 983042middot983042 is the Euclidean distance

14 The idea behind D1 is to ask for any off-path price whether one type of Seller would deviate for any

12

Even when our environment is specialized to monopoly pricing it is worth highlightingtwo contrasts between Theorem 11lowast and results of Bergemann et al (2015) and Roeslerand Szentes (2017) First we find that by not restricting the monopolist to be uninformedthe implementable payoff set typically expands rather dramatically trade can be efficientwith the monopolist securing none of the surplus beyond her payoff guarantee (πs) whichmay be zero (Roesler and Szentes establish implicitly that the implementable set withan uninformed monopolist is a superset of Bergemann et alrsquos which has a fully-informedconsumer) We will see in Subsection 32 that what is crucial to this expansion is price-dependent beliefs In particular the proof of Theorem 1lowast uses an information structure inwhich Buyer is not better informed than Seller mdash if he were then sequential equilibriumwould imply price-independent beliefs Second Theorem 1lowast establishes that for a givenε gt 0 a single information structure (and price grid) can be used to approximate the entirepayoff set Π(Γ) analogously to the construction described after Theorem 1 that used asingle information structure15 Bergemann et al (2015) and Roesler and Szentes (2017) onthe other hand vary information structures to span their payoff sets

32 More-informed Buyer and Price-independent Beliefs

In some economic settings it is plausible that Buyer is more informed than Seller Howdoes a restriction to such information structures ie τ isin Tmb affect the implementablepayoff set It turns out that what is in fact crucial is price-independent beliefs We have ex-plained earlier why it is desirable to impose this condition when Buyer is more informedbut that the condition is well defined even otherwise If Buyer is not more informed thanSeller price-independent beliefs ought to be viewed as an equilibrium restriction in gen-eral To reduce repetition readers should be bear in mind for the rest of this subsectionthe qualifier ldquowith price-independent beliefsrdquo applies unless stated explicitly otherwise

It is useful to define

πuss (Γ) equiv inf πs exist(πb πs) isin Πlowast

us(Γ) (1)

as the lowest payoff that an uninformed Seller can obtain no matter Buyerrsquos information(among equilibria with price-independent beliefs) Plainly πus

s (Γ) ge πs(Γ) In monopoly

Buyer mixed response that another type would Our construction has multiple Buyer types that are imper-fectly correlated with Seller types So different types of Seller have different beliefs about Buyer types Thisblunts the power of dominance considerations to the point where D1 does not exclude any Seller type fromthe support of Buyerrsquos off-path belief

15 In fact if one lets the price grid vary with ε then a single information structure implements exactlyrather than approximately in sequential equilibrium all payoffs ε-away from the boundary of Π(Γ) See in the Appendix for a formal statement

13

pricing with V = [v v] and c(middot) = v Roesler and Szentesrsquos (2017) characterization of theconsumer-optimal information structure identifies πus

s establishing that πuss gt πs If there

is no trade due to adverse selection when Seller is uniformed and Buyer has some informa-tion then πus

s = πs = 0 We do not have a general explicit formula for πuss Subsection 43

provides it for linear c(middot) Nonetheless we establish next that (i) the only additional restric-tion on equilibrium payoffs imposed by price-independent beliefs is a lower bound of πus

s

for Seller and (ii) uninformed-Seller information structures implement all such payoffs

Theorem 2 Consider equilibria with price-independent beliefs

1 Πlowast(Γ) = Πlowastmb(Γ) = Πlowast

us(Γ)

2 Πlowastus(Γ) = (πb πs) isin Π(Γ) πs ge πus

s (Γ)

3 For any (πb πs) isin Πlowastus(Γ) with πs gt πus

s (Γ) there is τ isin Tus with Π(Γ τ) = (πb πs)

Remark 1 We believe the substance of Theorem 2 would hold using discretizations and se-quential equilibria analogous to Theorem 1lowast As previously noted sequential equilibriumimplies price-independent beliefs when Buyer is more informed than Seller

To digest Theorem 2 note that Πlowast(Γ) sup Πlowastmb(Γ) sup Πlowast

us(Γ) is trivial So part 1 of thetheorem amounts to establishing the reverse inclusions The intuition for thismdashgiven part2rsquos characterization of Πlowast

usmdashis fairly straightforward with price-independent beliefs ad-ditional information cannot harm Seller even though it could alter the set of equilibria SoSellerrsquos lowest payoff obtains when she is uninformed

The characterization in part 2 of payoffs with an uninformed Seller corresponds to thetriangle ADE in Figure 1 Part 3 of the theorem assures ldquounique implementationrdquo of allimplementable payoffs satisfying πs gt πus

s (Γ) That is for any such payoff pair there is anuninformed-Seller information structure such that all equilibria (with price-independentbeliefs) induce exactly that payoff pair Unique implementation is appealing for multiplereasons one of which is that it obviates concerns about which among multiple payoff-distinct equilibria is more reasonable For example Ravid Roesler and Szentes (2019)bring to the fore the issue of equilibrium multiplicity in Roesler and Szentesrsquos (2017) infor-mation structure

Let us describe how we obtain the characterization of Πlowastus(Γ) and unique implementa-

tion There are two steps The first ensures that there is some information structure callit τ lowast isin Tus that implements Sellerrsquos payoff πus

s (Γ) That is we ensure that the infimum

14

in (1) is in fact a minimum16 While this argument is technical knowing τ lowast exists is usefulin what follows The second and economically insightful step is to construct informa-tion structures that implement every point in the triangle Πlowast

us(Γ) by suitably garbling theinformation structure τ lowast The construction is illustrated in Figure 2 Consider the distri-bution of Buyerrsquos posterior mean of his valuation v in information structure τ lowast (Givenprice-independent beliefs Buyerrsquos posterior mean is sufficient for his decision) For sim-plicity suppose this posterior-mean distribution has a density as depicted by the red curvein Figure 2 Fix any (πb πs) isin Πlowast

us(Γ)

First there is some number zlowast such that πb + πs is the total surplus from trading onlywhen Buyerrsquos posterior mean is greater than zlowast Next there is some price plowast ge zlowast suchthat Sellerrsquos payoff is πs if all these trades were to occur at price plowast17 Note that plowast must beno larger than the expected Buyer posterior mean conditional on that being above zlowast forotherwise πb lt 0 We claim that the information structure τ lowast can be garbled so that plowast isan equilibrium price and trade occurs only when Buyerrsquos posterior mean is greater thanzlowast The garbling is illustrated in Figure 2 by the blue distribution There is one signal that

pzv

ν(v)

Figure 2 Construction of information structure in Theorem 2

Buyer receives when the original posterior mean is between zlowast and plowast and also receiveswith some probability when the original posterior mean is above plowast The probability ischosen to make the posterior mean from this signal exactly plowast Apart from this one newsignal Buyer receives the original signal in τ lowast Plainly this is a garbling of τ lowast and hence isfeasible

16 The difficulty is in establishing suitable continuity Uninformed-Seller information structures can beviewed as probability measures over Buyerrsquos beliefs with convergence in the sense of the weak topologyThis topology ensures continuity with respect to probability measures of expectations of continuous orat least Lipschitz (and bounded) functions However Sellerrsquos expected payoff is not the expectation of aLipschitz function as Sellerrsquos profit is truncated at the price she charges

17 That plowast ge zlowast follows from πs ge πuss (Γ) as πs(Γ) itself is weakly larger than Sellerrsquos payoff from posting

price zlowast (and thus trading with the same set of Buyer posterior means) under information structure τlowast

15

Figure 2 makes clear why the new information structure has an equilibrium with priceplowast and Buyer breaking indifference in favor of trading (i) Sellerrsquos profit from posting anyprice below zlowast is the same as under τ lowast and hence no larger than πus

s (Γ) (ii) similarly Sellerrsquosprofit from posting any price above plowast is no higher than some fraction of πus

s (Γ) and (iii)any price between zlowast and plowast is worse that price plowast Moreover since Sellerrsquos profit fromoffering any price other than plowast is no more than πus

s (Γ) it follows that when πs gt πuss (Γ)

Buyer must break indifference as specified for Seller to have an optimal price and theequilibrium payoffs are unique

Remark 2 The above logic establishes that given any τ isin Tus that implements some (πb πs)τ can be garbled to uniquely implement any (πprime

b πprimes) isin Π(Γ) such that πprime

s gt πs That isan uninformed Sellerrsquos payoff can always be strictly raised and Buyerrsquos payoff reduced(strictly so long as it was not already zero) by garbling Buyerrsquos information

Remark 3 According to Theorem 11lowast and Theorem 2 uninformed-Seller information struc-tures cannot implement all implementable payoff pairs in an environment Γ if and only ifπuss (Γ) gt πs(Γ) This inequality fails in environment Γ if πus

s (Γ) = 0 since that impliesπuss (Γ) = πs(Γ) = 0 An example is when there is no trade due to adverse selection when

Seller is uniformed and Buyer has some information On the other hand it is sufficient forthe inequality that

v le E[c(v)] and forallv isin V c(v) lt v (2)

To see why (2) implies πuss (Γ) gt πs(Γ) notice that in any uninformed-Seller information

structure Seller can price at slightly less than Buyerrsquos highest posterior mean valuationand guarantee trade with only (a neighborhood of) that Buyer type If c(v) lt v for all v thisgives Seller a positive expected payoff and hence πus

s (Γ) gt 018 But the first inequality in (2)is equivalent to πs(Γ) = 0 We observe that Condition (2) is compatible with severe adverseselection resulting in very little trade when Seller is uninformed and Buyer (partially orfully) informed

33 Fully-Informed Buyer

We now turn to the third canonical class of information structures Buyer is fully in-formed of his value v As this is a special case of more-informed Buyer we maintain price-independent beliefs throughout this subsection

18 More precisely as V is compact c(v) lt v for all v implies there exists ε gt 0 such that v minus c(v) gt εGiven any uninformed-Seller information structure let mv be the highest posterior mean valuation in thesupport of the posterior means induced by Buyerrsquos signals So there is positive probability of Buyer signalswith posterior mean valuations at least mv minus ε2 By pricing at mv minus ε2 Sellerrsquos expected cost conditionalon trade is bounded above by mv minus ε and hence Sellerrsquos profit conditional on trade is at least ε2 gt 0 Itfollows that πus

s gt 0

16

Faced with a fully informed Buyer and any sequentially rational Buyer strategy an un-informed Seller can guarantee a profit level

πfbs (Γ) equiv sup

p

983133 v

p

(pminus c(v))micro(dv)

regardless of her information Plainly πfbs (Γ) ge πus

s (Γ) In monopoly pricing with c(middot) = vRoesler and Szentes (2017) have shown that πfb

s gt πuss if there is no trade due to adverse

selection when Buyer is fully informed and Seller is uniformed then πfbs = πus

s = 0 Belowwe establish that when Buyer is fully informed πfb

s is the only additional constraint onequilibrium payoffs Let Bε(πb πs) denote an ε-ball around (πb πs)

Theorem 3 Consider a fully informed Buyer and price-independent beliefs

1 Πlowastfb(Γ) = (πb πs) isin Π(Γ) πs ge πfb

s (Γ)

2 For any (πb πs) isin Πlowastfb(Γ) and any ε gt 0 there is τ isin Tfb with Π(Γ τ) sub Bε(πb πs)

The payoff set characterized in Theorem 3 corresponds to the triangle ABE in Figure 1Part 2 of the theorem establishes ldquoapproximately unique implementationrdquo for any point inthe triangle there is an information structure under which all equilibria yield payoff pairsin a neighborhood of that point

Here is the idea behind part 1 of Theorem 3 When Buyer is fully informed an informa-tion structure can be viewed as dividing vrsquos prior distribution micro into a set of microi that aver-age to micro with Seller informed of which microi she faces Theorem 3 is proven by establishingthat we can divide micro suitably so that against each microi Seller is indifferent between pricingat all prices in the support of microi including the price corresponding to πfb

s in that environ-ment We call such a microi a (Seller-price) incentive compatible distribution or ICD (Bergemannet al (2015) call their analogous construct an extreme market) The Appendix provides aldquogreedyrdquo algorithm to compute ICDs19

We can sketch how the algorithm works and construct a set of ICDs that average to theprior Suppose V = v1 v2 vK with vi lt vi+1 for i isin 1 K minus 1 and c(v) lt v forall v Given any small-enough mass of vK there is a unique mass of type vKminus1 that makesSeller indifferent between charging price vK and vKminus1 (If the mass is too low Seller prefersvK if it is too high she prefers vKminus1) Iterating down to keep Seller indifferent betweenall prices pins down an ICD Choose the maximum mass of type vK for which this worksRemove that ICD and repeat

19 The algorithm is defined for finite V and we take limits to handle the infinite case

17

Crucially whenever an ICD is removed the price corresponding to πfbs (Γ) remains op-

timal in the remaining ldquomarketrdquo this follows from the ICDrsquos defining property of Sellerindifference and an accounting identity Therefore Sellerrsquos profit in this segmentation ofICDs remains πfb

s (Γ) Moreover it is also optimal for Seller to always (ie for each microi) priceso that there is full trade or no trade Hence Buyerrsquos expected payoff can be either 0 or theentire surplus less πfb

s (Γ) It follows that the fully-informed Buyer information structuredefined by this set of ICDs implements point B and C in Figure 1 The entire triangle ABC

can then be implemented by convexification randomizing over this information structure(and two equilibria) and full information (where Seller obtains all the surplus)

The above logic of incentive compatible distributions reduces to that of Bergemann et al(2015) in the context of monopoly pricing Even in the monopoly pricing case Theorem 3part 2 strengthens the conclusions of Bergemann et al (2015) by ensuring approximatelyunique implementation

Remark 4 Theorems 1ndash3 imply that fully-informed-Buyer information structures imple-ment all implementable payoff pairs if and only if πfb

s (Γ) = πs(Γ) In that case trianglesAFG and ABC coincide in Figure 1 It follows that πfb

s (Γ) = πs(Γ) only when a fully-informed Buyer and uninformed Seller can result in full trade (v ge E[c(v)] and Sellerprices at v) or no trade (v le E[c(v)] and Seller prices at some p ge v) Interestinglywhen πfb

s (Γ) gt πs(Γ) fully-informed-Buyer information structures cannot even implementall payoff pairs implementable by uninformed-Seller information structures ie trianglesABC and ADE in Figure 1 are distinct That is when πfb

s (Γ) gt πs(Γ) there is an uninformed-Seller information structure that implements some πs lt πfb

s (Γ)20 Theorems 2ndash3 further im-ply in this case that Buyer can obtain a strictly higher payoff when he is not fully informed(and Seller is uninformed)

4 Discussion

41 Multidimensionality

Suppose Buyer and Sellerrsquos cost and valuation pair (c v) is a two-dimensional randomvariable distributed according to joint distribution micro with a compact support The ex-tension of our maintained assumption of commonly known gains from trade is for all

20 Pick any pprime gt v such that pprimeminusE[c(v)] lt πfbs (Γ) Following the construction described after Theorem 2 we

can mix all valuations v le pprime with a fraction λ gt 0 of valuations v gt pprime so that the mixture has posterior meanexactly pprime The remaining fraction 1 minus λ of valuations above pprime are revealed to Buyer With this uninformed-Seller information structure consider any equilibrium in which Buyer purchases when indifferent (Suchan equilibrium with price-independent belief exists) Sellerrsquos profit is at most (1 minus λ)πfb

s (Γ) from any pricep gt pprime and pprime minus E[c(v)] from price p = pprime Hence Sellerrsquos profit is strictly less than πfb

s (Γ)

18

(c v) isin Supp(micro) v ge c and E[v minus c] gt 0 An information structure is now a joint distribu-tion P (tb ts v c) whose marginal distribution on (v c) is micro

We claim that the content of our key results Theorem 1Theorem 1lowast Theorem 2 andTheorem 3 still hold To see why let v be the lowest valuation in the support of micro Sellerrsquosindividual rationality constraint is now maxv minus E[c] 0 as she can guarantee this by set-ting either a sufficiently high price or a price (arbitrarily close to) v regardless of her signalWe can define a cost function c(vprime) equiv Emicro[c|v = vprime] ge vprime This results in an environment sat-isfying all the maintained assumptions of our baseline model except that c(middot) may not becontinuous Recall that such continuity is not needed for proving Theorem 1 Theorem 1lowastBoth Theorem 2 and Theorem 3 use continuity of c(middot) to guarantee that Eν [c(v)] is a contin-uous function of ν isin ∆(V ) for a key convergence result However in the two-dimensionaltype environment Eν [c] is still a continuous function of ν isin ∆(V times C) Note that Theo-rem 3 uses upper semi-continuity of Sellerrsquos profit in price boundedness of c and c le v issufficient for such upper semi-continuity

42 Negative Trading Surplus

Returning to our baseline model we next discuss cases in which trade sometimes gen-erates negative surplus That is we drop the assumption that c(v) le v we do not requireE[v minus c(v)] gt 0 either Define Sλ(Γ) for λ isin [1infin) as

Sλ(Γ) equiv983133 v

v

[v minus c(v) + λ(v minus v)]+ micro(dv)

where [middot]+ equiv max middot 0 It is readily verified that S1(Γ) = E [[v minus c(v)]+] and limλrarr0

Sλ(Γ)λ =

E[v] minus v Allowing negative trading surplus does not affect our definition of wPBE Sothe notation Π(Γ) and πs(Γ) still have the same meanings as before The next propositionshows that Π(Γ) is now characterized by three constraints as before the two individualrationality constraints πs ge πs(Γ) and πb ge 0 and different now a Pareto frontier definedby all Sλ(Γ)

Proposition 1 Consider all information structures and equilibria when trade can generate negativesurplus

Π(Γ) =

983099983105983103

983105983101

πb ge 0

(πb πs) πs ge πs(Γ)

λπb + πs le Sλ(Γ) forallλ ge 1

983100983105983104

983105983102

Proposition 1 shows that when the trade can be inefficient it may be impossible to lower

19

Sellerrsquos profit without losing total surplus (ie πb + πs) This result is intuitive at the ex-tremes Suppose for simplicity v gt E[c(v)] If πs = πs(Γ) then since πs(Γ) = v minus E[c(v)]Seller must find it optimal to price at v and trade with probability one including whentrade generates negative surplus So Buyerrsquos payoff is E[v]minus v = limλrarrinfin Sλ(Γ)λ and totalsurplus is E[v minus c(v)] which includes the maximum amount of inefficient trade On theother hand an information structure that publicly reveals (only) whether trade is efficient(ie whether v ge c(v)) can implement efficient trade with all the surplus accruing to SellerIn this case Seller obtains S1(Γ) while Buyer obtains zero The proposition establishes thatin between S1(Γ) and πs(Γ) information structures and equilibria that lower Sellerrsquos profitmust reduce total surplus as the Pareto frontier Sλ(Γ) is a concave function with maximalslope minus1

The function capturing the Pareto frontier Sλ(Γ) is the maximal weighted sum of Buyerand Seller payoff assuming that Seller will trade at the low price p = v whenever it createspositive weighted total payoff Since we place higher weight on buyer (λ ge 1) S1(Γ) isthe upper bound for the weighted total payoff In the proof of Proposition 1 we showthat these bounds are tight mdash any payoff pair on the frontier can be implemented by someinformation structure The key observation is that whenever v minus c(v) + λ(v minus v) lt 0 itfollows that vminus c(v) lt (1minusλ)(vminus v) le 0 That is it is incentive compatible for Seller to notsell when it is common knowledge that v creates negative weighted total payoff Thereforewe can simply release a public signal indicating the sign of weighted total payoff and ifpositive induce Seller to sell at v which implies all bounds Sλ(Γ)

43 Linear Cost

When Sellerrsquos cost function c(v) is linear we can explicitly characterize an informationstructure that implements Sellerrsquos minimum implementable payoff πus

s (Γ) when Seller isuninformed (or per Theorem 2 when Buyer is better informed) subject to price-independentbeliefs We note that given the discussion in Subsection 41 a linear c(v) subsumes richerenvironments in which conditional expectations are linear including Gaussian environ-ments (cf Bar-Isaac et al 2018)

Condition 1 c(v) = λv + γ for some λ isin R

Let F (v) be the cumulative distribution function (CDF) corresponding to the prior mea-sure micro Let D(micro) be the set of all distributions whose CDF G satisfies

983133

V

vdG(v) =

983133

V

vdF (v) and983133 v

v

G(s)ds le983133 v

v

F (s)ds forallv isin V

20

That is D(micro) contains all distributions that are mean-preserving-contractions (MPC) of microIt is well known that D(micro) characterizes the set of distributions of Buyer posterior meansthat can be generated by any (uninformed-Seller) information structure

Definition 2 G is an incentive compatible distribution (ICD) if

983133

sgep

(pminus c(s))G(ds) = constant ge 0 forallp isin Supp(G)

That is when Buyerrsquos (mean) valuation is distributed according to an ICD G Seller isindifferent among all prices in the support of G assuming Buyer buys when indifferent Wefocus on a special family of ICDs whose supports are intervals [vlowast v

lowast] Given Condition 1such ICDs have analytical an expression

G(v) =

983099983105983105983103

983105983105983101

1minus983061

(1minus λ)v minus γ

(1minus λ)vlowast minus γ

983062 1λminus1

if λ ∕= 1

1minus evminusvlowast

γ if λ = 1

(3)

Given any vlowast and the above G(v) an upper bound vlowast is pinned down by the conditionEG[v] = EF [v] So a family of ICDs is parametrized by a single parameter vlowast and hence anICD is denoted by CDF 983144Gvlowast(v) with density 983144gvlowast(v) It is not difficult to verify that 983144Gvlowast(v) ismonotonic in vlowast for v in the support Consequently all ICDs are ordered according to theMPC order For vlowast isin [vE[v]] when vlowast increases the ICD increases in the MPC order

Proposition 2 Suppose Condition 1 is satisfied and let plowast equiv min983153vlowast983055983055 983144Gvlowast isin D(micro)

983154 It holds that

πuss (Γ) = plowast minus Emicro[c(v)]

Proposition 2 states that an uninformed Sellerrsquos minimum payoff (with price-independentbeliefs) is characterized by the ICDs that are MPCs of the prior distribution micro By defini-tion of ICDs Sellerrsquos profit when facing ICD 983144Gvlowast is vlowast minus E[v] Sellerrsquos minimum payoff isimplemented by the most dispersed ICD that is a MPC of the prior distribution

In proving Proposition 2 the key step is to show that given the prior micro garbling Buyerrsquosinformation such that the posterior mean distribution is an ICD makes Seller weakly worseoff It is then without loss to only consider ICDs to implement πus

s (Γ) We can thus workwith a tractable one-dimensional problem Consider the set D(micro) Suppose we find G(v)

such that (i) G isin D(micro) (ii) G is an ICD and (iii) existp such that983125 p

vG(s)ds =

983125 p

vF (s)ds Such

a G is the most-dispersed ICD that is a mean preserving contraction of F Consider the

21

following two manipulations

λ

983133 v

p

F (s)ds =

983133 v

p

(pminus c(s))dF (s)minus (1minus F (v)) (pminus c(v)) + λ(v minus p) (4)

λ

983133 v

p

G(s)ds =

983133 v

p

(pminus c(s))dG(s)minus (1minusG(v)) (pminus c(v)) + λ(v minus p) (5)

First by property (iii) above the LHS of Equation 4 equals the LHS of Equation 5 Secondsince

983125 v

p(F (s) minus G(s))ds is minimized at v this implies F (p) le G(p) Therefore the first

term on RHS of Equation 4 must be larger than that of Equation 5 Notice that the firstterm on RHS is Sellerrsquos profit when offering price p This implies that the profit fromoffering p given Buyer mean-valuation distribution G(v) is lower than that given F (v)which is lower than Sellerrsquos maximum profit given F (v) On the other hand by the ICDproperty p is already Sellerrsquos optimal price given G(v) The optimal profit given valuationdistribution F must be no lower than that of G Therefore we have proved that to minimizeSellerrsquos profit it is without loss to consider only ICDs within the set D(micro) which is a onedimensional subspace The detailed proof that such ICDs exist is relegated to the appendix

The monopoly-pricing environment with c(middot) = v is covered by Condition 1 with λ = 0

and γ = v The distribution G in (3) then reduces to that identified by Roesler and Szentes(2017)

If the prior micro has binary support then the cost function c(v) is linear In this special casewe can solve for πus

s (Γ) explicitly

Corollary 1 Suppose micro has binary support V = v1 v2 Let λ = c(v2)minusc(v1)v2minusv1

and let p be theunique solution to

(pminus c(p))1

λminus1 (pminus E[c(v)]) = (v2 minus c(v2))λ

λminus1 (6)

Then πuss (Γ) = max p v2minus E[c(v)]

References

AKERLOF G A (1970) ldquoThe Market for rdquoLemonsrdquo Quality Uncertainty and the MarketMechanismrdquo Quarterly Journal of Economics 84 488ndash500

BAGWELL K AND M H RIORDAN (1991) ldquoHigh and Declining Prices Signal ProductQualityrdquo American Economic Review 81 224ndash239

BAR-ISAAC H I JEWITT AND C LEAVER (2018) ldquoAdverse Selection Efficiency and theStructure of Informationrdquo Unpublished

22

BERGEMANN D B BROOKS AND S MORRIS (2015) ldquoThe Limits of Price Discrimina-tionrdquo American Economic Review 105 921ndash57

BERGEMANN D AND S MORRIS (2016) ldquoBayes Correlated Equilibrium and the Compar-ison of Information Structures in Gamesrdquo Theoretical Economics 11 487ndash522

BLACKWELL D (1953) ldquoEquivalent Comparisons of Experimentsrdquo Annals of MathematicalStatistics 24 265ndash272

CHO I-K AND D KREPS (1987) ldquoSignaling Games and Stable Equilibriardquo Quarterly Jour-nal of Economics 102 179ndash221

DU S (2018) ldquoRobust Mechanisms Under Common Valuationrdquo Econometrica 86 1569ndash1588

DURRETT R (1995) Probability Theory and Examples Duxbury Press 2nd ed

EINAV L AND A FINKELSTEIN (2011) ldquoSelection in Insurance Markets Theory and Em-pirics in Picturesrdquo Journal of Economic Perspectives 25 115ndash38

FUDENBERG D AND J TIROLE (1991) ldquoPerfect Bayesian Equilibrium and Sequential Equi-libriumrdquo Journal of Economic Theory 53 236ndash260

GARCIA D R TEPER AND M TSUR (2018) ldquoInformation Design in Insurance MarketsSelling Peaches in a Market for Lemonsrdquo Unpublished

GREENWALD B C (1986) ldquoAdverse Selection in the Labour Marketrdquo Review of EconomicStudies 53 325ndash347

JOVANOVIC B (1982) ldquoFavorable Selection with Asymmetric Informationrdquo QuarterlyJournal of Economics 97 535ndash539

KESSLER A S (2001) ldquoRevisiting the Lemons Marketrdquo International Economic Review 4225ndash41

KREPS D AND R WILSON (1982) ldquoSequential Equilibriardquo Econometrica 50 863ndash894

LEHMANN E L (1988) ldquoComparing Location Experimentsrdquo The Annals of Statistics 16521ndash533

LEVIN J (2001) ldquoInformation and the Market for Lemonsrdquo RAND Journal of Economics32 657ndash666

23

LIBGOBER J AND X MU (2019) ldquoInformational Robustness in Intertemporal PricingrdquoWorking paper

MAKRIS M AND L RENOU (2018) ldquoInformation Design in Multi-Stage Gamesrdquo Workingpaper

MILGROM P R AND R J WEBER (1985) ldquoDistributional Strategies for Games with In-complete Informationrdquo Mathematics of Operations Research 10 619ndash632

MUSSA M AND S ROSEN (1978) ldquoMonopoly and Product Qualityrdquo Journal of EconomicTheory 18 301ndash317

RAVID D A-K ROESLER AND B SZENTES (2019) ldquoLearning Before Trading On theInefficiency of Ignoring Free Informationrdquo Working paper

ROESLER A-K AND B SZENTES (2017) ldquoBuyer-Optimal Learning and Monopoly Pric-ingrdquo American Economic Review 107 2072ndash80

TERSTIEGE S AND C WASSER (2019) ldquoBuyer-optimal Extensionproof InformationrdquoWorking paper

Appendices omitted from this version but are available on request

24

To interpret this observe that it is optimal for Buyer to accept the price v no matter hisbelief Therefore Seller can guarantee herself the (expected) profit v minus E[c(v)] no matterwhat the information structure is More precisely she can guarantee v minus E[c(v)] minus ε forany ε gt 0 since sequential rationality requires Buyer to accept any price v minus ε SimilarlySeller can also guarantee zero profit offering a price p gt v Hence Sellerrsquos payoff in anyequilibrium of any information structure must be at least the payoff guarantee πs(Γ) above

On the other hand Buyer can guarantee himself the payoff πb = 0 by rejecting all pricesIt follows that the implementable set Π(Γ) must satisfy three simple constraints 1) Sellerrsquosldquoindividual rationalityrdquo constraint πs ge πs(Γ) 2) Buyerrsquos ldquoindividual rationalityrdquo con-straint πb ge 0 and 3) the feasibility constraint πb + πs le S(Γ)

Our first result is that these individual rationality and feasibility constraints are alsosufficient for a payoff pair to be implementable

Theorem 1 Consider all information structures and equilibria

Π(Γ) =

983099983105983103

983105983101

πb ge 0

(πb πs) πs ge πs(Γ)

πb + πs le S(Γ)

983100983105983104

983105983102

Theorem 1 says that the set Π(Γ) corresponds to the triangle AFG in Figure 1 In par-ticular Buyer can receive the entire surplus beyond Sellerrsquos payoff guarantee This is sur-prising as Seller has substantial bargaining power Note that when v le E[c(v)] an entirelyreasonable condition Sellerrsquos payoff guarantee is zero12

The proof of Theorem 1 is in fact straightforward Suppose for expositional simplicityv ge E[c(v)] Fix the trivial information structure in which neither player receives any infor-mation and consider the following family of strategy profiles Seller randomizes betweentwo prices some pl isin [vE[v]] and ph = E[v] Buyer accepts pl with probability one andaccepts ph with probability α(ph) where α(ph) isin [0 1] is specified to make Seller indifferentbetween the two prices That is α(ph)(ph minus E[c(v)]) = pl minus E[c(v)] The expected payoffsfrom this strategy profile are

πb = σ(pl)(E[v]minus pl) and πs = pl minus E[c(v)]

As pl traverses the interval [vE[v]] Sellerrsquos payoff πs traverses [πs(Γ) S(Γ)] Given any

12 In monopoly pricing with c(middot) = v Sellerrsquos payoff guarantee of zero is lower than the revenue guaranteeidentified in Du (2018 Section 5) which is typically positive Dursquos notion is different from ours

11

pl Buyerrsquos payoff πb traverses [0 S(Γ)minusπs] as σ(pl) traverses [0 1] Therefore the proposedstrategy profiles induce all the payoff pairs stated in Theorem 1

We are left to specify beliefs ν for Buyer After prices pl and ph Buyer holds the priorbelief micro After any other (necessarily off-path) price Buyerrsquos belief is that v = v and soBuyer rejects all prices p isin [vinfin)pl ph It is straightforward to confirm that the specified(σα ν) constitute a wPBE

To get more insight into this construction consider its implication for the monopolypricing with c(middot) = v The equilibrium with pl = v (hence α(ph) = 0) and σ(pl) = 1

corresponds to the monopolist deterministically pricing at v and Buyer purchasing Giventhat both sides of the market receive no information why doesnrsquot the monopolist deviate toany price in (vE[v]) The reason is that in this equilibrium the consumer will then not buybecause he updates to v = v Such belief is compatible with wPBE because the equilibriumconcept places no restrictions on off-path beliefs This may seem like a game-theoreticmisdirection Buyerrsquos belief is not consistent with ldquono signaling what you donrsquot knowrdquoPut differently since we have a (weakly) more-informed Buyer information structure weought to impose the price-independent belief condition described in Subsection 23 thatwould imply Buyer must purchase at any price p lt E[v]

But the message of Theorem 1 does not rely on the permissiveness of wPBE To illus-trate continue with the above monopoly-pricing environment and suppose v has positiveprior probability Consider Buyer remaining uninformed but Seller learning whether v = v

or v gt v Now Buyerrsquos off-path belief that v = v is consistent with ldquono signaling whatyou donrsquot knowrdquo More generally using richer information structures we can prove thatany payoff pair identified in Theorem 1 can be approximately implemented as a sequentialequilibrium (Kreps and Wilson 1982) in a suitably discretized game

Theorem 1lowast Fix any ε gt 0 There is a finite information structure and a finite price grid inducinga game with a set of sequential equilibrium payoffs that is an ε-net of Π(Γ)13

In fact the proof of Theorem 1lowast establishes even more the sequential equilibria in thediscretized games satisfy a natural version of the D1 refinement (Cho and Kreps 1987) Werelegate the logic to the Appendix but mention here that we use imperfectly-correlatedsignals for Buyer and Seller14

13 An information structure is finite if the signal spaces Tb and Ts are finite Sequential equilibrium isdefined in the obvious way for the ldquoinducedrdquo finite game where Nature directly draws (tb ts) rather thanfirst drawing v and playersrsquo payoffs are defined directly using (tb ts) For X sub R2 and ε gt 0 the set A sub Xis an ε-net of X if for each x isin X there is a isin A such that 983042xminus a983042 lt ε where 983042middot983042 is the Euclidean distance

14 The idea behind D1 is to ask for any off-path price whether one type of Seller would deviate for any

12

Even when our environment is specialized to monopoly pricing it is worth highlightingtwo contrasts between Theorem 11lowast and results of Bergemann et al (2015) and Roeslerand Szentes (2017) First we find that by not restricting the monopolist to be uninformedthe implementable payoff set typically expands rather dramatically trade can be efficientwith the monopolist securing none of the surplus beyond her payoff guarantee (πs) whichmay be zero (Roesler and Szentes establish implicitly that the implementable set withan uninformed monopolist is a superset of Bergemann et alrsquos which has a fully-informedconsumer) We will see in Subsection 32 that what is crucial to this expansion is price-dependent beliefs In particular the proof of Theorem 1lowast uses an information structure inwhich Buyer is not better informed than Seller mdash if he were then sequential equilibriumwould imply price-independent beliefs Second Theorem 1lowast establishes that for a givenε gt 0 a single information structure (and price grid) can be used to approximate the entirepayoff set Π(Γ) analogously to the construction described after Theorem 1 that used asingle information structure15 Bergemann et al (2015) and Roesler and Szentes (2017) onthe other hand vary information structures to span their payoff sets

32 More-informed Buyer and Price-independent Beliefs

In some economic settings it is plausible that Buyer is more informed than Seller Howdoes a restriction to such information structures ie τ isin Tmb affect the implementablepayoff set It turns out that what is in fact crucial is price-independent beliefs We have ex-plained earlier why it is desirable to impose this condition when Buyer is more informedbut that the condition is well defined even otherwise If Buyer is not more informed thanSeller price-independent beliefs ought to be viewed as an equilibrium restriction in gen-eral To reduce repetition readers should be bear in mind for the rest of this subsectionthe qualifier ldquowith price-independent beliefsrdquo applies unless stated explicitly otherwise

It is useful to define

πuss (Γ) equiv inf πs exist(πb πs) isin Πlowast

us(Γ) (1)

as the lowest payoff that an uninformed Seller can obtain no matter Buyerrsquos information(among equilibria with price-independent beliefs) Plainly πus

s (Γ) ge πs(Γ) In monopoly

Buyer mixed response that another type would Our construction has multiple Buyer types that are imper-fectly correlated with Seller types So different types of Seller have different beliefs about Buyer types Thisblunts the power of dominance considerations to the point where D1 does not exclude any Seller type fromthe support of Buyerrsquos off-path belief

15 In fact if one lets the price grid vary with ε then a single information structure implements exactlyrather than approximately in sequential equilibrium all payoffs ε-away from the boundary of Π(Γ) See in the Appendix for a formal statement

13

pricing with V = [v v] and c(middot) = v Roesler and Szentesrsquos (2017) characterization of theconsumer-optimal information structure identifies πus

s establishing that πuss gt πs If there

is no trade due to adverse selection when Seller is uniformed and Buyer has some informa-tion then πus

s = πs = 0 We do not have a general explicit formula for πuss Subsection 43

provides it for linear c(middot) Nonetheless we establish next that (i) the only additional restric-tion on equilibrium payoffs imposed by price-independent beliefs is a lower bound of πus

s

for Seller and (ii) uninformed-Seller information structures implement all such payoffs

Theorem 2 Consider equilibria with price-independent beliefs

1 Πlowast(Γ) = Πlowastmb(Γ) = Πlowast

us(Γ)

2 Πlowastus(Γ) = (πb πs) isin Π(Γ) πs ge πus

s (Γ)

3 For any (πb πs) isin Πlowastus(Γ) with πs gt πus

s (Γ) there is τ isin Tus with Π(Γ τ) = (πb πs)

Remark 1 We believe the substance of Theorem 2 would hold using discretizations and se-quential equilibria analogous to Theorem 1lowast As previously noted sequential equilibriumimplies price-independent beliefs when Buyer is more informed than Seller

To digest Theorem 2 note that Πlowast(Γ) sup Πlowastmb(Γ) sup Πlowast

us(Γ) is trivial So part 1 of thetheorem amounts to establishing the reverse inclusions The intuition for thismdashgiven part2rsquos characterization of Πlowast

usmdashis fairly straightforward with price-independent beliefs ad-ditional information cannot harm Seller even though it could alter the set of equilibria SoSellerrsquos lowest payoff obtains when she is uninformed

The characterization in part 2 of payoffs with an uninformed Seller corresponds to thetriangle ADE in Figure 1 Part 3 of the theorem assures ldquounique implementationrdquo of allimplementable payoffs satisfying πs gt πus

s (Γ) That is for any such payoff pair there is anuninformed-Seller information structure such that all equilibria (with price-independentbeliefs) induce exactly that payoff pair Unique implementation is appealing for multiplereasons one of which is that it obviates concerns about which among multiple payoff-distinct equilibria is more reasonable For example Ravid Roesler and Szentes (2019)bring to the fore the issue of equilibrium multiplicity in Roesler and Szentesrsquos (2017) infor-mation structure

Let us describe how we obtain the characterization of Πlowastus(Γ) and unique implementa-

tion There are two steps The first ensures that there is some information structure callit τ lowast isin Tus that implements Sellerrsquos payoff πus

s (Γ) That is we ensure that the infimum

14

in (1) is in fact a minimum16 While this argument is technical knowing τ lowast exists is usefulin what follows The second and economically insightful step is to construct informa-tion structures that implement every point in the triangle Πlowast

us(Γ) by suitably garbling theinformation structure τ lowast The construction is illustrated in Figure 2 Consider the distri-bution of Buyerrsquos posterior mean of his valuation v in information structure τ lowast (Givenprice-independent beliefs Buyerrsquos posterior mean is sufficient for his decision) For sim-plicity suppose this posterior-mean distribution has a density as depicted by the red curvein Figure 2 Fix any (πb πs) isin Πlowast

us(Γ)

First there is some number zlowast such that πb + πs is the total surplus from trading onlywhen Buyerrsquos posterior mean is greater than zlowast Next there is some price plowast ge zlowast suchthat Sellerrsquos payoff is πs if all these trades were to occur at price plowast17 Note that plowast must beno larger than the expected Buyer posterior mean conditional on that being above zlowast forotherwise πb lt 0 We claim that the information structure τ lowast can be garbled so that plowast isan equilibrium price and trade occurs only when Buyerrsquos posterior mean is greater thanzlowast The garbling is illustrated in Figure 2 by the blue distribution There is one signal that

pzv

ν(v)

Figure 2 Construction of information structure in Theorem 2

Buyer receives when the original posterior mean is between zlowast and plowast and also receiveswith some probability when the original posterior mean is above plowast The probability ischosen to make the posterior mean from this signal exactly plowast Apart from this one newsignal Buyer receives the original signal in τ lowast Plainly this is a garbling of τ lowast and hence isfeasible

16 The difficulty is in establishing suitable continuity Uninformed-Seller information structures can beviewed as probability measures over Buyerrsquos beliefs with convergence in the sense of the weak topologyThis topology ensures continuity with respect to probability measures of expectations of continuous orat least Lipschitz (and bounded) functions However Sellerrsquos expected payoff is not the expectation of aLipschitz function as Sellerrsquos profit is truncated at the price she charges

17 That plowast ge zlowast follows from πs ge πuss (Γ) as πs(Γ) itself is weakly larger than Sellerrsquos payoff from posting

price zlowast (and thus trading with the same set of Buyer posterior means) under information structure τlowast

15

Figure 2 makes clear why the new information structure has an equilibrium with priceplowast and Buyer breaking indifference in favor of trading (i) Sellerrsquos profit from posting anyprice below zlowast is the same as under τ lowast and hence no larger than πus

s (Γ) (ii) similarly Sellerrsquosprofit from posting any price above plowast is no higher than some fraction of πus

s (Γ) and (iii)any price between zlowast and plowast is worse that price plowast Moreover since Sellerrsquos profit fromoffering any price other than plowast is no more than πus

s (Γ) it follows that when πs gt πuss (Γ)

Buyer must break indifference as specified for Seller to have an optimal price and theequilibrium payoffs are unique

Remark 2 The above logic establishes that given any τ isin Tus that implements some (πb πs)τ can be garbled to uniquely implement any (πprime

b πprimes) isin Π(Γ) such that πprime

s gt πs That isan uninformed Sellerrsquos payoff can always be strictly raised and Buyerrsquos payoff reduced(strictly so long as it was not already zero) by garbling Buyerrsquos information

Remark 3 According to Theorem 11lowast and Theorem 2 uninformed-Seller information struc-tures cannot implement all implementable payoff pairs in an environment Γ if and only ifπuss (Γ) gt πs(Γ) This inequality fails in environment Γ if πus

s (Γ) = 0 since that impliesπuss (Γ) = πs(Γ) = 0 An example is when there is no trade due to adverse selection when

Seller is uniformed and Buyer has some information On the other hand it is sufficient forthe inequality that

v le E[c(v)] and forallv isin V c(v) lt v (2)

To see why (2) implies πuss (Γ) gt πs(Γ) notice that in any uninformed-Seller information

structure Seller can price at slightly less than Buyerrsquos highest posterior mean valuationand guarantee trade with only (a neighborhood of) that Buyer type If c(v) lt v for all v thisgives Seller a positive expected payoff and hence πus

s (Γ) gt 018 But the first inequality in (2)is equivalent to πs(Γ) = 0 We observe that Condition (2) is compatible with severe adverseselection resulting in very little trade when Seller is uninformed and Buyer (partially orfully) informed

33 Fully-Informed Buyer

We now turn to the third canonical class of information structures Buyer is fully in-formed of his value v As this is a special case of more-informed Buyer we maintain price-independent beliefs throughout this subsection

18 More precisely as V is compact c(v) lt v for all v implies there exists ε gt 0 such that v minus c(v) gt εGiven any uninformed-Seller information structure let mv be the highest posterior mean valuation in thesupport of the posterior means induced by Buyerrsquos signals So there is positive probability of Buyer signalswith posterior mean valuations at least mv minus ε2 By pricing at mv minus ε2 Sellerrsquos expected cost conditionalon trade is bounded above by mv minus ε and hence Sellerrsquos profit conditional on trade is at least ε2 gt 0 Itfollows that πus

s gt 0

16

Faced with a fully informed Buyer and any sequentially rational Buyer strategy an un-informed Seller can guarantee a profit level

πfbs (Γ) equiv sup

p

983133 v

p

(pminus c(v))micro(dv)

regardless of her information Plainly πfbs (Γ) ge πus

s (Γ) In monopoly pricing with c(middot) = vRoesler and Szentes (2017) have shown that πfb

s gt πuss if there is no trade due to adverse

selection when Buyer is fully informed and Seller is uniformed then πfbs = πus

s = 0 Belowwe establish that when Buyer is fully informed πfb

s is the only additional constraint onequilibrium payoffs Let Bε(πb πs) denote an ε-ball around (πb πs)

Theorem 3 Consider a fully informed Buyer and price-independent beliefs

1 Πlowastfb(Γ) = (πb πs) isin Π(Γ) πs ge πfb

s (Γ)

2 For any (πb πs) isin Πlowastfb(Γ) and any ε gt 0 there is τ isin Tfb with Π(Γ τ) sub Bε(πb πs)

The payoff set characterized in Theorem 3 corresponds to the triangle ABE in Figure 1Part 2 of the theorem establishes ldquoapproximately unique implementationrdquo for any point inthe triangle there is an information structure under which all equilibria yield payoff pairsin a neighborhood of that point

Here is the idea behind part 1 of Theorem 3 When Buyer is fully informed an informa-tion structure can be viewed as dividing vrsquos prior distribution micro into a set of microi that aver-age to micro with Seller informed of which microi she faces Theorem 3 is proven by establishingthat we can divide micro suitably so that against each microi Seller is indifferent between pricingat all prices in the support of microi including the price corresponding to πfb

s in that environ-ment We call such a microi a (Seller-price) incentive compatible distribution or ICD (Bergemannet al (2015) call their analogous construct an extreme market) The Appendix provides aldquogreedyrdquo algorithm to compute ICDs19

We can sketch how the algorithm works and construct a set of ICDs that average to theprior Suppose V = v1 v2 vK with vi lt vi+1 for i isin 1 K minus 1 and c(v) lt v forall v Given any small-enough mass of vK there is a unique mass of type vKminus1 that makesSeller indifferent between charging price vK and vKminus1 (If the mass is too low Seller prefersvK if it is too high she prefers vKminus1) Iterating down to keep Seller indifferent betweenall prices pins down an ICD Choose the maximum mass of type vK for which this worksRemove that ICD and repeat

19 The algorithm is defined for finite V and we take limits to handle the infinite case

17

Crucially whenever an ICD is removed the price corresponding to πfbs (Γ) remains op-

timal in the remaining ldquomarketrdquo this follows from the ICDrsquos defining property of Sellerindifference and an accounting identity Therefore Sellerrsquos profit in this segmentation ofICDs remains πfb

s (Γ) Moreover it is also optimal for Seller to always (ie for each microi) priceso that there is full trade or no trade Hence Buyerrsquos expected payoff can be either 0 or theentire surplus less πfb

s (Γ) It follows that the fully-informed Buyer information structuredefined by this set of ICDs implements point B and C in Figure 1 The entire triangle ABC

can then be implemented by convexification randomizing over this information structure(and two equilibria) and full information (where Seller obtains all the surplus)

The above logic of incentive compatible distributions reduces to that of Bergemann et al(2015) in the context of monopoly pricing Even in the monopoly pricing case Theorem 3part 2 strengthens the conclusions of Bergemann et al (2015) by ensuring approximatelyunique implementation

Remark 4 Theorems 1ndash3 imply that fully-informed-Buyer information structures imple-ment all implementable payoff pairs if and only if πfb

s (Γ) = πs(Γ) In that case trianglesAFG and ABC coincide in Figure 1 It follows that πfb

s (Γ) = πs(Γ) only when a fully-informed Buyer and uninformed Seller can result in full trade (v ge E[c(v)] and Sellerprices at v) or no trade (v le E[c(v)] and Seller prices at some p ge v) Interestinglywhen πfb

s (Γ) gt πs(Γ) fully-informed-Buyer information structures cannot even implementall payoff pairs implementable by uninformed-Seller information structures ie trianglesABC and ADE in Figure 1 are distinct That is when πfb

s (Γ) gt πs(Γ) there is an uninformed-Seller information structure that implements some πs lt πfb

s (Γ)20 Theorems 2ndash3 further im-ply in this case that Buyer can obtain a strictly higher payoff when he is not fully informed(and Seller is uninformed)

4 Discussion

41 Multidimensionality

Suppose Buyer and Sellerrsquos cost and valuation pair (c v) is a two-dimensional randomvariable distributed according to joint distribution micro with a compact support The ex-tension of our maintained assumption of commonly known gains from trade is for all

20 Pick any pprime gt v such that pprimeminusE[c(v)] lt πfbs (Γ) Following the construction described after Theorem 2 we

can mix all valuations v le pprime with a fraction λ gt 0 of valuations v gt pprime so that the mixture has posterior meanexactly pprime The remaining fraction 1 minus λ of valuations above pprime are revealed to Buyer With this uninformed-Seller information structure consider any equilibrium in which Buyer purchases when indifferent (Suchan equilibrium with price-independent belief exists) Sellerrsquos profit is at most (1 minus λ)πfb

s (Γ) from any pricep gt pprime and pprime minus E[c(v)] from price p = pprime Hence Sellerrsquos profit is strictly less than πfb

s (Γ)

18

(c v) isin Supp(micro) v ge c and E[v minus c] gt 0 An information structure is now a joint distribu-tion P (tb ts v c) whose marginal distribution on (v c) is micro

We claim that the content of our key results Theorem 1Theorem 1lowast Theorem 2 andTheorem 3 still hold To see why let v be the lowest valuation in the support of micro Sellerrsquosindividual rationality constraint is now maxv minus E[c] 0 as she can guarantee this by set-ting either a sufficiently high price or a price (arbitrarily close to) v regardless of her signalWe can define a cost function c(vprime) equiv Emicro[c|v = vprime] ge vprime This results in an environment sat-isfying all the maintained assumptions of our baseline model except that c(middot) may not becontinuous Recall that such continuity is not needed for proving Theorem 1 Theorem 1lowastBoth Theorem 2 and Theorem 3 use continuity of c(middot) to guarantee that Eν [c(v)] is a contin-uous function of ν isin ∆(V ) for a key convergence result However in the two-dimensionaltype environment Eν [c] is still a continuous function of ν isin ∆(V times C) Note that Theo-rem 3 uses upper semi-continuity of Sellerrsquos profit in price boundedness of c and c le v issufficient for such upper semi-continuity

42 Negative Trading Surplus

Returning to our baseline model we next discuss cases in which trade sometimes gen-erates negative surplus That is we drop the assumption that c(v) le v we do not requireE[v minus c(v)] gt 0 either Define Sλ(Γ) for λ isin [1infin) as

Sλ(Γ) equiv983133 v

v

[v minus c(v) + λ(v minus v)]+ micro(dv)

where [middot]+ equiv max middot 0 It is readily verified that S1(Γ) = E [[v minus c(v)]+] and limλrarr0

Sλ(Γ)λ =

E[v] minus v Allowing negative trading surplus does not affect our definition of wPBE Sothe notation Π(Γ) and πs(Γ) still have the same meanings as before The next propositionshows that Π(Γ) is now characterized by three constraints as before the two individualrationality constraints πs ge πs(Γ) and πb ge 0 and different now a Pareto frontier definedby all Sλ(Γ)

Proposition 1 Consider all information structures and equilibria when trade can generate negativesurplus

Π(Γ) =

983099983105983103

983105983101

πb ge 0

(πb πs) πs ge πs(Γ)

λπb + πs le Sλ(Γ) forallλ ge 1

983100983105983104

983105983102

Proposition 1 shows that when the trade can be inefficient it may be impossible to lower

19

Sellerrsquos profit without losing total surplus (ie πb + πs) This result is intuitive at the ex-tremes Suppose for simplicity v gt E[c(v)] If πs = πs(Γ) then since πs(Γ) = v minus E[c(v)]Seller must find it optimal to price at v and trade with probability one including whentrade generates negative surplus So Buyerrsquos payoff is E[v]minus v = limλrarrinfin Sλ(Γ)λ and totalsurplus is E[v minus c(v)] which includes the maximum amount of inefficient trade On theother hand an information structure that publicly reveals (only) whether trade is efficient(ie whether v ge c(v)) can implement efficient trade with all the surplus accruing to SellerIn this case Seller obtains S1(Γ) while Buyer obtains zero The proposition establishes thatin between S1(Γ) and πs(Γ) information structures and equilibria that lower Sellerrsquos profitmust reduce total surplus as the Pareto frontier Sλ(Γ) is a concave function with maximalslope minus1

The function capturing the Pareto frontier Sλ(Γ) is the maximal weighted sum of Buyerand Seller payoff assuming that Seller will trade at the low price p = v whenever it createspositive weighted total payoff Since we place higher weight on buyer (λ ge 1) S1(Γ) isthe upper bound for the weighted total payoff In the proof of Proposition 1 we showthat these bounds are tight mdash any payoff pair on the frontier can be implemented by someinformation structure The key observation is that whenever v minus c(v) + λ(v minus v) lt 0 itfollows that vminus c(v) lt (1minusλ)(vminus v) le 0 That is it is incentive compatible for Seller to notsell when it is common knowledge that v creates negative weighted total payoff Thereforewe can simply release a public signal indicating the sign of weighted total payoff and ifpositive induce Seller to sell at v which implies all bounds Sλ(Γ)

43 Linear Cost

When Sellerrsquos cost function c(v) is linear we can explicitly characterize an informationstructure that implements Sellerrsquos minimum implementable payoff πus

s (Γ) when Seller isuninformed (or per Theorem 2 when Buyer is better informed) subject to price-independentbeliefs We note that given the discussion in Subsection 41 a linear c(v) subsumes richerenvironments in which conditional expectations are linear including Gaussian environ-ments (cf Bar-Isaac et al 2018)

Condition 1 c(v) = λv + γ for some λ isin R

Let F (v) be the cumulative distribution function (CDF) corresponding to the prior mea-sure micro Let D(micro) be the set of all distributions whose CDF G satisfies

983133

V

vdG(v) =

983133

V

vdF (v) and983133 v

v

G(s)ds le983133 v

v

F (s)ds forallv isin V

20

That is D(micro) contains all distributions that are mean-preserving-contractions (MPC) of microIt is well known that D(micro) characterizes the set of distributions of Buyer posterior meansthat can be generated by any (uninformed-Seller) information structure

Definition 2 G is an incentive compatible distribution (ICD) if

983133

sgep

(pminus c(s))G(ds) = constant ge 0 forallp isin Supp(G)

That is when Buyerrsquos (mean) valuation is distributed according to an ICD G Seller isindifferent among all prices in the support of G assuming Buyer buys when indifferent Wefocus on a special family of ICDs whose supports are intervals [vlowast v

lowast] Given Condition 1such ICDs have analytical an expression

G(v) =

983099983105983105983103

983105983105983101

1minus983061

(1minus λ)v minus γ

(1minus λ)vlowast minus γ

983062 1λminus1

if λ ∕= 1

1minus evminusvlowast

γ if λ = 1

(3)

Given any vlowast and the above G(v) an upper bound vlowast is pinned down by the conditionEG[v] = EF [v] So a family of ICDs is parametrized by a single parameter vlowast and hence anICD is denoted by CDF 983144Gvlowast(v) with density 983144gvlowast(v) It is not difficult to verify that 983144Gvlowast(v) ismonotonic in vlowast for v in the support Consequently all ICDs are ordered according to theMPC order For vlowast isin [vE[v]] when vlowast increases the ICD increases in the MPC order

Proposition 2 Suppose Condition 1 is satisfied and let plowast equiv min983153vlowast983055983055 983144Gvlowast isin D(micro)

983154 It holds that

πuss (Γ) = plowast minus Emicro[c(v)]

Proposition 2 states that an uninformed Sellerrsquos minimum payoff (with price-independentbeliefs) is characterized by the ICDs that are MPCs of the prior distribution micro By defini-tion of ICDs Sellerrsquos profit when facing ICD 983144Gvlowast is vlowast minus E[v] Sellerrsquos minimum payoff isimplemented by the most dispersed ICD that is a MPC of the prior distribution

In proving Proposition 2 the key step is to show that given the prior micro garbling Buyerrsquosinformation such that the posterior mean distribution is an ICD makes Seller weakly worseoff It is then without loss to only consider ICDs to implement πus

s (Γ) We can thus workwith a tractable one-dimensional problem Consider the set D(micro) Suppose we find G(v)

such that (i) G isin D(micro) (ii) G is an ICD and (iii) existp such that983125 p

vG(s)ds =

983125 p

vF (s)ds Such

a G is the most-dispersed ICD that is a mean preserving contraction of F Consider the

21

following two manipulations

λ

983133 v

p

F (s)ds =

983133 v

p

(pminus c(s))dF (s)minus (1minus F (v)) (pminus c(v)) + λ(v minus p) (4)

λ

983133 v

p

G(s)ds =

983133 v

p

(pminus c(s))dG(s)minus (1minusG(v)) (pminus c(v)) + λ(v minus p) (5)

First by property (iii) above the LHS of Equation 4 equals the LHS of Equation 5 Secondsince

983125 v

p(F (s) minus G(s))ds is minimized at v this implies F (p) le G(p) Therefore the first

term on RHS of Equation 4 must be larger than that of Equation 5 Notice that the firstterm on RHS is Sellerrsquos profit when offering price p This implies that the profit fromoffering p given Buyer mean-valuation distribution G(v) is lower than that given F (v)which is lower than Sellerrsquos maximum profit given F (v) On the other hand by the ICDproperty p is already Sellerrsquos optimal price given G(v) The optimal profit given valuationdistribution F must be no lower than that of G Therefore we have proved that to minimizeSellerrsquos profit it is without loss to consider only ICDs within the set D(micro) which is a onedimensional subspace The detailed proof that such ICDs exist is relegated to the appendix

The monopoly-pricing environment with c(middot) = v is covered by Condition 1 with λ = 0

and γ = v The distribution G in (3) then reduces to that identified by Roesler and Szentes(2017)

If the prior micro has binary support then the cost function c(v) is linear In this special casewe can solve for πus

s (Γ) explicitly

Corollary 1 Suppose micro has binary support V = v1 v2 Let λ = c(v2)minusc(v1)v2minusv1

and let p be theunique solution to

(pminus c(p))1

λminus1 (pminus E[c(v)]) = (v2 minus c(v2))λ

λminus1 (6)

Then πuss (Γ) = max p v2minus E[c(v)]

References

AKERLOF G A (1970) ldquoThe Market for rdquoLemonsrdquo Quality Uncertainty and the MarketMechanismrdquo Quarterly Journal of Economics 84 488ndash500

BAGWELL K AND M H RIORDAN (1991) ldquoHigh and Declining Prices Signal ProductQualityrdquo American Economic Review 81 224ndash239

BAR-ISAAC H I JEWITT AND C LEAVER (2018) ldquoAdverse Selection Efficiency and theStructure of Informationrdquo Unpublished

22

BERGEMANN D B BROOKS AND S MORRIS (2015) ldquoThe Limits of Price Discrimina-tionrdquo American Economic Review 105 921ndash57

BERGEMANN D AND S MORRIS (2016) ldquoBayes Correlated Equilibrium and the Compar-ison of Information Structures in Gamesrdquo Theoretical Economics 11 487ndash522

BLACKWELL D (1953) ldquoEquivalent Comparisons of Experimentsrdquo Annals of MathematicalStatistics 24 265ndash272

CHO I-K AND D KREPS (1987) ldquoSignaling Games and Stable Equilibriardquo Quarterly Jour-nal of Economics 102 179ndash221

DU S (2018) ldquoRobust Mechanisms Under Common Valuationrdquo Econometrica 86 1569ndash1588

DURRETT R (1995) Probability Theory and Examples Duxbury Press 2nd ed

EINAV L AND A FINKELSTEIN (2011) ldquoSelection in Insurance Markets Theory and Em-pirics in Picturesrdquo Journal of Economic Perspectives 25 115ndash38

FUDENBERG D AND J TIROLE (1991) ldquoPerfect Bayesian Equilibrium and Sequential Equi-libriumrdquo Journal of Economic Theory 53 236ndash260

GARCIA D R TEPER AND M TSUR (2018) ldquoInformation Design in Insurance MarketsSelling Peaches in a Market for Lemonsrdquo Unpublished

GREENWALD B C (1986) ldquoAdverse Selection in the Labour Marketrdquo Review of EconomicStudies 53 325ndash347

JOVANOVIC B (1982) ldquoFavorable Selection with Asymmetric Informationrdquo QuarterlyJournal of Economics 97 535ndash539

KESSLER A S (2001) ldquoRevisiting the Lemons Marketrdquo International Economic Review 4225ndash41

KREPS D AND R WILSON (1982) ldquoSequential Equilibriardquo Econometrica 50 863ndash894

LEHMANN E L (1988) ldquoComparing Location Experimentsrdquo The Annals of Statistics 16521ndash533

LEVIN J (2001) ldquoInformation and the Market for Lemonsrdquo RAND Journal of Economics32 657ndash666

23

LIBGOBER J AND X MU (2019) ldquoInformational Robustness in Intertemporal PricingrdquoWorking paper

MAKRIS M AND L RENOU (2018) ldquoInformation Design in Multi-Stage Gamesrdquo Workingpaper

MILGROM P R AND R J WEBER (1985) ldquoDistributional Strategies for Games with In-complete Informationrdquo Mathematics of Operations Research 10 619ndash632

MUSSA M AND S ROSEN (1978) ldquoMonopoly and Product Qualityrdquo Journal of EconomicTheory 18 301ndash317

RAVID D A-K ROESLER AND B SZENTES (2019) ldquoLearning Before Trading On theInefficiency of Ignoring Free Informationrdquo Working paper

ROESLER A-K AND B SZENTES (2017) ldquoBuyer-Optimal Learning and Monopoly Pric-ingrdquo American Economic Review 107 2072ndash80

TERSTIEGE S AND C WASSER (2019) ldquoBuyer-optimal Extensionproof InformationrdquoWorking paper

Appendices omitted from this version but are available on request

24

pl Buyerrsquos payoff πb traverses [0 S(Γ)minusπs] as σ(pl) traverses [0 1] Therefore the proposedstrategy profiles induce all the payoff pairs stated in Theorem 1

We are left to specify beliefs ν for Buyer After prices pl and ph Buyer holds the priorbelief micro After any other (necessarily off-path) price Buyerrsquos belief is that v = v and soBuyer rejects all prices p isin [vinfin)pl ph It is straightforward to confirm that the specified(σα ν) constitute a wPBE

To get more insight into this construction consider its implication for the monopolypricing with c(middot) = v The equilibrium with pl = v (hence α(ph) = 0) and σ(pl) = 1

corresponds to the monopolist deterministically pricing at v and Buyer purchasing Giventhat both sides of the market receive no information why doesnrsquot the monopolist deviate toany price in (vE[v]) The reason is that in this equilibrium the consumer will then not buybecause he updates to v = v Such belief is compatible with wPBE because the equilibriumconcept places no restrictions on off-path beliefs This may seem like a game-theoreticmisdirection Buyerrsquos belief is not consistent with ldquono signaling what you donrsquot knowrdquoPut differently since we have a (weakly) more-informed Buyer information structure weought to impose the price-independent belief condition described in Subsection 23 thatwould imply Buyer must purchase at any price p lt E[v]

But the message of Theorem 1 does not rely on the permissiveness of wPBE To illus-trate continue with the above monopoly-pricing environment and suppose v has positiveprior probability Consider Buyer remaining uninformed but Seller learning whether v = v

or v gt v Now Buyerrsquos off-path belief that v = v is consistent with ldquono signaling whatyou donrsquot knowrdquo More generally using richer information structures we can prove thatany payoff pair identified in Theorem 1 can be approximately implemented as a sequentialequilibrium (Kreps and Wilson 1982) in a suitably discretized game

Theorem 1lowast Fix any ε gt 0 There is a finite information structure and a finite price grid inducinga game with a set of sequential equilibrium payoffs that is an ε-net of Π(Γ)13

In fact the proof of Theorem 1lowast establishes even more the sequential equilibria in thediscretized games satisfy a natural version of the D1 refinement (Cho and Kreps 1987) Werelegate the logic to the Appendix but mention here that we use imperfectly-correlatedsignals for Buyer and Seller14

13 An information structure is finite if the signal spaces Tb and Ts are finite Sequential equilibrium isdefined in the obvious way for the ldquoinducedrdquo finite game where Nature directly draws (tb ts) rather thanfirst drawing v and playersrsquo payoffs are defined directly using (tb ts) For X sub R2 and ε gt 0 the set A sub Xis an ε-net of X if for each x isin X there is a isin A such that 983042xminus a983042 lt ε where 983042middot983042 is the Euclidean distance

14 The idea behind D1 is to ask for any off-path price whether one type of Seller would deviate for any

12

Even when our environment is specialized to monopoly pricing it is worth highlightingtwo contrasts between Theorem 11lowast and results of Bergemann et al (2015) and Roeslerand Szentes (2017) First we find that by not restricting the monopolist to be uninformedthe implementable payoff set typically expands rather dramatically trade can be efficientwith the monopolist securing none of the surplus beyond her payoff guarantee (πs) whichmay be zero (Roesler and Szentes establish implicitly that the implementable set withan uninformed monopolist is a superset of Bergemann et alrsquos which has a fully-informedconsumer) We will see in Subsection 32 that what is crucial to this expansion is price-dependent beliefs In particular the proof of Theorem 1lowast uses an information structure inwhich Buyer is not better informed than Seller mdash if he were then sequential equilibriumwould imply price-independent beliefs Second Theorem 1lowast establishes that for a givenε gt 0 a single information structure (and price grid) can be used to approximate the entirepayoff set Π(Γ) analogously to the construction described after Theorem 1 that used asingle information structure15 Bergemann et al (2015) and Roesler and Szentes (2017) onthe other hand vary information structures to span their payoff sets

32 More-informed Buyer and Price-independent Beliefs

In some economic settings it is plausible that Buyer is more informed than Seller Howdoes a restriction to such information structures ie τ isin Tmb affect the implementablepayoff set It turns out that what is in fact crucial is price-independent beliefs We have ex-plained earlier why it is desirable to impose this condition when Buyer is more informedbut that the condition is well defined even otherwise If Buyer is not more informed thanSeller price-independent beliefs ought to be viewed as an equilibrium restriction in gen-eral To reduce repetition readers should be bear in mind for the rest of this subsectionthe qualifier ldquowith price-independent beliefsrdquo applies unless stated explicitly otherwise

It is useful to define

πuss (Γ) equiv inf πs exist(πb πs) isin Πlowast

us(Γ) (1)

as the lowest payoff that an uninformed Seller can obtain no matter Buyerrsquos information(among equilibria with price-independent beliefs) Plainly πus

s (Γ) ge πs(Γ) In monopoly

Buyer mixed response that another type would Our construction has multiple Buyer types that are imper-fectly correlated with Seller types So different types of Seller have different beliefs about Buyer types Thisblunts the power of dominance considerations to the point where D1 does not exclude any Seller type fromthe support of Buyerrsquos off-path belief

15 In fact if one lets the price grid vary with ε then a single information structure implements exactlyrather than approximately in sequential equilibrium all payoffs ε-away from the boundary of Π(Γ) See in the Appendix for a formal statement

13

pricing with V = [v v] and c(middot) = v Roesler and Szentesrsquos (2017) characterization of theconsumer-optimal information structure identifies πus

s establishing that πuss gt πs If there

is no trade due to adverse selection when Seller is uniformed and Buyer has some informa-tion then πus

s = πs = 0 We do not have a general explicit formula for πuss Subsection 43

provides it for linear c(middot) Nonetheless we establish next that (i) the only additional restric-tion on equilibrium payoffs imposed by price-independent beliefs is a lower bound of πus

s

for Seller and (ii) uninformed-Seller information structures implement all such payoffs

Theorem 2 Consider equilibria with price-independent beliefs

1 Πlowast(Γ) = Πlowastmb(Γ) = Πlowast

us(Γ)

2 Πlowastus(Γ) = (πb πs) isin Π(Γ) πs ge πus

s (Γ)

3 For any (πb πs) isin Πlowastus(Γ) with πs gt πus

s (Γ) there is τ isin Tus with Π(Γ τ) = (πb πs)

Remark 1 We believe the substance of Theorem 2 would hold using discretizations and se-quential equilibria analogous to Theorem 1lowast As previously noted sequential equilibriumimplies price-independent beliefs when Buyer is more informed than Seller

To digest Theorem 2 note that Πlowast(Γ) sup Πlowastmb(Γ) sup Πlowast

us(Γ) is trivial So part 1 of thetheorem amounts to establishing the reverse inclusions The intuition for thismdashgiven part2rsquos characterization of Πlowast

usmdashis fairly straightforward with price-independent beliefs ad-ditional information cannot harm Seller even though it could alter the set of equilibria SoSellerrsquos lowest payoff obtains when she is uninformed

The characterization in part 2 of payoffs with an uninformed Seller corresponds to thetriangle ADE in Figure 1 Part 3 of the theorem assures ldquounique implementationrdquo of allimplementable payoffs satisfying πs gt πus

s (Γ) That is for any such payoff pair there is anuninformed-Seller information structure such that all equilibria (with price-independentbeliefs) induce exactly that payoff pair Unique implementation is appealing for multiplereasons one of which is that it obviates concerns about which among multiple payoff-distinct equilibria is more reasonable For example Ravid Roesler and Szentes (2019)bring to the fore the issue of equilibrium multiplicity in Roesler and Szentesrsquos (2017) infor-mation structure

Let us describe how we obtain the characterization of Πlowastus(Γ) and unique implementa-

tion There are two steps The first ensures that there is some information structure callit τ lowast isin Tus that implements Sellerrsquos payoff πus

s (Γ) That is we ensure that the infimum

14

in (1) is in fact a minimum16 While this argument is technical knowing τ lowast exists is usefulin what follows The second and economically insightful step is to construct informa-tion structures that implement every point in the triangle Πlowast

us(Γ) by suitably garbling theinformation structure τ lowast The construction is illustrated in Figure 2 Consider the distri-bution of Buyerrsquos posterior mean of his valuation v in information structure τ lowast (Givenprice-independent beliefs Buyerrsquos posterior mean is sufficient for his decision) For sim-plicity suppose this posterior-mean distribution has a density as depicted by the red curvein Figure 2 Fix any (πb πs) isin Πlowast

us(Γ)

First there is some number zlowast such that πb + πs is the total surplus from trading onlywhen Buyerrsquos posterior mean is greater than zlowast Next there is some price plowast ge zlowast suchthat Sellerrsquos payoff is πs if all these trades were to occur at price plowast17 Note that plowast must beno larger than the expected Buyer posterior mean conditional on that being above zlowast forotherwise πb lt 0 We claim that the information structure τ lowast can be garbled so that plowast isan equilibrium price and trade occurs only when Buyerrsquos posterior mean is greater thanzlowast The garbling is illustrated in Figure 2 by the blue distribution There is one signal that

pzv

ν(v)

Figure 2 Construction of information structure in Theorem 2

Buyer receives when the original posterior mean is between zlowast and plowast and also receiveswith some probability when the original posterior mean is above plowast The probability ischosen to make the posterior mean from this signal exactly plowast Apart from this one newsignal Buyer receives the original signal in τ lowast Plainly this is a garbling of τ lowast and hence isfeasible

16 The difficulty is in establishing suitable continuity Uninformed-Seller information structures can beviewed as probability measures over Buyerrsquos beliefs with convergence in the sense of the weak topologyThis topology ensures continuity with respect to probability measures of expectations of continuous orat least Lipschitz (and bounded) functions However Sellerrsquos expected payoff is not the expectation of aLipschitz function as Sellerrsquos profit is truncated at the price she charges

17 That plowast ge zlowast follows from πs ge πuss (Γ) as πs(Γ) itself is weakly larger than Sellerrsquos payoff from posting

price zlowast (and thus trading with the same set of Buyer posterior means) under information structure τlowast

15

Figure 2 makes clear why the new information structure has an equilibrium with priceplowast and Buyer breaking indifference in favor of trading (i) Sellerrsquos profit from posting anyprice below zlowast is the same as under τ lowast and hence no larger than πus

s (Γ) (ii) similarly Sellerrsquosprofit from posting any price above plowast is no higher than some fraction of πus

s (Γ) and (iii)any price between zlowast and plowast is worse that price plowast Moreover since Sellerrsquos profit fromoffering any price other than plowast is no more than πus

s (Γ) it follows that when πs gt πuss (Γ)

Buyer must break indifference as specified for Seller to have an optimal price and theequilibrium payoffs are unique

Remark 2 The above logic establishes that given any τ isin Tus that implements some (πb πs)τ can be garbled to uniquely implement any (πprime

b πprimes) isin Π(Γ) such that πprime

s gt πs That isan uninformed Sellerrsquos payoff can always be strictly raised and Buyerrsquos payoff reduced(strictly so long as it was not already zero) by garbling Buyerrsquos information

Remark 3 According to Theorem 11lowast and Theorem 2 uninformed-Seller information struc-tures cannot implement all implementable payoff pairs in an environment Γ if and only ifπuss (Γ) gt πs(Γ) This inequality fails in environment Γ if πus

s (Γ) = 0 since that impliesπuss (Γ) = πs(Γ) = 0 An example is when there is no trade due to adverse selection when

Seller is uniformed and Buyer has some information On the other hand it is sufficient forthe inequality that

v le E[c(v)] and forallv isin V c(v) lt v (2)

To see why (2) implies πuss (Γ) gt πs(Γ) notice that in any uninformed-Seller information

structure Seller can price at slightly less than Buyerrsquos highest posterior mean valuationand guarantee trade with only (a neighborhood of) that Buyer type If c(v) lt v for all v thisgives Seller a positive expected payoff and hence πus

s (Γ) gt 018 But the first inequality in (2)is equivalent to πs(Γ) = 0 We observe that Condition (2) is compatible with severe adverseselection resulting in very little trade when Seller is uninformed and Buyer (partially orfully) informed

33 Fully-Informed Buyer

We now turn to the third canonical class of information structures Buyer is fully in-formed of his value v As this is a special case of more-informed Buyer we maintain price-independent beliefs throughout this subsection

18 More precisely as V is compact c(v) lt v for all v implies there exists ε gt 0 such that v minus c(v) gt εGiven any uninformed-Seller information structure let mv be the highest posterior mean valuation in thesupport of the posterior means induced by Buyerrsquos signals So there is positive probability of Buyer signalswith posterior mean valuations at least mv minus ε2 By pricing at mv minus ε2 Sellerrsquos expected cost conditionalon trade is bounded above by mv minus ε and hence Sellerrsquos profit conditional on trade is at least ε2 gt 0 Itfollows that πus

s gt 0

16

Faced with a fully informed Buyer and any sequentially rational Buyer strategy an un-informed Seller can guarantee a profit level

πfbs (Γ) equiv sup

p

983133 v

p

(pminus c(v))micro(dv)

regardless of her information Plainly πfbs (Γ) ge πus

s (Γ) In monopoly pricing with c(middot) = vRoesler and Szentes (2017) have shown that πfb

s gt πuss if there is no trade due to adverse

selection when Buyer is fully informed and Seller is uniformed then πfbs = πus

s = 0 Belowwe establish that when Buyer is fully informed πfb

s is the only additional constraint onequilibrium payoffs Let Bε(πb πs) denote an ε-ball around (πb πs)

Theorem 3 Consider a fully informed Buyer and price-independent beliefs

1 Πlowastfb(Γ) = (πb πs) isin Π(Γ) πs ge πfb

s (Γ)

2 For any (πb πs) isin Πlowastfb(Γ) and any ε gt 0 there is τ isin Tfb with Π(Γ τ) sub Bε(πb πs)

The payoff set characterized in Theorem 3 corresponds to the triangle ABE in Figure 1Part 2 of the theorem establishes ldquoapproximately unique implementationrdquo for any point inthe triangle there is an information structure under which all equilibria yield payoff pairsin a neighborhood of that point

Here is the idea behind part 1 of Theorem 3 When Buyer is fully informed an informa-tion structure can be viewed as dividing vrsquos prior distribution micro into a set of microi that aver-age to micro with Seller informed of which microi she faces Theorem 3 is proven by establishingthat we can divide micro suitably so that against each microi Seller is indifferent between pricingat all prices in the support of microi including the price corresponding to πfb

s in that environ-ment We call such a microi a (Seller-price) incentive compatible distribution or ICD (Bergemannet al (2015) call their analogous construct an extreme market) The Appendix provides aldquogreedyrdquo algorithm to compute ICDs19

We can sketch how the algorithm works and construct a set of ICDs that average to theprior Suppose V = v1 v2 vK with vi lt vi+1 for i isin 1 K minus 1 and c(v) lt v forall v Given any small-enough mass of vK there is a unique mass of type vKminus1 that makesSeller indifferent between charging price vK and vKminus1 (If the mass is too low Seller prefersvK if it is too high she prefers vKminus1) Iterating down to keep Seller indifferent betweenall prices pins down an ICD Choose the maximum mass of type vK for which this worksRemove that ICD and repeat

19 The algorithm is defined for finite V and we take limits to handle the infinite case

17

Crucially whenever an ICD is removed the price corresponding to πfbs (Γ) remains op-

timal in the remaining ldquomarketrdquo this follows from the ICDrsquos defining property of Sellerindifference and an accounting identity Therefore Sellerrsquos profit in this segmentation ofICDs remains πfb

s (Γ) Moreover it is also optimal for Seller to always (ie for each microi) priceso that there is full trade or no trade Hence Buyerrsquos expected payoff can be either 0 or theentire surplus less πfb

s (Γ) It follows that the fully-informed Buyer information structuredefined by this set of ICDs implements point B and C in Figure 1 The entire triangle ABC

can then be implemented by convexification randomizing over this information structure(and two equilibria) and full information (where Seller obtains all the surplus)

The above logic of incentive compatible distributions reduces to that of Bergemann et al(2015) in the context of monopoly pricing Even in the monopoly pricing case Theorem 3part 2 strengthens the conclusions of Bergemann et al (2015) by ensuring approximatelyunique implementation

Remark 4 Theorems 1ndash3 imply that fully-informed-Buyer information structures imple-ment all implementable payoff pairs if and only if πfb

s (Γ) = πs(Γ) In that case trianglesAFG and ABC coincide in Figure 1 It follows that πfb

s (Γ) = πs(Γ) only when a fully-informed Buyer and uninformed Seller can result in full trade (v ge E[c(v)] and Sellerprices at v) or no trade (v le E[c(v)] and Seller prices at some p ge v) Interestinglywhen πfb

s (Γ) gt πs(Γ) fully-informed-Buyer information structures cannot even implementall payoff pairs implementable by uninformed-Seller information structures ie trianglesABC and ADE in Figure 1 are distinct That is when πfb

s (Γ) gt πs(Γ) there is an uninformed-Seller information structure that implements some πs lt πfb

s (Γ)20 Theorems 2ndash3 further im-ply in this case that Buyer can obtain a strictly higher payoff when he is not fully informed(and Seller is uninformed)

4 Discussion

41 Multidimensionality

Suppose Buyer and Sellerrsquos cost and valuation pair (c v) is a two-dimensional randomvariable distributed according to joint distribution micro with a compact support The ex-tension of our maintained assumption of commonly known gains from trade is for all

20 Pick any pprime gt v such that pprimeminusE[c(v)] lt πfbs (Γ) Following the construction described after Theorem 2 we

can mix all valuations v le pprime with a fraction λ gt 0 of valuations v gt pprime so that the mixture has posterior meanexactly pprime The remaining fraction 1 minus λ of valuations above pprime are revealed to Buyer With this uninformed-Seller information structure consider any equilibrium in which Buyer purchases when indifferent (Suchan equilibrium with price-independent belief exists) Sellerrsquos profit is at most (1 minus λ)πfb

s (Γ) from any pricep gt pprime and pprime minus E[c(v)] from price p = pprime Hence Sellerrsquos profit is strictly less than πfb

s (Γ)

18

(c v) isin Supp(micro) v ge c and E[v minus c] gt 0 An information structure is now a joint distribu-tion P (tb ts v c) whose marginal distribution on (v c) is micro

We claim that the content of our key results Theorem 1Theorem 1lowast Theorem 2 andTheorem 3 still hold To see why let v be the lowest valuation in the support of micro Sellerrsquosindividual rationality constraint is now maxv minus E[c] 0 as she can guarantee this by set-ting either a sufficiently high price or a price (arbitrarily close to) v regardless of her signalWe can define a cost function c(vprime) equiv Emicro[c|v = vprime] ge vprime This results in an environment sat-isfying all the maintained assumptions of our baseline model except that c(middot) may not becontinuous Recall that such continuity is not needed for proving Theorem 1 Theorem 1lowastBoth Theorem 2 and Theorem 3 use continuity of c(middot) to guarantee that Eν [c(v)] is a contin-uous function of ν isin ∆(V ) for a key convergence result However in the two-dimensionaltype environment Eν [c] is still a continuous function of ν isin ∆(V times C) Note that Theo-rem 3 uses upper semi-continuity of Sellerrsquos profit in price boundedness of c and c le v issufficient for such upper semi-continuity

42 Negative Trading Surplus

Returning to our baseline model we next discuss cases in which trade sometimes gen-erates negative surplus That is we drop the assumption that c(v) le v we do not requireE[v minus c(v)] gt 0 either Define Sλ(Γ) for λ isin [1infin) as

Sλ(Γ) equiv983133 v

v

[v minus c(v) + λ(v minus v)]+ micro(dv)

where [middot]+ equiv max middot 0 It is readily verified that S1(Γ) = E [[v minus c(v)]+] and limλrarr0

Sλ(Γ)λ =

E[v] minus v Allowing negative trading surplus does not affect our definition of wPBE Sothe notation Π(Γ) and πs(Γ) still have the same meanings as before The next propositionshows that Π(Γ) is now characterized by three constraints as before the two individualrationality constraints πs ge πs(Γ) and πb ge 0 and different now a Pareto frontier definedby all Sλ(Γ)

Proposition 1 Consider all information structures and equilibria when trade can generate negativesurplus

Π(Γ) =

983099983105983103

983105983101

πb ge 0

(πb πs) πs ge πs(Γ)

λπb + πs le Sλ(Γ) forallλ ge 1

983100983105983104

983105983102

Proposition 1 shows that when the trade can be inefficient it may be impossible to lower

19

Sellerrsquos profit without losing total surplus (ie πb + πs) This result is intuitive at the ex-tremes Suppose for simplicity v gt E[c(v)] If πs = πs(Γ) then since πs(Γ) = v minus E[c(v)]Seller must find it optimal to price at v and trade with probability one including whentrade generates negative surplus So Buyerrsquos payoff is E[v]minus v = limλrarrinfin Sλ(Γ)λ and totalsurplus is E[v minus c(v)] which includes the maximum amount of inefficient trade On theother hand an information structure that publicly reveals (only) whether trade is efficient(ie whether v ge c(v)) can implement efficient trade with all the surplus accruing to SellerIn this case Seller obtains S1(Γ) while Buyer obtains zero The proposition establishes thatin between S1(Γ) and πs(Γ) information structures and equilibria that lower Sellerrsquos profitmust reduce total surplus as the Pareto frontier Sλ(Γ) is a concave function with maximalslope minus1

The function capturing the Pareto frontier Sλ(Γ) is the maximal weighted sum of Buyerand Seller payoff assuming that Seller will trade at the low price p = v whenever it createspositive weighted total payoff Since we place higher weight on buyer (λ ge 1) S1(Γ) isthe upper bound for the weighted total payoff In the proof of Proposition 1 we showthat these bounds are tight mdash any payoff pair on the frontier can be implemented by someinformation structure The key observation is that whenever v minus c(v) + λ(v minus v) lt 0 itfollows that vminus c(v) lt (1minusλ)(vminus v) le 0 That is it is incentive compatible for Seller to notsell when it is common knowledge that v creates negative weighted total payoff Thereforewe can simply release a public signal indicating the sign of weighted total payoff and ifpositive induce Seller to sell at v which implies all bounds Sλ(Γ)

43 Linear Cost

When Sellerrsquos cost function c(v) is linear we can explicitly characterize an informationstructure that implements Sellerrsquos minimum implementable payoff πus

s (Γ) when Seller isuninformed (or per Theorem 2 when Buyer is better informed) subject to price-independentbeliefs We note that given the discussion in Subsection 41 a linear c(v) subsumes richerenvironments in which conditional expectations are linear including Gaussian environ-ments (cf Bar-Isaac et al 2018)

Condition 1 c(v) = λv + γ for some λ isin R

Let F (v) be the cumulative distribution function (CDF) corresponding to the prior mea-sure micro Let D(micro) be the set of all distributions whose CDF G satisfies

983133

V

vdG(v) =

983133

V

vdF (v) and983133 v

v

G(s)ds le983133 v

v

F (s)ds forallv isin V

20

That is D(micro) contains all distributions that are mean-preserving-contractions (MPC) of microIt is well known that D(micro) characterizes the set of distributions of Buyer posterior meansthat can be generated by any (uninformed-Seller) information structure

Definition 2 G is an incentive compatible distribution (ICD) if

983133

sgep

(pminus c(s))G(ds) = constant ge 0 forallp isin Supp(G)

That is when Buyerrsquos (mean) valuation is distributed according to an ICD G Seller isindifferent among all prices in the support of G assuming Buyer buys when indifferent Wefocus on a special family of ICDs whose supports are intervals [vlowast v

lowast] Given Condition 1such ICDs have analytical an expression

G(v) =

983099983105983105983103

983105983105983101

1minus983061

(1minus λ)v minus γ

(1minus λ)vlowast minus γ

983062 1λminus1

if λ ∕= 1

1minus evminusvlowast

γ if λ = 1

(3)

Given any vlowast and the above G(v) an upper bound vlowast is pinned down by the conditionEG[v] = EF [v] So a family of ICDs is parametrized by a single parameter vlowast and hence anICD is denoted by CDF 983144Gvlowast(v) with density 983144gvlowast(v) It is not difficult to verify that 983144Gvlowast(v) ismonotonic in vlowast for v in the support Consequently all ICDs are ordered according to theMPC order For vlowast isin [vE[v]] when vlowast increases the ICD increases in the MPC order

Proposition 2 Suppose Condition 1 is satisfied and let plowast equiv min983153vlowast983055983055 983144Gvlowast isin D(micro)

983154 It holds that

πuss (Γ) = plowast minus Emicro[c(v)]

Proposition 2 states that an uninformed Sellerrsquos minimum payoff (with price-independentbeliefs) is characterized by the ICDs that are MPCs of the prior distribution micro By defini-tion of ICDs Sellerrsquos profit when facing ICD 983144Gvlowast is vlowast minus E[v] Sellerrsquos minimum payoff isimplemented by the most dispersed ICD that is a MPC of the prior distribution

In proving Proposition 2 the key step is to show that given the prior micro garbling Buyerrsquosinformation such that the posterior mean distribution is an ICD makes Seller weakly worseoff It is then without loss to only consider ICDs to implement πus

s (Γ) We can thus workwith a tractable one-dimensional problem Consider the set D(micro) Suppose we find G(v)

such that (i) G isin D(micro) (ii) G is an ICD and (iii) existp such that983125 p

vG(s)ds =

983125 p

vF (s)ds Such

a G is the most-dispersed ICD that is a mean preserving contraction of F Consider the

21

following two manipulations

λ

983133 v

p

F (s)ds =

983133 v

p

(pminus c(s))dF (s)minus (1minus F (v)) (pminus c(v)) + λ(v minus p) (4)

λ

983133 v

p

G(s)ds =

983133 v

p

(pminus c(s))dG(s)minus (1minusG(v)) (pminus c(v)) + λ(v minus p) (5)

First by property (iii) above the LHS of Equation 4 equals the LHS of Equation 5 Secondsince

983125 v

p(F (s) minus G(s))ds is minimized at v this implies F (p) le G(p) Therefore the first

term on RHS of Equation 4 must be larger than that of Equation 5 Notice that the firstterm on RHS is Sellerrsquos profit when offering price p This implies that the profit fromoffering p given Buyer mean-valuation distribution G(v) is lower than that given F (v)which is lower than Sellerrsquos maximum profit given F (v) On the other hand by the ICDproperty p is already Sellerrsquos optimal price given G(v) The optimal profit given valuationdistribution F must be no lower than that of G Therefore we have proved that to minimizeSellerrsquos profit it is without loss to consider only ICDs within the set D(micro) which is a onedimensional subspace The detailed proof that such ICDs exist is relegated to the appendix

The monopoly-pricing environment with c(middot) = v is covered by Condition 1 with λ = 0

and γ = v The distribution G in (3) then reduces to that identified by Roesler and Szentes(2017)

If the prior micro has binary support then the cost function c(v) is linear In this special casewe can solve for πus

s (Γ) explicitly

Corollary 1 Suppose micro has binary support V = v1 v2 Let λ = c(v2)minusc(v1)v2minusv1

and let p be theunique solution to

(pminus c(p))1

λminus1 (pminus E[c(v)]) = (v2 minus c(v2))λ

λminus1 (6)

Then πuss (Γ) = max p v2minus E[c(v)]

References

AKERLOF G A (1970) ldquoThe Market for rdquoLemonsrdquo Quality Uncertainty and the MarketMechanismrdquo Quarterly Journal of Economics 84 488ndash500

BAGWELL K AND M H RIORDAN (1991) ldquoHigh and Declining Prices Signal ProductQualityrdquo American Economic Review 81 224ndash239

BAR-ISAAC H I JEWITT AND C LEAVER (2018) ldquoAdverse Selection Efficiency and theStructure of Informationrdquo Unpublished

22

BERGEMANN D B BROOKS AND S MORRIS (2015) ldquoThe Limits of Price Discrimina-tionrdquo American Economic Review 105 921ndash57

BERGEMANN D AND S MORRIS (2016) ldquoBayes Correlated Equilibrium and the Compar-ison of Information Structures in Gamesrdquo Theoretical Economics 11 487ndash522

BLACKWELL D (1953) ldquoEquivalent Comparisons of Experimentsrdquo Annals of MathematicalStatistics 24 265ndash272

CHO I-K AND D KREPS (1987) ldquoSignaling Games and Stable Equilibriardquo Quarterly Jour-nal of Economics 102 179ndash221

DU S (2018) ldquoRobust Mechanisms Under Common Valuationrdquo Econometrica 86 1569ndash1588

DURRETT R (1995) Probability Theory and Examples Duxbury Press 2nd ed

EINAV L AND A FINKELSTEIN (2011) ldquoSelection in Insurance Markets Theory and Em-pirics in Picturesrdquo Journal of Economic Perspectives 25 115ndash38

FUDENBERG D AND J TIROLE (1991) ldquoPerfect Bayesian Equilibrium and Sequential Equi-libriumrdquo Journal of Economic Theory 53 236ndash260

GARCIA D R TEPER AND M TSUR (2018) ldquoInformation Design in Insurance MarketsSelling Peaches in a Market for Lemonsrdquo Unpublished

GREENWALD B C (1986) ldquoAdverse Selection in the Labour Marketrdquo Review of EconomicStudies 53 325ndash347

JOVANOVIC B (1982) ldquoFavorable Selection with Asymmetric Informationrdquo QuarterlyJournal of Economics 97 535ndash539

KESSLER A S (2001) ldquoRevisiting the Lemons Marketrdquo International Economic Review 4225ndash41

KREPS D AND R WILSON (1982) ldquoSequential Equilibriardquo Econometrica 50 863ndash894

LEHMANN E L (1988) ldquoComparing Location Experimentsrdquo The Annals of Statistics 16521ndash533

LEVIN J (2001) ldquoInformation and the Market for Lemonsrdquo RAND Journal of Economics32 657ndash666

23

LIBGOBER J AND X MU (2019) ldquoInformational Robustness in Intertemporal PricingrdquoWorking paper

MAKRIS M AND L RENOU (2018) ldquoInformation Design in Multi-Stage Gamesrdquo Workingpaper

MILGROM P R AND R J WEBER (1985) ldquoDistributional Strategies for Games with In-complete Informationrdquo Mathematics of Operations Research 10 619ndash632

MUSSA M AND S ROSEN (1978) ldquoMonopoly and Product Qualityrdquo Journal of EconomicTheory 18 301ndash317

RAVID D A-K ROESLER AND B SZENTES (2019) ldquoLearning Before Trading On theInefficiency of Ignoring Free Informationrdquo Working paper

ROESLER A-K AND B SZENTES (2017) ldquoBuyer-Optimal Learning and Monopoly Pric-ingrdquo American Economic Review 107 2072ndash80

TERSTIEGE S AND C WASSER (2019) ldquoBuyer-optimal Extensionproof InformationrdquoWorking paper

Appendices omitted from this version but are available on request

24

Even when our environment is specialized to monopoly pricing it is worth highlightingtwo contrasts between Theorem 11lowast and results of Bergemann et al (2015) and Roeslerand Szentes (2017) First we find that by not restricting the monopolist to be uninformedthe implementable payoff set typically expands rather dramatically trade can be efficientwith the monopolist securing none of the surplus beyond her payoff guarantee (πs) whichmay be zero (Roesler and Szentes establish implicitly that the implementable set withan uninformed monopolist is a superset of Bergemann et alrsquos which has a fully-informedconsumer) We will see in Subsection 32 that what is crucial to this expansion is price-dependent beliefs In particular the proof of Theorem 1lowast uses an information structure inwhich Buyer is not better informed than Seller mdash if he were then sequential equilibriumwould imply price-independent beliefs Second Theorem 1lowast establishes that for a givenε gt 0 a single information structure (and price grid) can be used to approximate the entirepayoff set Π(Γ) analogously to the construction described after Theorem 1 that used asingle information structure15 Bergemann et al (2015) and Roesler and Szentes (2017) onthe other hand vary information structures to span their payoff sets

32 More-informed Buyer and Price-independent Beliefs

In some economic settings it is plausible that Buyer is more informed than Seller Howdoes a restriction to such information structures ie τ isin Tmb affect the implementablepayoff set It turns out that what is in fact crucial is price-independent beliefs We have ex-plained earlier why it is desirable to impose this condition when Buyer is more informedbut that the condition is well defined even otherwise If Buyer is not more informed thanSeller price-independent beliefs ought to be viewed as an equilibrium restriction in gen-eral To reduce repetition readers should be bear in mind for the rest of this subsectionthe qualifier ldquowith price-independent beliefsrdquo applies unless stated explicitly otherwise

It is useful to define

πuss (Γ) equiv inf πs exist(πb πs) isin Πlowast

us(Γ) (1)

as the lowest payoff that an uninformed Seller can obtain no matter Buyerrsquos information(among equilibria with price-independent beliefs) Plainly πus

s (Γ) ge πs(Γ) In monopoly

Buyer mixed response that another type would Our construction has multiple Buyer types that are imper-fectly correlated with Seller types So different types of Seller have different beliefs about Buyer types Thisblunts the power of dominance considerations to the point where D1 does not exclude any Seller type fromthe support of Buyerrsquos off-path belief

15 In fact if one lets the price grid vary with ε then a single information structure implements exactlyrather than approximately in sequential equilibrium all payoffs ε-away from the boundary of Π(Γ) See in the Appendix for a formal statement

13

pricing with V = [v v] and c(middot) = v Roesler and Szentesrsquos (2017) characterization of theconsumer-optimal information structure identifies πus

s establishing that πuss gt πs If there

is no trade due to adverse selection when Seller is uniformed and Buyer has some informa-tion then πus

s = πs = 0 We do not have a general explicit formula for πuss Subsection 43

provides it for linear c(middot) Nonetheless we establish next that (i) the only additional restric-tion on equilibrium payoffs imposed by price-independent beliefs is a lower bound of πus

s

for Seller and (ii) uninformed-Seller information structures implement all such payoffs

Theorem 2 Consider equilibria with price-independent beliefs

1 Πlowast(Γ) = Πlowastmb(Γ) = Πlowast

us(Γ)

2 Πlowastus(Γ) = (πb πs) isin Π(Γ) πs ge πus

s (Γ)

3 For any (πb πs) isin Πlowastus(Γ) with πs gt πus

s (Γ) there is τ isin Tus with Π(Γ τ) = (πb πs)

Remark 1 We believe the substance of Theorem 2 would hold using discretizations and se-quential equilibria analogous to Theorem 1lowast As previously noted sequential equilibriumimplies price-independent beliefs when Buyer is more informed than Seller

To digest Theorem 2 note that Πlowast(Γ) sup Πlowastmb(Γ) sup Πlowast

us(Γ) is trivial So part 1 of thetheorem amounts to establishing the reverse inclusions The intuition for thismdashgiven part2rsquos characterization of Πlowast

usmdashis fairly straightforward with price-independent beliefs ad-ditional information cannot harm Seller even though it could alter the set of equilibria SoSellerrsquos lowest payoff obtains when she is uninformed

The characterization in part 2 of payoffs with an uninformed Seller corresponds to thetriangle ADE in Figure 1 Part 3 of the theorem assures ldquounique implementationrdquo of allimplementable payoffs satisfying πs gt πus

s (Γ) That is for any such payoff pair there is anuninformed-Seller information structure such that all equilibria (with price-independentbeliefs) induce exactly that payoff pair Unique implementation is appealing for multiplereasons one of which is that it obviates concerns about which among multiple payoff-distinct equilibria is more reasonable For example Ravid Roesler and Szentes (2019)bring to the fore the issue of equilibrium multiplicity in Roesler and Szentesrsquos (2017) infor-mation structure

Let us describe how we obtain the characterization of Πlowastus(Γ) and unique implementa-

tion There are two steps The first ensures that there is some information structure callit τ lowast isin Tus that implements Sellerrsquos payoff πus

s (Γ) That is we ensure that the infimum

14

in (1) is in fact a minimum16 While this argument is technical knowing τ lowast exists is usefulin what follows The second and economically insightful step is to construct informa-tion structures that implement every point in the triangle Πlowast

us(Γ) by suitably garbling theinformation structure τ lowast The construction is illustrated in Figure 2 Consider the distri-bution of Buyerrsquos posterior mean of his valuation v in information structure τ lowast (Givenprice-independent beliefs Buyerrsquos posterior mean is sufficient for his decision) For sim-plicity suppose this posterior-mean distribution has a density as depicted by the red curvein Figure 2 Fix any (πb πs) isin Πlowast

us(Γ)

First there is some number zlowast such that πb + πs is the total surplus from trading onlywhen Buyerrsquos posterior mean is greater than zlowast Next there is some price plowast ge zlowast suchthat Sellerrsquos payoff is πs if all these trades were to occur at price plowast17 Note that plowast must beno larger than the expected Buyer posterior mean conditional on that being above zlowast forotherwise πb lt 0 We claim that the information structure τ lowast can be garbled so that plowast isan equilibrium price and trade occurs only when Buyerrsquos posterior mean is greater thanzlowast The garbling is illustrated in Figure 2 by the blue distribution There is one signal that

pzv

ν(v)

Figure 2 Construction of information structure in Theorem 2

Buyer receives when the original posterior mean is between zlowast and plowast and also receiveswith some probability when the original posterior mean is above plowast The probability ischosen to make the posterior mean from this signal exactly plowast Apart from this one newsignal Buyer receives the original signal in τ lowast Plainly this is a garbling of τ lowast and hence isfeasible

16 The difficulty is in establishing suitable continuity Uninformed-Seller information structures can beviewed as probability measures over Buyerrsquos beliefs with convergence in the sense of the weak topologyThis topology ensures continuity with respect to probability measures of expectations of continuous orat least Lipschitz (and bounded) functions However Sellerrsquos expected payoff is not the expectation of aLipschitz function as Sellerrsquos profit is truncated at the price she charges

17 That plowast ge zlowast follows from πs ge πuss (Γ) as πs(Γ) itself is weakly larger than Sellerrsquos payoff from posting

price zlowast (and thus trading with the same set of Buyer posterior means) under information structure τlowast

15

Figure 2 makes clear why the new information structure has an equilibrium with priceplowast and Buyer breaking indifference in favor of trading (i) Sellerrsquos profit from posting anyprice below zlowast is the same as under τ lowast and hence no larger than πus

s (Γ) (ii) similarly Sellerrsquosprofit from posting any price above plowast is no higher than some fraction of πus

s (Γ) and (iii)any price between zlowast and plowast is worse that price plowast Moreover since Sellerrsquos profit fromoffering any price other than plowast is no more than πus

s (Γ) it follows that when πs gt πuss (Γ)

Buyer must break indifference as specified for Seller to have an optimal price and theequilibrium payoffs are unique

Remark 2 The above logic establishes that given any τ isin Tus that implements some (πb πs)τ can be garbled to uniquely implement any (πprime

b πprimes) isin Π(Γ) such that πprime

s gt πs That isan uninformed Sellerrsquos payoff can always be strictly raised and Buyerrsquos payoff reduced(strictly so long as it was not already zero) by garbling Buyerrsquos information

Remark 3 According to Theorem 11lowast and Theorem 2 uninformed-Seller information struc-tures cannot implement all implementable payoff pairs in an environment Γ if and only ifπuss (Γ) gt πs(Γ) This inequality fails in environment Γ if πus

s (Γ) = 0 since that impliesπuss (Γ) = πs(Γ) = 0 An example is when there is no trade due to adverse selection when

Seller is uniformed and Buyer has some information On the other hand it is sufficient forthe inequality that

v le E[c(v)] and forallv isin V c(v) lt v (2)

To see why (2) implies πuss (Γ) gt πs(Γ) notice that in any uninformed-Seller information

structure Seller can price at slightly less than Buyerrsquos highest posterior mean valuationand guarantee trade with only (a neighborhood of) that Buyer type If c(v) lt v for all v thisgives Seller a positive expected payoff and hence πus

s (Γ) gt 018 But the first inequality in (2)is equivalent to πs(Γ) = 0 We observe that Condition (2) is compatible with severe adverseselection resulting in very little trade when Seller is uninformed and Buyer (partially orfully) informed

33 Fully-Informed Buyer

We now turn to the third canonical class of information structures Buyer is fully in-formed of his value v As this is a special case of more-informed Buyer we maintain price-independent beliefs throughout this subsection

18 More precisely as V is compact c(v) lt v for all v implies there exists ε gt 0 such that v minus c(v) gt εGiven any uninformed-Seller information structure let mv be the highest posterior mean valuation in thesupport of the posterior means induced by Buyerrsquos signals So there is positive probability of Buyer signalswith posterior mean valuations at least mv minus ε2 By pricing at mv minus ε2 Sellerrsquos expected cost conditionalon trade is bounded above by mv minus ε and hence Sellerrsquos profit conditional on trade is at least ε2 gt 0 Itfollows that πus

s gt 0

16

Faced with a fully informed Buyer and any sequentially rational Buyer strategy an un-informed Seller can guarantee a profit level

πfbs (Γ) equiv sup

p

983133 v

p

(pminus c(v))micro(dv)

regardless of her information Plainly πfbs (Γ) ge πus

s (Γ) In monopoly pricing with c(middot) = vRoesler and Szentes (2017) have shown that πfb

s gt πuss if there is no trade due to adverse

selection when Buyer is fully informed and Seller is uniformed then πfbs = πus

s = 0 Belowwe establish that when Buyer is fully informed πfb

s is the only additional constraint onequilibrium payoffs Let Bε(πb πs) denote an ε-ball around (πb πs)

Theorem 3 Consider a fully informed Buyer and price-independent beliefs

1 Πlowastfb(Γ) = (πb πs) isin Π(Γ) πs ge πfb

s (Γ)

2 For any (πb πs) isin Πlowastfb(Γ) and any ε gt 0 there is τ isin Tfb with Π(Γ τ) sub Bε(πb πs)

The payoff set characterized in Theorem 3 corresponds to the triangle ABE in Figure 1Part 2 of the theorem establishes ldquoapproximately unique implementationrdquo for any point inthe triangle there is an information structure under which all equilibria yield payoff pairsin a neighborhood of that point

Here is the idea behind part 1 of Theorem 3 When Buyer is fully informed an informa-tion structure can be viewed as dividing vrsquos prior distribution micro into a set of microi that aver-age to micro with Seller informed of which microi she faces Theorem 3 is proven by establishingthat we can divide micro suitably so that against each microi Seller is indifferent between pricingat all prices in the support of microi including the price corresponding to πfb

s in that environ-ment We call such a microi a (Seller-price) incentive compatible distribution or ICD (Bergemannet al (2015) call their analogous construct an extreme market) The Appendix provides aldquogreedyrdquo algorithm to compute ICDs19

We can sketch how the algorithm works and construct a set of ICDs that average to theprior Suppose V = v1 v2 vK with vi lt vi+1 for i isin 1 K minus 1 and c(v) lt v forall v Given any small-enough mass of vK there is a unique mass of type vKminus1 that makesSeller indifferent between charging price vK and vKminus1 (If the mass is too low Seller prefersvK if it is too high she prefers vKminus1) Iterating down to keep Seller indifferent betweenall prices pins down an ICD Choose the maximum mass of type vK for which this worksRemove that ICD and repeat

19 The algorithm is defined for finite V and we take limits to handle the infinite case

17

Crucially whenever an ICD is removed the price corresponding to πfbs (Γ) remains op-

timal in the remaining ldquomarketrdquo this follows from the ICDrsquos defining property of Sellerindifference and an accounting identity Therefore Sellerrsquos profit in this segmentation ofICDs remains πfb

s (Γ) Moreover it is also optimal for Seller to always (ie for each microi) priceso that there is full trade or no trade Hence Buyerrsquos expected payoff can be either 0 or theentire surplus less πfb

s (Γ) It follows that the fully-informed Buyer information structuredefined by this set of ICDs implements point B and C in Figure 1 The entire triangle ABC

can then be implemented by convexification randomizing over this information structure(and two equilibria) and full information (where Seller obtains all the surplus)

The above logic of incentive compatible distributions reduces to that of Bergemann et al(2015) in the context of monopoly pricing Even in the monopoly pricing case Theorem 3part 2 strengthens the conclusions of Bergemann et al (2015) by ensuring approximatelyunique implementation

Remark 4 Theorems 1ndash3 imply that fully-informed-Buyer information structures imple-ment all implementable payoff pairs if and only if πfb

s (Γ) = πs(Γ) In that case trianglesAFG and ABC coincide in Figure 1 It follows that πfb

s (Γ) = πs(Γ) only when a fully-informed Buyer and uninformed Seller can result in full trade (v ge E[c(v)] and Sellerprices at v) or no trade (v le E[c(v)] and Seller prices at some p ge v) Interestinglywhen πfb

s (Γ) gt πs(Γ) fully-informed-Buyer information structures cannot even implementall payoff pairs implementable by uninformed-Seller information structures ie trianglesABC and ADE in Figure 1 are distinct That is when πfb

s (Γ) gt πs(Γ) there is an uninformed-Seller information structure that implements some πs lt πfb

s (Γ)20 Theorems 2ndash3 further im-ply in this case that Buyer can obtain a strictly higher payoff when he is not fully informed(and Seller is uninformed)

4 Discussion

41 Multidimensionality

Suppose Buyer and Sellerrsquos cost and valuation pair (c v) is a two-dimensional randomvariable distributed according to joint distribution micro with a compact support The ex-tension of our maintained assumption of commonly known gains from trade is for all

20 Pick any pprime gt v such that pprimeminusE[c(v)] lt πfbs (Γ) Following the construction described after Theorem 2 we

can mix all valuations v le pprime with a fraction λ gt 0 of valuations v gt pprime so that the mixture has posterior meanexactly pprime The remaining fraction 1 minus λ of valuations above pprime are revealed to Buyer With this uninformed-Seller information structure consider any equilibrium in which Buyer purchases when indifferent (Suchan equilibrium with price-independent belief exists) Sellerrsquos profit is at most (1 minus λ)πfb

s (Γ) from any pricep gt pprime and pprime minus E[c(v)] from price p = pprime Hence Sellerrsquos profit is strictly less than πfb

s (Γ)

18

(c v) isin Supp(micro) v ge c and E[v minus c] gt 0 An information structure is now a joint distribu-tion P (tb ts v c) whose marginal distribution on (v c) is micro

We claim that the content of our key results Theorem 1Theorem 1lowast Theorem 2 andTheorem 3 still hold To see why let v be the lowest valuation in the support of micro Sellerrsquosindividual rationality constraint is now maxv minus E[c] 0 as she can guarantee this by set-ting either a sufficiently high price or a price (arbitrarily close to) v regardless of her signalWe can define a cost function c(vprime) equiv Emicro[c|v = vprime] ge vprime This results in an environment sat-isfying all the maintained assumptions of our baseline model except that c(middot) may not becontinuous Recall that such continuity is not needed for proving Theorem 1 Theorem 1lowastBoth Theorem 2 and Theorem 3 use continuity of c(middot) to guarantee that Eν [c(v)] is a contin-uous function of ν isin ∆(V ) for a key convergence result However in the two-dimensionaltype environment Eν [c] is still a continuous function of ν isin ∆(V times C) Note that Theo-rem 3 uses upper semi-continuity of Sellerrsquos profit in price boundedness of c and c le v issufficient for such upper semi-continuity

42 Negative Trading Surplus

Returning to our baseline model we next discuss cases in which trade sometimes gen-erates negative surplus That is we drop the assumption that c(v) le v we do not requireE[v minus c(v)] gt 0 either Define Sλ(Γ) for λ isin [1infin) as

Sλ(Γ) equiv983133 v

v

[v minus c(v) + λ(v minus v)]+ micro(dv)

where [middot]+ equiv max middot 0 It is readily verified that S1(Γ) = E [[v minus c(v)]+] and limλrarr0

Sλ(Γ)λ =

E[v] minus v Allowing negative trading surplus does not affect our definition of wPBE Sothe notation Π(Γ) and πs(Γ) still have the same meanings as before The next propositionshows that Π(Γ) is now characterized by three constraints as before the two individualrationality constraints πs ge πs(Γ) and πb ge 0 and different now a Pareto frontier definedby all Sλ(Γ)

Proposition 1 Consider all information structures and equilibria when trade can generate negativesurplus

Π(Γ) =

983099983105983103

983105983101

πb ge 0

(πb πs) πs ge πs(Γ)

λπb + πs le Sλ(Γ) forallλ ge 1

983100983105983104

983105983102

Proposition 1 shows that when the trade can be inefficient it may be impossible to lower

19

Sellerrsquos profit without losing total surplus (ie πb + πs) This result is intuitive at the ex-tremes Suppose for simplicity v gt E[c(v)] If πs = πs(Γ) then since πs(Γ) = v minus E[c(v)]Seller must find it optimal to price at v and trade with probability one including whentrade generates negative surplus So Buyerrsquos payoff is E[v]minus v = limλrarrinfin Sλ(Γ)λ and totalsurplus is E[v minus c(v)] which includes the maximum amount of inefficient trade On theother hand an information structure that publicly reveals (only) whether trade is efficient(ie whether v ge c(v)) can implement efficient trade with all the surplus accruing to SellerIn this case Seller obtains S1(Γ) while Buyer obtains zero The proposition establishes thatin between S1(Γ) and πs(Γ) information structures and equilibria that lower Sellerrsquos profitmust reduce total surplus as the Pareto frontier Sλ(Γ) is a concave function with maximalslope minus1

The function capturing the Pareto frontier Sλ(Γ) is the maximal weighted sum of Buyerand Seller payoff assuming that Seller will trade at the low price p = v whenever it createspositive weighted total payoff Since we place higher weight on buyer (λ ge 1) S1(Γ) isthe upper bound for the weighted total payoff In the proof of Proposition 1 we showthat these bounds are tight mdash any payoff pair on the frontier can be implemented by someinformation structure The key observation is that whenever v minus c(v) + λ(v minus v) lt 0 itfollows that vminus c(v) lt (1minusλ)(vminus v) le 0 That is it is incentive compatible for Seller to notsell when it is common knowledge that v creates negative weighted total payoff Thereforewe can simply release a public signal indicating the sign of weighted total payoff and ifpositive induce Seller to sell at v which implies all bounds Sλ(Γ)

43 Linear Cost

When Sellerrsquos cost function c(v) is linear we can explicitly characterize an informationstructure that implements Sellerrsquos minimum implementable payoff πus

s (Γ) when Seller isuninformed (or per Theorem 2 when Buyer is better informed) subject to price-independentbeliefs We note that given the discussion in Subsection 41 a linear c(v) subsumes richerenvironments in which conditional expectations are linear including Gaussian environ-ments (cf Bar-Isaac et al 2018)

Condition 1 c(v) = λv + γ for some λ isin R

Let F (v) be the cumulative distribution function (CDF) corresponding to the prior mea-sure micro Let D(micro) be the set of all distributions whose CDF G satisfies

983133

V

vdG(v) =

983133

V

vdF (v) and983133 v

v

G(s)ds le983133 v

v

F (s)ds forallv isin V

20

That is D(micro) contains all distributions that are mean-preserving-contractions (MPC) of microIt is well known that D(micro) characterizes the set of distributions of Buyer posterior meansthat can be generated by any (uninformed-Seller) information structure

Definition 2 G is an incentive compatible distribution (ICD) if

983133

sgep

(pminus c(s))G(ds) = constant ge 0 forallp isin Supp(G)

That is when Buyerrsquos (mean) valuation is distributed according to an ICD G Seller isindifferent among all prices in the support of G assuming Buyer buys when indifferent Wefocus on a special family of ICDs whose supports are intervals [vlowast v

lowast] Given Condition 1such ICDs have analytical an expression

G(v) =

983099983105983105983103

983105983105983101

1minus983061

(1minus λ)v minus γ

(1minus λ)vlowast minus γ

983062 1λminus1

if λ ∕= 1

1minus evminusvlowast

γ if λ = 1

(3)

Given any vlowast and the above G(v) an upper bound vlowast is pinned down by the conditionEG[v] = EF [v] So a family of ICDs is parametrized by a single parameter vlowast and hence anICD is denoted by CDF 983144Gvlowast(v) with density 983144gvlowast(v) It is not difficult to verify that 983144Gvlowast(v) ismonotonic in vlowast for v in the support Consequently all ICDs are ordered according to theMPC order For vlowast isin [vE[v]] when vlowast increases the ICD increases in the MPC order

Proposition 2 Suppose Condition 1 is satisfied and let plowast equiv min983153vlowast983055983055 983144Gvlowast isin D(micro)

983154 It holds that

πuss (Γ) = plowast minus Emicro[c(v)]

Proposition 2 states that an uninformed Sellerrsquos minimum payoff (with price-independentbeliefs) is characterized by the ICDs that are MPCs of the prior distribution micro By defini-tion of ICDs Sellerrsquos profit when facing ICD 983144Gvlowast is vlowast minus E[v] Sellerrsquos minimum payoff isimplemented by the most dispersed ICD that is a MPC of the prior distribution

In proving Proposition 2 the key step is to show that given the prior micro garbling Buyerrsquosinformation such that the posterior mean distribution is an ICD makes Seller weakly worseoff It is then without loss to only consider ICDs to implement πus

s (Γ) We can thus workwith a tractable one-dimensional problem Consider the set D(micro) Suppose we find G(v)

such that (i) G isin D(micro) (ii) G is an ICD and (iii) existp such that983125 p

vG(s)ds =

983125 p

vF (s)ds Such

a G is the most-dispersed ICD that is a mean preserving contraction of F Consider the

21

following two manipulations

λ

983133 v

p

F (s)ds =

983133 v

p

(pminus c(s))dF (s)minus (1minus F (v)) (pminus c(v)) + λ(v minus p) (4)

λ

983133 v

p

G(s)ds =

983133 v

p

(pminus c(s))dG(s)minus (1minusG(v)) (pminus c(v)) + λ(v minus p) (5)

First by property (iii) above the LHS of Equation 4 equals the LHS of Equation 5 Secondsince

983125 v

p(F (s) minus G(s))ds is minimized at v this implies F (p) le G(p) Therefore the first

term on RHS of Equation 4 must be larger than that of Equation 5 Notice that the firstterm on RHS is Sellerrsquos profit when offering price p This implies that the profit fromoffering p given Buyer mean-valuation distribution G(v) is lower than that given F (v)which is lower than Sellerrsquos maximum profit given F (v) On the other hand by the ICDproperty p is already Sellerrsquos optimal price given G(v) The optimal profit given valuationdistribution F must be no lower than that of G Therefore we have proved that to minimizeSellerrsquos profit it is without loss to consider only ICDs within the set D(micro) which is a onedimensional subspace The detailed proof that such ICDs exist is relegated to the appendix

The monopoly-pricing environment with c(middot) = v is covered by Condition 1 with λ = 0

and γ = v The distribution G in (3) then reduces to that identified by Roesler and Szentes(2017)

If the prior micro has binary support then the cost function c(v) is linear In this special casewe can solve for πus

s (Γ) explicitly

Corollary 1 Suppose micro has binary support V = v1 v2 Let λ = c(v2)minusc(v1)v2minusv1

and let p be theunique solution to

(pminus c(p))1

λminus1 (pminus E[c(v)]) = (v2 minus c(v2))λ

λminus1 (6)

Then πuss (Γ) = max p v2minus E[c(v)]

References

AKERLOF G A (1970) ldquoThe Market for rdquoLemonsrdquo Quality Uncertainty and the MarketMechanismrdquo Quarterly Journal of Economics 84 488ndash500

BAGWELL K AND M H RIORDAN (1991) ldquoHigh and Declining Prices Signal ProductQualityrdquo American Economic Review 81 224ndash239

BAR-ISAAC H I JEWITT AND C LEAVER (2018) ldquoAdverse Selection Efficiency and theStructure of Informationrdquo Unpublished

22

BERGEMANN D B BROOKS AND S MORRIS (2015) ldquoThe Limits of Price Discrimina-tionrdquo American Economic Review 105 921ndash57

BERGEMANN D AND S MORRIS (2016) ldquoBayes Correlated Equilibrium and the Compar-ison of Information Structures in Gamesrdquo Theoretical Economics 11 487ndash522

BLACKWELL D (1953) ldquoEquivalent Comparisons of Experimentsrdquo Annals of MathematicalStatistics 24 265ndash272

CHO I-K AND D KREPS (1987) ldquoSignaling Games and Stable Equilibriardquo Quarterly Jour-nal of Economics 102 179ndash221

DU S (2018) ldquoRobust Mechanisms Under Common Valuationrdquo Econometrica 86 1569ndash1588

DURRETT R (1995) Probability Theory and Examples Duxbury Press 2nd ed

EINAV L AND A FINKELSTEIN (2011) ldquoSelection in Insurance Markets Theory and Em-pirics in Picturesrdquo Journal of Economic Perspectives 25 115ndash38

FUDENBERG D AND J TIROLE (1991) ldquoPerfect Bayesian Equilibrium and Sequential Equi-libriumrdquo Journal of Economic Theory 53 236ndash260

GARCIA D R TEPER AND M TSUR (2018) ldquoInformation Design in Insurance MarketsSelling Peaches in a Market for Lemonsrdquo Unpublished

GREENWALD B C (1986) ldquoAdverse Selection in the Labour Marketrdquo Review of EconomicStudies 53 325ndash347

JOVANOVIC B (1982) ldquoFavorable Selection with Asymmetric Informationrdquo QuarterlyJournal of Economics 97 535ndash539

KESSLER A S (2001) ldquoRevisiting the Lemons Marketrdquo International Economic Review 4225ndash41

KREPS D AND R WILSON (1982) ldquoSequential Equilibriardquo Econometrica 50 863ndash894

LEHMANN E L (1988) ldquoComparing Location Experimentsrdquo The Annals of Statistics 16521ndash533

LEVIN J (2001) ldquoInformation and the Market for Lemonsrdquo RAND Journal of Economics32 657ndash666

23

LIBGOBER J AND X MU (2019) ldquoInformational Robustness in Intertemporal PricingrdquoWorking paper

MAKRIS M AND L RENOU (2018) ldquoInformation Design in Multi-Stage Gamesrdquo Workingpaper

MILGROM P R AND R J WEBER (1985) ldquoDistributional Strategies for Games with In-complete Informationrdquo Mathematics of Operations Research 10 619ndash632

MUSSA M AND S ROSEN (1978) ldquoMonopoly and Product Qualityrdquo Journal of EconomicTheory 18 301ndash317

RAVID D A-K ROESLER AND B SZENTES (2019) ldquoLearning Before Trading On theInefficiency of Ignoring Free Informationrdquo Working paper

ROESLER A-K AND B SZENTES (2017) ldquoBuyer-Optimal Learning and Monopoly Pric-ingrdquo American Economic Review 107 2072ndash80

TERSTIEGE S AND C WASSER (2019) ldquoBuyer-optimal Extensionproof InformationrdquoWorking paper

Appendices omitted from this version but are available on request

24

pricing with V = [v v] and c(middot) = v Roesler and Szentesrsquos (2017) characterization of theconsumer-optimal information structure identifies πus

s establishing that πuss gt πs If there

is no trade due to adverse selection when Seller is uniformed and Buyer has some informa-tion then πus

s = πs = 0 We do not have a general explicit formula for πuss Subsection 43

provides it for linear c(middot) Nonetheless we establish next that (i) the only additional restric-tion on equilibrium payoffs imposed by price-independent beliefs is a lower bound of πus

s

for Seller and (ii) uninformed-Seller information structures implement all such payoffs

Theorem 2 Consider equilibria with price-independent beliefs

1 Πlowast(Γ) = Πlowastmb(Γ) = Πlowast

us(Γ)

2 Πlowastus(Γ) = (πb πs) isin Π(Γ) πs ge πus

s (Γ)

3 For any (πb πs) isin Πlowastus(Γ) with πs gt πus

s (Γ) there is τ isin Tus with Π(Γ τ) = (πb πs)

Remark 1 We believe the substance of Theorem 2 would hold using discretizations and se-quential equilibria analogous to Theorem 1lowast As previously noted sequential equilibriumimplies price-independent beliefs when Buyer is more informed than Seller

To digest Theorem 2 note that Πlowast(Γ) sup Πlowastmb(Γ) sup Πlowast

us(Γ) is trivial So part 1 of thetheorem amounts to establishing the reverse inclusions The intuition for thismdashgiven part2rsquos characterization of Πlowast

usmdashis fairly straightforward with price-independent beliefs ad-ditional information cannot harm Seller even though it could alter the set of equilibria SoSellerrsquos lowest payoff obtains when she is uninformed

The characterization in part 2 of payoffs with an uninformed Seller corresponds to thetriangle ADE in Figure 1 Part 3 of the theorem assures ldquounique implementationrdquo of allimplementable payoffs satisfying πs gt πus

s (Γ) That is for any such payoff pair there is anuninformed-Seller information structure such that all equilibria (with price-independentbeliefs) induce exactly that payoff pair Unique implementation is appealing for multiplereasons one of which is that it obviates concerns about which among multiple payoff-distinct equilibria is more reasonable For example Ravid Roesler and Szentes (2019)bring to the fore the issue of equilibrium multiplicity in Roesler and Szentesrsquos (2017) infor-mation structure

Let us describe how we obtain the characterization of Πlowastus(Γ) and unique implementa-

tion There are two steps The first ensures that there is some information structure callit τ lowast isin Tus that implements Sellerrsquos payoff πus

s (Γ) That is we ensure that the infimum

14

in (1) is in fact a minimum16 While this argument is technical knowing τ lowast exists is usefulin what follows The second and economically insightful step is to construct informa-tion structures that implement every point in the triangle Πlowast

us(Γ) by suitably garbling theinformation structure τ lowast The construction is illustrated in Figure 2 Consider the distri-bution of Buyerrsquos posterior mean of his valuation v in information structure τ lowast (Givenprice-independent beliefs Buyerrsquos posterior mean is sufficient for his decision) For sim-plicity suppose this posterior-mean distribution has a density as depicted by the red curvein Figure 2 Fix any (πb πs) isin Πlowast

us(Γ)

First there is some number zlowast such that πb + πs is the total surplus from trading onlywhen Buyerrsquos posterior mean is greater than zlowast Next there is some price plowast ge zlowast suchthat Sellerrsquos payoff is πs if all these trades were to occur at price plowast17 Note that plowast must beno larger than the expected Buyer posterior mean conditional on that being above zlowast forotherwise πb lt 0 We claim that the information structure τ lowast can be garbled so that plowast isan equilibrium price and trade occurs only when Buyerrsquos posterior mean is greater thanzlowast The garbling is illustrated in Figure 2 by the blue distribution There is one signal that

pzv

ν(v)

Figure 2 Construction of information structure in Theorem 2

Buyer receives when the original posterior mean is between zlowast and plowast and also receiveswith some probability when the original posterior mean is above plowast The probability ischosen to make the posterior mean from this signal exactly plowast Apart from this one newsignal Buyer receives the original signal in τ lowast Plainly this is a garbling of τ lowast and hence isfeasible

16 The difficulty is in establishing suitable continuity Uninformed-Seller information structures can beviewed as probability measures over Buyerrsquos beliefs with convergence in the sense of the weak topologyThis topology ensures continuity with respect to probability measures of expectations of continuous orat least Lipschitz (and bounded) functions However Sellerrsquos expected payoff is not the expectation of aLipschitz function as Sellerrsquos profit is truncated at the price she charges

17 That plowast ge zlowast follows from πs ge πuss (Γ) as πs(Γ) itself is weakly larger than Sellerrsquos payoff from posting

price zlowast (and thus trading with the same set of Buyer posterior means) under information structure τlowast

15

Figure 2 makes clear why the new information structure has an equilibrium with priceplowast and Buyer breaking indifference in favor of trading (i) Sellerrsquos profit from posting anyprice below zlowast is the same as under τ lowast and hence no larger than πus

s (Γ) (ii) similarly Sellerrsquosprofit from posting any price above plowast is no higher than some fraction of πus

s (Γ) and (iii)any price between zlowast and plowast is worse that price plowast Moreover since Sellerrsquos profit fromoffering any price other than plowast is no more than πus

s (Γ) it follows that when πs gt πuss (Γ)

Buyer must break indifference as specified for Seller to have an optimal price and theequilibrium payoffs are unique

Remark 2 The above logic establishes that given any τ isin Tus that implements some (πb πs)τ can be garbled to uniquely implement any (πprime

b πprimes) isin Π(Γ) such that πprime

s gt πs That isan uninformed Sellerrsquos payoff can always be strictly raised and Buyerrsquos payoff reduced(strictly so long as it was not already zero) by garbling Buyerrsquos information

Remark 3 According to Theorem 11lowast and Theorem 2 uninformed-Seller information struc-tures cannot implement all implementable payoff pairs in an environment Γ if and only ifπuss (Γ) gt πs(Γ) This inequality fails in environment Γ if πus

s (Γ) = 0 since that impliesπuss (Γ) = πs(Γ) = 0 An example is when there is no trade due to adverse selection when

Seller is uniformed and Buyer has some information On the other hand it is sufficient forthe inequality that

v le E[c(v)] and forallv isin V c(v) lt v (2)

To see why (2) implies πuss (Γ) gt πs(Γ) notice that in any uninformed-Seller information

structure Seller can price at slightly less than Buyerrsquos highest posterior mean valuationand guarantee trade with only (a neighborhood of) that Buyer type If c(v) lt v for all v thisgives Seller a positive expected payoff and hence πus

s (Γ) gt 018 But the first inequality in (2)is equivalent to πs(Γ) = 0 We observe that Condition (2) is compatible with severe adverseselection resulting in very little trade when Seller is uninformed and Buyer (partially orfully) informed

33 Fully-Informed Buyer

We now turn to the third canonical class of information structures Buyer is fully in-formed of his value v As this is a special case of more-informed Buyer we maintain price-independent beliefs throughout this subsection

18 More precisely as V is compact c(v) lt v for all v implies there exists ε gt 0 such that v minus c(v) gt εGiven any uninformed-Seller information structure let mv be the highest posterior mean valuation in thesupport of the posterior means induced by Buyerrsquos signals So there is positive probability of Buyer signalswith posterior mean valuations at least mv minus ε2 By pricing at mv minus ε2 Sellerrsquos expected cost conditionalon trade is bounded above by mv minus ε and hence Sellerrsquos profit conditional on trade is at least ε2 gt 0 Itfollows that πus

s gt 0

16

Faced with a fully informed Buyer and any sequentially rational Buyer strategy an un-informed Seller can guarantee a profit level

πfbs (Γ) equiv sup

p

983133 v

p

(pminus c(v))micro(dv)

regardless of her information Plainly πfbs (Γ) ge πus

s (Γ) In monopoly pricing with c(middot) = vRoesler and Szentes (2017) have shown that πfb

s gt πuss if there is no trade due to adverse

selection when Buyer is fully informed and Seller is uniformed then πfbs = πus

s = 0 Belowwe establish that when Buyer is fully informed πfb

s is the only additional constraint onequilibrium payoffs Let Bε(πb πs) denote an ε-ball around (πb πs)

Theorem 3 Consider a fully informed Buyer and price-independent beliefs

1 Πlowastfb(Γ) = (πb πs) isin Π(Γ) πs ge πfb

s (Γ)

2 For any (πb πs) isin Πlowastfb(Γ) and any ε gt 0 there is τ isin Tfb with Π(Γ τ) sub Bε(πb πs)

The payoff set characterized in Theorem 3 corresponds to the triangle ABE in Figure 1Part 2 of the theorem establishes ldquoapproximately unique implementationrdquo for any point inthe triangle there is an information structure under which all equilibria yield payoff pairsin a neighborhood of that point

Here is the idea behind part 1 of Theorem 3 When Buyer is fully informed an informa-tion structure can be viewed as dividing vrsquos prior distribution micro into a set of microi that aver-age to micro with Seller informed of which microi she faces Theorem 3 is proven by establishingthat we can divide micro suitably so that against each microi Seller is indifferent between pricingat all prices in the support of microi including the price corresponding to πfb

s in that environ-ment We call such a microi a (Seller-price) incentive compatible distribution or ICD (Bergemannet al (2015) call their analogous construct an extreme market) The Appendix provides aldquogreedyrdquo algorithm to compute ICDs19

We can sketch how the algorithm works and construct a set of ICDs that average to theprior Suppose V = v1 v2 vK with vi lt vi+1 for i isin 1 K minus 1 and c(v) lt v forall v Given any small-enough mass of vK there is a unique mass of type vKminus1 that makesSeller indifferent between charging price vK and vKminus1 (If the mass is too low Seller prefersvK if it is too high she prefers vKminus1) Iterating down to keep Seller indifferent betweenall prices pins down an ICD Choose the maximum mass of type vK for which this worksRemove that ICD and repeat

19 The algorithm is defined for finite V and we take limits to handle the infinite case

17

Crucially whenever an ICD is removed the price corresponding to πfbs (Γ) remains op-

timal in the remaining ldquomarketrdquo this follows from the ICDrsquos defining property of Sellerindifference and an accounting identity Therefore Sellerrsquos profit in this segmentation ofICDs remains πfb

s (Γ) Moreover it is also optimal for Seller to always (ie for each microi) priceso that there is full trade or no trade Hence Buyerrsquos expected payoff can be either 0 or theentire surplus less πfb

s (Γ) It follows that the fully-informed Buyer information structuredefined by this set of ICDs implements point B and C in Figure 1 The entire triangle ABC

can then be implemented by convexification randomizing over this information structure(and two equilibria) and full information (where Seller obtains all the surplus)

The above logic of incentive compatible distributions reduces to that of Bergemann et al(2015) in the context of monopoly pricing Even in the monopoly pricing case Theorem 3part 2 strengthens the conclusions of Bergemann et al (2015) by ensuring approximatelyunique implementation

Remark 4 Theorems 1ndash3 imply that fully-informed-Buyer information structures imple-ment all implementable payoff pairs if and only if πfb

s (Γ) = πs(Γ) In that case trianglesAFG and ABC coincide in Figure 1 It follows that πfb

s (Γ) = πs(Γ) only when a fully-informed Buyer and uninformed Seller can result in full trade (v ge E[c(v)] and Sellerprices at v) or no trade (v le E[c(v)] and Seller prices at some p ge v) Interestinglywhen πfb

s (Γ) gt πs(Γ) fully-informed-Buyer information structures cannot even implementall payoff pairs implementable by uninformed-Seller information structures ie trianglesABC and ADE in Figure 1 are distinct That is when πfb

s (Γ) gt πs(Γ) there is an uninformed-Seller information structure that implements some πs lt πfb

s (Γ)20 Theorems 2ndash3 further im-ply in this case that Buyer can obtain a strictly higher payoff when he is not fully informed(and Seller is uninformed)

4 Discussion

41 Multidimensionality

Suppose Buyer and Sellerrsquos cost and valuation pair (c v) is a two-dimensional randomvariable distributed according to joint distribution micro with a compact support The ex-tension of our maintained assumption of commonly known gains from trade is for all

20 Pick any pprime gt v such that pprimeminusE[c(v)] lt πfbs (Γ) Following the construction described after Theorem 2 we

can mix all valuations v le pprime with a fraction λ gt 0 of valuations v gt pprime so that the mixture has posterior meanexactly pprime The remaining fraction 1 minus λ of valuations above pprime are revealed to Buyer With this uninformed-Seller information structure consider any equilibrium in which Buyer purchases when indifferent (Suchan equilibrium with price-independent belief exists) Sellerrsquos profit is at most (1 minus λ)πfb

s (Γ) from any pricep gt pprime and pprime minus E[c(v)] from price p = pprime Hence Sellerrsquos profit is strictly less than πfb

s (Γ)

18

(c v) isin Supp(micro) v ge c and E[v minus c] gt 0 An information structure is now a joint distribu-tion P (tb ts v c) whose marginal distribution on (v c) is micro

We claim that the content of our key results Theorem 1Theorem 1lowast Theorem 2 andTheorem 3 still hold To see why let v be the lowest valuation in the support of micro Sellerrsquosindividual rationality constraint is now maxv minus E[c] 0 as she can guarantee this by set-ting either a sufficiently high price or a price (arbitrarily close to) v regardless of her signalWe can define a cost function c(vprime) equiv Emicro[c|v = vprime] ge vprime This results in an environment sat-isfying all the maintained assumptions of our baseline model except that c(middot) may not becontinuous Recall that such continuity is not needed for proving Theorem 1 Theorem 1lowastBoth Theorem 2 and Theorem 3 use continuity of c(middot) to guarantee that Eν [c(v)] is a contin-uous function of ν isin ∆(V ) for a key convergence result However in the two-dimensionaltype environment Eν [c] is still a continuous function of ν isin ∆(V times C) Note that Theo-rem 3 uses upper semi-continuity of Sellerrsquos profit in price boundedness of c and c le v issufficient for such upper semi-continuity

42 Negative Trading Surplus

Returning to our baseline model we next discuss cases in which trade sometimes gen-erates negative surplus That is we drop the assumption that c(v) le v we do not requireE[v minus c(v)] gt 0 either Define Sλ(Γ) for λ isin [1infin) as

Sλ(Γ) equiv983133 v

v

[v minus c(v) + λ(v minus v)]+ micro(dv)

where [middot]+ equiv max middot 0 It is readily verified that S1(Γ) = E [[v minus c(v)]+] and limλrarr0

Sλ(Γ)λ =

E[v] minus v Allowing negative trading surplus does not affect our definition of wPBE Sothe notation Π(Γ) and πs(Γ) still have the same meanings as before The next propositionshows that Π(Γ) is now characterized by three constraints as before the two individualrationality constraints πs ge πs(Γ) and πb ge 0 and different now a Pareto frontier definedby all Sλ(Γ)

Proposition 1 Consider all information structures and equilibria when trade can generate negativesurplus

Π(Γ) =

983099983105983103

983105983101

πb ge 0

(πb πs) πs ge πs(Γ)

λπb + πs le Sλ(Γ) forallλ ge 1

983100983105983104

983105983102

Proposition 1 shows that when the trade can be inefficient it may be impossible to lower

19

Sellerrsquos profit without losing total surplus (ie πb + πs) This result is intuitive at the ex-tremes Suppose for simplicity v gt E[c(v)] If πs = πs(Γ) then since πs(Γ) = v minus E[c(v)]Seller must find it optimal to price at v and trade with probability one including whentrade generates negative surplus So Buyerrsquos payoff is E[v]minus v = limλrarrinfin Sλ(Γ)λ and totalsurplus is E[v minus c(v)] which includes the maximum amount of inefficient trade On theother hand an information structure that publicly reveals (only) whether trade is efficient(ie whether v ge c(v)) can implement efficient trade with all the surplus accruing to SellerIn this case Seller obtains S1(Γ) while Buyer obtains zero The proposition establishes thatin between S1(Γ) and πs(Γ) information structures and equilibria that lower Sellerrsquos profitmust reduce total surplus as the Pareto frontier Sλ(Γ) is a concave function with maximalslope minus1

The function capturing the Pareto frontier Sλ(Γ) is the maximal weighted sum of Buyerand Seller payoff assuming that Seller will trade at the low price p = v whenever it createspositive weighted total payoff Since we place higher weight on buyer (λ ge 1) S1(Γ) isthe upper bound for the weighted total payoff In the proof of Proposition 1 we showthat these bounds are tight mdash any payoff pair on the frontier can be implemented by someinformation structure The key observation is that whenever v minus c(v) + λ(v minus v) lt 0 itfollows that vminus c(v) lt (1minusλ)(vminus v) le 0 That is it is incentive compatible for Seller to notsell when it is common knowledge that v creates negative weighted total payoff Thereforewe can simply release a public signal indicating the sign of weighted total payoff and ifpositive induce Seller to sell at v which implies all bounds Sλ(Γ)

43 Linear Cost

When Sellerrsquos cost function c(v) is linear we can explicitly characterize an informationstructure that implements Sellerrsquos minimum implementable payoff πus

s (Γ) when Seller isuninformed (or per Theorem 2 when Buyer is better informed) subject to price-independentbeliefs We note that given the discussion in Subsection 41 a linear c(v) subsumes richerenvironments in which conditional expectations are linear including Gaussian environ-ments (cf Bar-Isaac et al 2018)

Condition 1 c(v) = λv + γ for some λ isin R

Let F (v) be the cumulative distribution function (CDF) corresponding to the prior mea-sure micro Let D(micro) be the set of all distributions whose CDF G satisfies

983133

V

vdG(v) =

983133

V

vdF (v) and983133 v

v

G(s)ds le983133 v

v

F (s)ds forallv isin V

20

That is D(micro) contains all distributions that are mean-preserving-contractions (MPC) of microIt is well known that D(micro) characterizes the set of distributions of Buyer posterior meansthat can be generated by any (uninformed-Seller) information structure

Definition 2 G is an incentive compatible distribution (ICD) if

983133

sgep

(pminus c(s))G(ds) = constant ge 0 forallp isin Supp(G)

That is when Buyerrsquos (mean) valuation is distributed according to an ICD G Seller isindifferent among all prices in the support of G assuming Buyer buys when indifferent Wefocus on a special family of ICDs whose supports are intervals [vlowast v

lowast] Given Condition 1such ICDs have analytical an expression

G(v) =

983099983105983105983103

983105983105983101

1minus983061

(1minus λ)v minus γ

(1minus λ)vlowast minus γ

983062 1λminus1

if λ ∕= 1

1minus evminusvlowast

γ if λ = 1

(3)

Given any vlowast and the above G(v) an upper bound vlowast is pinned down by the conditionEG[v] = EF [v] So a family of ICDs is parametrized by a single parameter vlowast and hence anICD is denoted by CDF 983144Gvlowast(v) with density 983144gvlowast(v) It is not difficult to verify that 983144Gvlowast(v) ismonotonic in vlowast for v in the support Consequently all ICDs are ordered according to theMPC order For vlowast isin [vE[v]] when vlowast increases the ICD increases in the MPC order

Proposition 2 Suppose Condition 1 is satisfied and let plowast equiv min983153vlowast983055983055 983144Gvlowast isin D(micro)

983154 It holds that

πuss (Γ) = plowast minus Emicro[c(v)]

Proposition 2 states that an uninformed Sellerrsquos minimum payoff (with price-independentbeliefs) is characterized by the ICDs that are MPCs of the prior distribution micro By defini-tion of ICDs Sellerrsquos profit when facing ICD 983144Gvlowast is vlowast minus E[v] Sellerrsquos minimum payoff isimplemented by the most dispersed ICD that is a MPC of the prior distribution

In proving Proposition 2 the key step is to show that given the prior micro garbling Buyerrsquosinformation such that the posterior mean distribution is an ICD makes Seller weakly worseoff It is then without loss to only consider ICDs to implement πus

s (Γ) We can thus workwith a tractable one-dimensional problem Consider the set D(micro) Suppose we find G(v)

such that (i) G isin D(micro) (ii) G is an ICD and (iii) existp such that983125 p

vG(s)ds =

983125 p

vF (s)ds Such

a G is the most-dispersed ICD that is a mean preserving contraction of F Consider the

21

following two manipulations

λ

983133 v

p

F (s)ds =

983133 v

p

(pminus c(s))dF (s)minus (1minus F (v)) (pminus c(v)) + λ(v minus p) (4)

λ

983133 v

p

G(s)ds =

983133 v

p

(pminus c(s))dG(s)minus (1minusG(v)) (pminus c(v)) + λ(v minus p) (5)

First by property (iii) above the LHS of Equation 4 equals the LHS of Equation 5 Secondsince

983125 v

p(F (s) minus G(s))ds is minimized at v this implies F (p) le G(p) Therefore the first

term on RHS of Equation 4 must be larger than that of Equation 5 Notice that the firstterm on RHS is Sellerrsquos profit when offering price p This implies that the profit fromoffering p given Buyer mean-valuation distribution G(v) is lower than that given F (v)which is lower than Sellerrsquos maximum profit given F (v) On the other hand by the ICDproperty p is already Sellerrsquos optimal price given G(v) The optimal profit given valuationdistribution F must be no lower than that of G Therefore we have proved that to minimizeSellerrsquos profit it is without loss to consider only ICDs within the set D(micro) which is a onedimensional subspace The detailed proof that such ICDs exist is relegated to the appendix

The monopoly-pricing environment with c(middot) = v is covered by Condition 1 with λ = 0

and γ = v The distribution G in (3) then reduces to that identified by Roesler and Szentes(2017)

If the prior micro has binary support then the cost function c(v) is linear In this special casewe can solve for πus

s (Γ) explicitly

Corollary 1 Suppose micro has binary support V = v1 v2 Let λ = c(v2)minusc(v1)v2minusv1

and let p be theunique solution to

(pminus c(p))1

λminus1 (pminus E[c(v)]) = (v2 minus c(v2))λ

λminus1 (6)

Then πuss (Γ) = max p v2minus E[c(v)]

References

AKERLOF G A (1970) ldquoThe Market for rdquoLemonsrdquo Quality Uncertainty and the MarketMechanismrdquo Quarterly Journal of Economics 84 488ndash500

BAGWELL K AND M H RIORDAN (1991) ldquoHigh and Declining Prices Signal ProductQualityrdquo American Economic Review 81 224ndash239

BAR-ISAAC H I JEWITT AND C LEAVER (2018) ldquoAdverse Selection Efficiency and theStructure of Informationrdquo Unpublished

22

BERGEMANN D B BROOKS AND S MORRIS (2015) ldquoThe Limits of Price Discrimina-tionrdquo American Economic Review 105 921ndash57

BERGEMANN D AND S MORRIS (2016) ldquoBayes Correlated Equilibrium and the Compar-ison of Information Structures in Gamesrdquo Theoretical Economics 11 487ndash522

BLACKWELL D (1953) ldquoEquivalent Comparisons of Experimentsrdquo Annals of MathematicalStatistics 24 265ndash272

CHO I-K AND D KREPS (1987) ldquoSignaling Games and Stable Equilibriardquo Quarterly Jour-nal of Economics 102 179ndash221

DU S (2018) ldquoRobust Mechanisms Under Common Valuationrdquo Econometrica 86 1569ndash1588

DURRETT R (1995) Probability Theory and Examples Duxbury Press 2nd ed

EINAV L AND A FINKELSTEIN (2011) ldquoSelection in Insurance Markets Theory and Em-pirics in Picturesrdquo Journal of Economic Perspectives 25 115ndash38

FUDENBERG D AND J TIROLE (1991) ldquoPerfect Bayesian Equilibrium and Sequential Equi-libriumrdquo Journal of Economic Theory 53 236ndash260

GARCIA D R TEPER AND M TSUR (2018) ldquoInformation Design in Insurance MarketsSelling Peaches in a Market for Lemonsrdquo Unpublished

GREENWALD B C (1986) ldquoAdverse Selection in the Labour Marketrdquo Review of EconomicStudies 53 325ndash347

JOVANOVIC B (1982) ldquoFavorable Selection with Asymmetric Informationrdquo QuarterlyJournal of Economics 97 535ndash539

KESSLER A S (2001) ldquoRevisiting the Lemons Marketrdquo International Economic Review 4225ndash41

KREPS D AND R WILSON (1982) ldquoSequential Equilibriardquo Econometrica 50 863ndash894

LEHMANN E L (1988) ldquoComparing Location Experimentsrdquo The Annals of Statistics 16521ndash533

LEVIN J (2001) ldquoInformation and the Market for Lemonsrdquo RAND Journal of Economics32 657ndash666

23

LIBGOBER J AND X MU (2019) ldquoInformational Robustness in Intertemporal PricingrdquoWorking paper

MAKRIS M AND L RENOU (2018) ldquoInformation Design in Multi-Stage Gamesrdquo Workingpaper

MILGROM P R AND R J WEBER (1985) ldquoDistributional Strategies for Games with In-complete Informationrdquo Mathematics of Operations Research 10 619ndash632

MUSSA M AND S ROSEN (1978) ldquoMonopoly and Product Qualityrdquo Journal of EconomicTheory 18 301ndash317

RAVID D A-K ROESLER AND B SZENTES (2019) ldquoLearning Before Trading On theInefficiency of Ignoring Free Informationrdquo Working paper

ROESLER A-K AND B SZENTES (2017) ldquoBuyer-Optimal Learning and Monopoly Pric-ingrdquo American Economic Review 107 2072ndash80

TERSTIEGE S AND C WASSER (2019) ldquoBuyer-optimal Extensionproof InformationrdquoWorking paper

Appendices omitted from this version but are available on request

24

in (1) is in fact a minimum16 While this argument is technical knowing τ lowast exists is usefulin what follows The second and economically insightful step is to construct informa-tion structures that implement every point in the triangle Πlowast

us(Γ) by suitably garbling theinformation structure τ lowast The construction is illustrated in Figure 2 Consider the distri-bution of Buyerrsquos posterior mean of his valuation v in information structure τ lowast (Givenprice-independent beliefs Buyerrsquos posterior mean is sufficient for his decision) For sim-plicity suppose this posterior-mean distribution has a density as depicted by the red curvein Figure 2 Fix any (πb πs) isin Πlowast

us(Γ)

First there is some number zlowast such that πb + πs is the total surplus from trading onlywhen Buyerrsquos posterior mean is greater than zlowast Next there is some price plowast ge zlowast suchthat Sellerrsquos payoff is πs if all these trades were to occur at price plowast17 Note that plowast must beno larger than the expected Buyer posterior mean conditional on that being above zlowast forotherwise πb lt 0 We claim that the information structure τ lowast can be garbled so that plowast isan equilibrium price and trade occurs only when Buyerrsquos posterior mean is greater thanzlowast The garbling is illustrated in Figure 2 by the blue distribution There is one signal that

pzv

ν(v)

Figure 2 Construction of information structure in Theorem 2

Buyer receives when the original posterior mean is between zlowast and plowast and also receiveswith some probability when the original posterior mean is above plowast The probability ischosen to make the posterior mean from this signal exactly plowast Apart from this one newsignal Buyer receives the original signal in τ lowast Plainly this is a garbling of τ lowast and hence isfeasible

16 The difficulty is in establishing suitable continuity Uninformed-Seller information structures can beviewed as probability measures over Buyerrsquos beliefs with convergence in the sense of the weak topologyThis topology ensures continuity with respect to probability measures of expectations of continuous orat least Lipschitz (and bounded) functions However Sellerrsquos expected payoff is not the expectation of aLipschitz function as Sellerrsquos profit is truncated at the price she charges

17 That plowast ge zlowast follows from πs ge πuss (Γ) as πs(Γ) itself is weakly larger than Sellerrsquos payoff from posting

price zlowast (and thus trading with the same set of Buyer posterior means) under information structure τlowast

15

Figure 2 makes clear why the new information structure has an equilibrium with priceplowast and Buyer breaking indifference in favor of trading (i) Sellerrsquos profit from posting anyprice below zlowast is the same as under τ lowast and hence no larger than πus

s (Γ) (ii) similarly Sellerrsquosprofit from posting any price above plowast is no higher than some fraction of πus

s (Γ) and (iii)any price between zlowast and plowast is worse that price plowast Moreover since Sellerrsquos profit fromoffering any price other than plowast is no more than πus

s (Γ) it follows that when πs gt πuss (Γ)

Buyer must break indifference as specified for Seller to have an optimal price and theequilibrium payoffs are unique

Remark 2 The above logic establishes that given any τ isin Tus that implements some (πb πs)τ can be garbled to uniquely implement any (πprime

b πprimes) isin Π(Γ) such that πprime

s gt πs That isan uninformed Sellerrsquos payoff can always be strictly raised and Buyerrsquos payoff reduced(strictly so long as it was not already zero) by garbling Buyerrsquos information

Remark 3 According to Theorem 11lowast and Theorem 2 uninformed-Seller information struc-tures cannot implement all implementable payoff pairs in an environment Γ if and only ifπuss (Γ) gt πs(Γ) This inequality fails in environment Γ if πus

s (Γ) = 0 since that impliesπuss (Γ) = πs(Γ) = 0 An example is when there is no trade due to adverse selection when

Seller is uniformed and Buyer has some information On the other hand it is sufficient forthe inequality that

v le E[c(v)] and forallv isin V c(v) lt v (2)

To see why (2) implies πuss (Γ) gt πs(Γ) notice that in any uninformed-Seller information

structure Seller can price at slightly less than Buyerrsquos highest posterior mean valuationand guarantee trade with only (a neighborhood of) that Buyer type If c(v) lt v for all v thisgives Seller a positive expected payoff and hence πus

s (Γ) gt 018 But the first inequality in (2)is equivalent to πs(Γ) = 0 We observe that Condition (2) is compatible with severe adverseselection resulting in very little trade when Seller is uninformed and Buyer (partially orfully) informed

33 Fully-Informed Buyer

We now turn to the third canonical class of information structures Buyer is fully in-formed of his value v As this is a special case of more-informed Buyer we maintain price-independent beliefs throughout this subsection

18 More precisely as V is compact c(v) lt v for all v implies there exists ε gt 0 such that v minus c(v) gt εGiven any uninformed-Seller information structure let mv be the highest posterior mean valuation in thesupport of the posterior means induced by Buyerrsquos signals So there is positive probability of Buyer signalswith posterior mean valuations at least mv minus ε2 By pricing at mv minus ε2 Sellerrsquos expected cost conditionalon trade is bounded above by mv minus ε and hence Sellerrsquos profit conditional on trade is at least ε2 gt 0 Itfollows that πus

s gt 0

16

Faced with a fully informed Buyer and any sequentially rational Buyer strategy an un-informed Seller can guarantee a profit level

πfbs (Γ) equiv sup

p

983133 v

p

(pminus c(v))micro(dv)

regardless of her information Plainly πfbs (Γ) ge πus

s (Γ) In monopoly pricing with c(middot) = vRoesler and Szentes (2017) have shown that πfb

s gt πuss if there is no trade due to adverse

selection when Buyer is fully informed and Seller is uniformed then πfbs = πus

s = 0 Belowwe establish that when Buyer is fully informed πfb

s is the only additional constraint onequilibrium payoffs Let Bε(πb πs) denote an ε-ball around (πb πs)

Theorem 3 Consider a fully informed Buyer and price-independent beliefs

1 Πlowastfb(Γ) = (πb πs) isin Π(Γ) πs ge πfb

s (Γ)

2 For any (πb πs) isin Πlowastfb(Γ) and any ε gt 0 there is τ isin Tfb with Π(Γ τ) sub Bε(πb πs)

The payoff set characterized in Theorem 3 corresponds to the triangle ABE in Figure 1Part 2 of the theorem establishes ldquoapproximately unique implementationrdquo for any point inthe triangle there is an information structure under which all equilibria yield payoff pairsin a neighborhood of that point

Here is the idea behind part 1 of Theorem 3 When Buyer is fully informed an informa-tion structure can be viewed as dividing vrsquos prior distribution micro into a set of microi that aver-age to micro with Seller informed of which microi she faces Theorem 3 is proven by establishingthat we can divide micro suitably so that against each microi Seller is indifferent between pricingat all prices in the support of microi including the price corresponding to πfb

s in that environ-ment We call such a microi a (Seller-price) incentive compatible distribution or ICD (Bergemannet al (2015) call their analogous construct an extreme market) The Appendix provides aldquogreedyrdquo algorithm to compute ICDs19

We can sketch how the algorithm works and construct a set of ICDs that average to theprior Suppose V = v1 v2 vK with vi lt vi+1 for i isin 1 K minus 1 and c(v) lt v forall v Given any small-enough mass of vK there is a unique mass of type vKminus1 that makesSeller indifferent between charging price vK and vKminus1 (If the mass is too low Seller prefersvK if it is too high she prefers vKminus1) Iterating down to keep Seller indifferent betweenall prices pins down an ICD Choose the maximum mass of type vK for which this worksRemove that ICD and repeat

19 The algorithm is defined for finite V and we take limits to handle the infinite case

17

Crucially whenever an ICD is removed the price corresponding to πfbs (Γ) remains op-

timal in the remaining ldquomarketrdquo this follows from the ICDrsquos defining property of Sellerindifference and an accounting identity Therefore Sellerrsquos profit in this segmentation ofICDs remains πfb

s (Γ) Moreover it is also optimal for Seller to always (ie for each microi) priceso that there is full trade or no trade Hence Buyerrsquos expected payoff can be either 0 or theentire surplus less πfb

s (Γ) It follows that the fully-informed Buyer information structuredefined by this set of ICDs implements point B and C in Figure 1 The entire triangle ABC

can then be implemented by convexification randomizing over this information structure(and two equilibria) and full information (where Seller obtains all the surplus)

The above logic of incentive compatible distributions reduces to that of Bergemann et al(2015) in the context of monopoly pricing Even in the monopoly pricing case Theorem 3part 2 strengthens the conclusions of Bergemann et al (2015) by ensuring approximatelyunique implementation

Remark 4 Theorems 1ndash3 imply that fully-informed-Buyer information structures imple-ment all implementable payoff pairs if and only if πfb

s (Γ) = πs(Γ) In that case trianglesAFG and ABC coincide in Figure 1 It follows that πfb

s (Γ) = πs(Γ) only when a fully-informed Buyer and uninformed Seller can result in full trade (v ge E[c(v)] and Sellerprices at v) or no trade (v le E[c(v)] and Seller prices at some p ge v) Interestinglywhen πfb

s (Γ) gt πs(Γ) fully-informed-Buyer information structures cannot even implementall payoff pairs implementable by uninformed-Seller information structures ie trianglesABC and ADE in Figure 1 are distinct That is when πfb

s (Γ) gt πs(Γ) there is an uninformed-Seller information structure that implements some πs lt πfb

s (Γ)20 Theorems 2ndash3 further im-ply in this case that Buyer can obtain a strictly higher payoff when he is not fully informed(and Seller is uninformed)

4 Discussion

41 Multidimensionality

Suppose Buyer and Sellerrsquos cost and valuation pair (c v) is a two-dimensional randomvariable distributed according to joint distribution micro with a compact support The ex-tension of our maintained assumption of commonly known gains from trade is for all

20 Pick any pprime gt v such that pprimeminusE[c(v)] lt πfbs (Γ) Following the construction described after Theorem 2 we

can mix all valuations v le pprime with a fraction λ gt 0 of valuations v gt pprime so that the mixture has posterior meanexactly pprime The remaining fraction 1 minus λ of valuations above pprime are revealed to Buyer With this uninformed-Seller information structure consider any equilibrium in which Buyer purchases when indifferent (Suchan equilibrium with price-independent belief exists) Sellerrsquos profit is at most (1 minus λ)πfb

s (Γ) from any pricep gt pprime and pprime minus E[c(v)] from price p = pprime Hence Sellerrsquos profit is strictly less than πfb

s (Γ)

18

(c v) isin Supp(micro) v ge c and E[v minus c] gt 0 An information structure is now a joint distribu-tion P (tb ts v c) whose marginal distribution on (v c) is micro

We claim that the content of our key results Theorem 1Theorem 1lowast Theorem 2 andTheorem 3 still hold To see why let v be the lowest valuation in the support of micro Sellerrsquosindividual rationality constraint is now maxv minus E[c] 0 as she can guarantee this by set-ting either a sufficiently high price or a price (arbitrarily close to) v regardless of her signalWe can define a cost function c(vprime) equiv Emicro[c|v = vprime] ge vprime This results in an environment sat-isfying all the maintained assumptions of our baseline model except that c(middot) may not becontinuous Recall that such continuity is not needed for proving Theorem 1 Theorem 1lowastBoth Theorem 2 and Theorem 3 use continuity of c(middot) to guarantee that Eν [c(v)] is a contin-uous function of ν isin ∆(V ) for a key convergence result However in the two-dimensionaltype environment Eν [c] is still a continuous function of ν isin ∆(V times C) Note that Theo-rem 3 uses upper semi-continuity of Sellerrsquos profit in price boundedness of c and c le v issufficient for such upper semi-continuity

42 Negative Trading Surplus

Returning to our baseline model we next discuss cases in which trade sometimes gen-erates negative surplus That is we drop the assumption that c(v) le v we do not requireE[v minus c(v)] gt 0 either Define Sλ(Γ) for λ isin [1infin) as

Sλ(Γ) equiv983133 v

v

[v minus c(v) + λ(v minus v)]+ micro(dv)

where [middot]+ equiv max middot 0 It is readily verified that S1(Γ) = E [[v minus c(v)]+] and limλrarr0

Sλ(Γ)λ =

E[v] minus v Allowing negative trading surplus does not affect our definition of wPBE Sothe notation Π(Γ) and πs(Γ) still have the same meanings as before The next propositionshows that Π(Γ) is now characterized by three constraints as before the two individualrationality constraints πs ge πs(Γ) and πb ge 0 and different now a Pareto frontier definedby all Sλ(Γ)

Proposition 1 Consider all information structures and equilibria when trade can generate negativesurplus

Π(Γ) =

983099983105983103

983105983101

πb ge 0

(πb πs) πs ge πs(Γ)

λπb + πs le Sλ(Γ) forallλ ge 1

983100983105983104

983105983102

Proposition 1 shows that when the trade can be inefficient it may be impossible to lower

19

Sellerrsquos profit without losing total surplus (ie πb + πs) This result is intuitive at the ex-tremes Suppose for simplicity v gt E[c(v)] If πs = πs(Γ) then since πs(Γ) = v minus E[c(v)]Seller must find it optimal to price at v and trade with probability one including whentrade generates negative surplus So Buyerrsquos payoff is E[v]minus v = limλrarrinfin Sλ(Γ)λ and totalsurplus is E[v minus c(v)] which includes the maximum amount of inefficient trade On theother hand an information structure that publicly reveals (only) whether trade is efficient(ie whether v ge c(v)) can implement efficient trade with all the surplus accruing to SellerIn this case Seller obtains S1(Γ) while Buyer obtains zero The proposition establishes thatin between S1(Γ) and πs(Γ) information structures and equilibria that lower Sellerrsquos profitmust reduce total surplus as the Pareto frontier Sλ(Γ) is a concave function with maximalslope minus1

The function capturing the Pareto frontier Sλ(Γ) is the maximal weighted sum of Buyerand Seller payoff assuming that Seller will trade at the low price p = v whenever it createspositive weighted total payoff Since we place higher weight on buyer (λ ge 1) S1(Γ) isthe upper bound for the weighted total payoff In the proof of Proposition 1 we showthat these bounds are tight mdash any payoff pair on the frontier can be implemented by someinformation structure The key observation is that whenever v minus c(v) + λ(v minus v) lt 0 itfollows that vminus c(v) lt (1minusλ)(vminus v) le 0 That is it is incentive compatible for Seller to notsell when it is common knowledge that v creates negative weighted total payoff Thereforewe can simply release a public signal indicating the sign of weighted total payoff and ifpositive induce Seller to sell at v which implies all bounds Sλ(Γ)

43 Linear Cost

When Sellerrsquos cost function c(v) is linear we can explicitly characterize an informationstructure that implements Sellerrsquos minimum implementable payoff πus

s (Γ) when Seller isuninformed (or per Theorem 2 when Buyer is better informed) subject to price-independentbeliefs We note that given the discussion in Subsection 41 a linear c(v) subsumes richerenvironments in which conditional expectations are linear including Gaussian environ-ments (cf Bar-Isaac et al 2018)

Condition 1 c(v) = λv + γ for some λ isin R

Let F (v) be the cumulative distribution function (CDF) corresponding to the prior mea-sure micro Let D(micro) be the set of all distributions whose CDF G satisfies

983133

V

vdG(v) =

983133

V

vdF (v) and983133 v

v

G(s)ds le983133 v

v

F (s)ds forallv isin V

20

That is D(micro) contains all distributions that are mean-preserving-contractions (MPC) of microIt is well known that D(micro) characterizes the set of distributions of Buyer posterior meansthat can be generated by any (uninformed-Seller) information structure

Definition 2 G is an incentive compatible distribution (ICD) if

983133

sgep

(pminus c(s))G(ds) = constant ge 0 forallp isin Supp(G)

That is when Buyerrsquos (mean) valuation is distributed according to an ICD G Seller isindifferent among all prices in the support of G assuming Buyer buys when indifferent Wefocus on a special family of ICDs whose supports are intervals [vlowast v

lowast] Given Condition 1such ICDs have analytical an expression

G(v) =

983099983105983105983103

983105983105983101

1minus983061

(1minus λ)v minus γ

(1minus λ)vlowast minus γ

983062 1λminus1

if λ ∕= 1

1minus evminusvlowast

γ if λ = 1

(3)

Given any vlowast and the above G(v) an upper bound vlowast is pinned down by the conditionEG[v] = EF [v] So a family of ICDs is parametrized by a single parameter vlowast and hence anICD is denoted by CDF 983144Gvlowast(v) with density 983144gvlowast(v) It is not difficult to verify that 983144Gvlowast(v) ismonotonic in vlowast for v in the support Consequently all ICDs are ordered according to theMPC order For vlowast isin [vE[v]] when vlowast increases the ICD increases in the MPC order

Proposition 2 Suppose Condition 1 is satisfied and let plowast equiv min983153vlowast983055983055 983144Gvlowast isin D(micro)

983154 It holds that

πuss (Γ) = plowast minus Emicro[c(v)]

Proposition 2 states that an uninformed Sellerrsquos minimum payoff (with price-independentbeliefs) is characterized by the ICDs that are MPCs of the prior distribution micro By defini-tion of ICDs Sellerrsquos profit when facing ICD 983144Gvlowast is vlowast minus E[v] Sellerrsquos minimum payoff isimplemented by the most dispersed ICD that is a MPC of the prior distribution

In proving Proposition 2 the key step is to show that given the prior micro garbling Buyerrsquosinformation such that the posterior mean distribution is an ICD makes Seller weakly worseoff It is then without loss to only consider ICDs to implement πus

s (Γ) We can thus workwith a tractable one-dimensional problem Consider the set D(micro) Suppose we find G(v)

such that (i) G isin D(micro) (ii) G is an ICD and (iii) existp such that983125 p

vG(s)ds =

983125 p

vF (s)ds Such

a G is the most-dispersed ICD that is a mean preserving contraction of F Consider the

21

following two manipulations

λ

983133 v

p

F (s)ds =

983133 v

p

(pminus c(s))dF (s)minus (1minus F (v)) (pminus c(v)) + λ(v minus p) (4)

λ

983133 v

p

G(s)ds =

983133 v

p

(pminus c(s))dG(s)minus (1minusG(v)) (pminus c(v)) + λ(v minus p) (5)

First by property (iii) above the LHS of Equation 4 equals the LHS of Equation 5 Secondsince

983125 v

p(F (s) minus G(s))ds is minimized at v this implies F (p) le G(p) Therefore the first

term on RHS of Equation 4 must be larger than that of Equation 5 Notice that the firstterm on RHS is Sellerrsquos profit when offering price p This implies that the profit fromoffering p given Buyer mean-valuation distribution G(v) is lower than that given F (v)which is lower than Sellerrsquos maximum profit given F (v) On the other hand by the ICDproperty p is already Sellerrsquos optimal price given G(v) The optimal profit given valuationdistribution F must be no lower than that of G Therefore we have proved that to minimizeSellerrsquos profit it is without loss to consider only ICDs within the set D(micro) which is a onedimensional subspace The detailed proof that such ICDs exist is relegated to the appendix

The monopoly-pricing environment with c(middot) = v is covered by Condition 1 with λ = 0

and γ = v The distribution G in (3) then reduces to that identified by Roesler and Szentes(2017)

If the prior micro has binary support then the cost function c(v) is linear In this special casewe can solve for πus

s (Γ) explicitly

Corollary 1 Suppose micro has binary support V = v1 v2 Let λ = c(v2)minusc(v1)v2minusv1

and let p be theunique solution to

(pminus c(p))1

λminus1 (pminus E[c(v)]) = (v2 minus c(v2))λ

λminus1 (6)

Then πuss (Γ) = max p v2minus E[c(v)]

References

AKERLOF G A (1970) ldquoThe Market for rdquoLemonsrdquo Quality Uncertainty and the MarketMechanismrdquo Quarterly Journal of Economics 84 488ndash500

BAGWELL K AND M H RIORDAN (1991) ldquoHigh and Declining Prices Signal ProductQualityrdquo American Economic Review 81 224ndash239

BAR-ISAAC H I JEWITT AND C LEAVER (2018) ldquoAdverse Selection Efficiency and theStructure of Informationrdquo Unpublished

22

BERGEMANN D B BROOKS AND S MORRIS (2015) ldquoThe Limits of Price Discrimina-tionrdquo American Economic Review 105 921ndash57

BERGEMANN D AND S MORRIS (2016) ldquoBayes Correlated Equilibrium and the Compar-ison of Information Structures in Gamesrdquo Theoretical Economics 11 487ndash522

BLACKWELL D (1953) ldquoEquivalent Comparisons of Experimentsrdquo Annals of MathematicalStatistics 24 265ndash272

CHO I-K AND D KREPS (1987) ldquoSignaling Games and Stable Equilibriardquo Quarterly Jour-nal of Economics 102 179ndash221

DU S (2018) ldquoRobust Mechanisms Under Common Valuationrdquo Econometrica 86 1569ndash1588

DURRETT R (1995) Probability Theory and Examples Duxbury Press 2nd ed

EINAV L AND A FINKELSTEIN (2011) ldquoSelection in Insurance Markets Theory and Em-pirics in Picturesrdquo Journal of Economic Perspectives 25 115ndash38

FUDENBERG D AND J TIROLE (1991) ldquoPerfect Bayesian Equilibrium and Sequential Equi-libriumrdquo Journal of Economic Theory 53 236ndash260

GARCIA D R TEPER AND M TSUR (2018) ldquoInformation Design in Insurance MarketsSelling Peaches in a Market for Lemonsrdquo Unpublished

GREENWALD B C (1986) ldquoAdverse Selection in the Labour Marketrdquo Review of EconomicStudies 53 325ndash347

JOVANOVIC B (1982) ldquoFavorable Selection with Asymmetric Informationrdquo QuarterlyJournal of Economics 97 535ndash539

KESSLER A S (2001) ldquoRevisiting the Lemons Marketrdquo International Economic Review 4225ndash41

KREPS D AND R WILSON (1982) ldquoSequential Equilibriardquo Econometrica 50 863ndash894

LEHMANN E L (1988) ldquoComparing Location Experimentsrdquo The Annals of Statistics 16521ndash533

LEVIN J (2001) ldquoInformation and the Market for Lemonsrdquo RAND Journal of Economics32 657ndash666

23

LIBGOBER J AND X MU (2019) ldquoInformational Robustness in Intertemporal PricingrdquoWorking paper

MAKRIS M AND L RENOU (2018) ldquoInformation Design in Multi-Stage Gamesrdquo Workingpaper

MILGROM P R AND R J WEBER (1985) ldquoDistributional Strategies for Games with In-complete Informationrdquo Mathematics of Operations Research 10 619ndash632

MUSSA M AND S ROSEN (1978) ldquoMonopoly and Product Qualityrdquo Journal of EconomicTheory 18 301ndash317

RAVID D A-K ROESLER AND B SZENTES (2019) ldquoLearning Before Trading On theInefficiency of Ignoring Free Informationrdquo Working paper

ROESLER A-K AND B SZENTES (2017) ldquoBuyer-Optimal Learning and Monopoly Pric-ingrdquo American Economic Review 107 2072ndash80

TERSTIEGE S AND C WASSER (2019) ldquoBuyer-optimal Extensionproof InformationrdquoWorking paper

Appendices omitted from this version but are available on request

24

Figure 2 makes clear why the new information structure has an equilibrium with priceplowast and Buyer breaking indifference in favor of trading (i) Sellerrsquos profit from posting anyprice below zlowast is the same as under τ lowast and hence no larger than πus

s (Γ) (ii) similarly Sellerrsquosprofit from posting any price above plowast is no higher than some fraction of πus

s (Γ) and (iii)any price between zlowast and plowast is worse that price plowast Moreover since Sellerrsquos profit fromoffering any price other than plowast is no more than πus

s (Γ) it follows that when πs gt πuss (Γ)

Buyer must break indifference as specified for Seller to have an optimal price and theequilibrium payoffs are unique

Remark 2 The above logic establishes that given any τ isin Tus that implements some (πb πs)τ can be garbled to uniquely implement any (πprime

b πprimes) isin Π(Γ) such that πprime

s gt πs That isan uninformed Sellerrsquos payoff can always be strictly raised and Buyerrsquos payoff reduced(strictly so long as it was not already zero) by garbling Buyerrsquos information

Remark 3 According to Theorem 11lowast and Theorem 2 uninformed-Seller information struc-tures cannot implement all implementable payoff pairs in an environment Γ if and only ifπuss (Γ) gt πs(Γ) This inequality fails in environment Γ if πus

s (Γ) = 0 since that impliesπuss (Γ) = πs(Γ) = 0 An example is when there is no trade due to adverse selection when

Seller is uniformed and Buyer has some information On the other hand it is sufficient forthe inequality that

v le E[c(v)] and forallv isin V c(v) lt v (2)

To see why (2) implies πuss (Γ) gt πs(Γ) notice that in any uninformed-Seller information

structure Seller can price at slightly less than Buyerrsquos highest posterior mean valuationand guarantee trade with only (a neighborhood of) that Buyer type If c(v) lt v for all v thisgives Seller a positive expected payoff and hence πus

s (Γ) gt 018 But the first inequality in (2)is equivalent to πs(Γ) = 0 We observe that Condition (2) is compatible with severe adverseselection resulting in very little trade when Seller is uninformed and Buyer (partially orfully) informed

33 Fully-Informed Buyer

We now turn to the third canonical class of information structures Buyer is fully in-formed of his value v As this is a special case of more-informed Buyer we maintain price-independent beliefs throughout this subsection

18 More precisely as V is compact c(v) lt v for all v implies there exists ε gt 0 such that v minus c(v) gt εGiven any uninformed-Seller information structure let mv be the highest posterior mean valuation in thesupport of the posterior means induced by Buyerrsquos signals So there is positive probability of Buyer signalswith posterior mean valuations at least mv minus ε2 By pricing at mv minus ε2 Sellerrsquos expected cost conditionalon trade is bounded above by mv minus ε and hence Sellerrsquos profit conditional on trade is at least ε2 gt 0 Itfollows that πus

s gt 0

16

Faced with a fully informed Buyer and any sequentially rational Buyer strategy an un-informed Seller can guarantee a profit level

πfbs (Γ) equiv sup

p

983133 v

p

(pminus c(v))micro(dv)

regardless of her information Plainly πfbs (Γ) ge πus

s (Γ) In monopoly pricing with c(middot) = vRoesler and Szentes (2017) have shown that πfb

s gt πuss if there is no trade due to adverse

selection when Buyer is fully informed and Seller is uniformed then πfbs = πus

s = 0 Belowwe establish that when Buyer is fully informed πfb

s is the only additional constraint onequilibrium payoffs Let Bε(πb πs) denote an ε-ball around (πb πs)

Theorem 3 Consider a fully informed Buyer and price-independent beliefs

1 Πlowastfb(Γ) = (πb πs) isin Π(Γ) πs ge πfb

s (Γ)

2 For any (πb πs) isin Πlowastfb(Γ) and any ε gt 0 there is τ isin Tfb with Π(Γ τ) sub Bε(πb πs)

The payoff set characterized in Theorem 3 corresponds to the triangle ABE in Figure 1Part 2 of the theorem establishes ldquoapproximately unique implementationrdquo for any point inthe triangle there is an information structure under which all equilibria yield payoff pairsin a neighborhood of that point

Here is the idea behind part 1 of Theorem 3 When Buyer is fully informed an informa-tion structure can be viewed as dividing vrsquos prior distribution micro into a set of microi that aver-age to micro with Seller informed of which microi she faces Theorem 3 is proven by establishingthat we can divide micro suitably so that against each microi Seller is indifferent between pricingat all prices in the support of microi including the price corresponding to πfb

s in that environ-ment We call such a microi a (Seller-price) incentive compatible distribution or ICD (Bergemannet al (2015) call their analogous construct an extreme market) The Appendix provides aldquogreedyrdquo algorithm to compute ICDs19

We can sketch how the algorithm works and construct a set of ICDs that average to theprior Suppose V = v1 v2 vK with vi lt vi+1 for i isin 1 K minus 1 and c(v) lt v forall v Given any small-enough mass of vK there is a unique mass of type vKminus1 that makesSeller indifferent between charging price vK and vKminus1 (If the mass is too low Seller prefersvK if it is too high she prefers vKminus1) Iterating down to keep Seller indifferent betweenall prices pins down an ICD Choose the maximum mass of type vK for which this worksRemove that ICD and repeat

19 The algorithm is defined for finite V and we take limits to handle the infinite case

17

Crucially whenever an ICD is removed the price corresponding to πfbs (Γ) remains op-

timal in the remaining ldquomarketrdquo this follows from the ICDrsquos defining property of Sellerindifference and an accounting identity Therefore Sellerrsquos profit in this segmentation ofICDs remains πfb

s (Γ) Moreover it is also optimal for Seller to always (ie for each microi) priceso that there is full trade or no trade Hence Buyerrsquos expected payoff can be either 0 or theentire surplus less πfb

s (Γ) It follows that the fully-informed Buyer information structuredefined by this set of ICDs implements point B and C in Figure 1 The entire triangle ABC

can then be implemented by convexification randomizing over this information structure(and two equilibria) and full information (where Seller obtains all the surplus)

The above logic of incentive compatible distributions reduces to that of Bergemann et al(2015) in the context of monopoly pricing Even in the monopoly pricing case Theorem 3part 2 strengthens the conclusions of Bergemann et al (2015) by ensuring approximatelyunique implementation

Remark 4 Theorems 1ndash3 imply that fully-informed-Buyer information structures imple-ment all implementable payoff pairs if and only if πfb

s (Γ) = πs(Γ) In that case trianglesAFG and ABC coincide in Figure 1 It follows that πfb

s (Γ) = πs(Γ) only when a fully-informed Buyer and uninformed Seller can result in full trade (v ge E[c(v)] and Sellerprices at v) or no trade (v le E[c(v)] and Seller prices at some p ge v) Interestinglywhen πfb

s (Γ) gt πs(Γ) fully-informed-Buyer information structures cannot even implementall payoff pairs implementable by uninformed-Seller information structures ie trianglesABC and ADE in Figure 1 are distinct That is when πfb

s (Γ) gt πs(Γ) there is an uninformed-Seller information structure that implements some πs lt πfb

s (Γ)20 Theorems 2ndash3 further im-ply in this case that Buyer can obtain a strictly higher payoff when he is not fully informed(and Seller is uninformed)

4 Discussion

41 Multidimensionality

Suppose Buyer and Sellerrsquos cost and valuation pair (c v) is a two-dimensional randomvariable distributed according to joint distribution micro with a compact support The ex-tension of our maintained assumption of commonly known gains from trade is for all

20 Pick any pprime gt v such that pprimeminusE[c(v)] lt πfbs (Γ) Following the construction described after Theorem 2 we

can mix all valuations v le pprime with a fraction λ gt 0 of valuations v gt pprime so that the mixture has posterior meanexactly pprime The remaining fraction 1 minus λ of valuations above pprime are revealed to Buyer With this uninformed-Seller information structure consider any equilibrium in which Buyer purchases when indifferent (Suchan equilibrium with price-independent belief exists) Sellerrsquos profit is at most (1 minus λ)πfb

s (Γ) from any pricep gt pprime and pprime minus E[c(v)] from price p = pprime Hence Sellerrsquos profit is strictly less than πfb

s (Γ)

18

(c v) isin Supp(micro) v ge c and E[v minus c] gt 0 An information structure is now a joint distribu-tion P (tb ts v c) whose marginal distribution on (v c) is micro

We claim that the content of our key results Theorem 1Theorem 1lowast Theorem 2 andTheorem 3 still hold To see why let v be the lowest valuation in the support of micro Sellerrsquosindividual rationality constraint is now maxv minus E[c] 0 as she can guarantee this by set-ting either a sufficiently high price or a price (arbitrarily close to) v regardless of her signalWe can define a cost function c(vprime) equiv Emicro[c|v = vprime] ge vprime This results in an environment sat-isfying all the maintained assumptions of our baseline model except that c(middot) may not becontinuous Recall that such continuity is not needed for proving Theorem 1 Theorem 1lowastBoth Theorem 2 and Theorem 3 use continuity of c(middot) to guarantee that Eν [c(v)] is a contin-uous function of ν isin ∆(V ) for a key convergence result However in the two-dimensionaltype environment Eν [c] is still a continuous function of ν isin ∆(V times C) Note that Theo-rem 3 uses upper semi-continuity of Sellerrsquos profit in price boundedness of c and c le v issufficient for such upper semi-continuity

42 Negative Trading Surplus

Returning to our baseline model we next discuss cases in which trade sometimes gen-erates negative surplus That is we drop the assumption that c(v) le v we do not requireE[v minus c(v)] gt 0 either Define Sλ(Γ) for λ isin [1infin) as

Sλ(Γ) equiv983133 v

v

[v minus c(v) + λ(v minus v)]+ micro(dv)

where [middot]+ equiv max middot 0 It is readily verified that S1(Γ) = E [[v minus c(v)]+] and limλrarr0

Sλ(Γ)λ =

E[v] minus v Allowing negative trading surplus does not affect our definition of wPBE Sothe notation Π(Γ) and πs(Γ) still have the same meanings as before The next propositionshows that Π(Γ) is now characterized by three constraints as before the two individualrationality constraints πs ge πs(Γ) and πb ge 0 and different now a Pareto frontier definedby all Sλ(Γ)

Proposition 1 Consider all information structures and equilibria when trade can generate negativesurplus

Π(Γ) =

983099983105983103

983105983101

πb ge 0

(πb πs) πs ge πs(Γ)

λπb + πs le Sλ(Γ) forallλ ge 1

983100983105983104

983105983102

Proposition 1 shows that when the trade can be inefficient it may be impossible to lower

19

Sellerrsquos profit without losing total surplus (ie πb + πs) This result is intuitive at the ex-tremes Suppose for simplicity v gt E[c(v)] If πs = πs(Γ) then since πs(Γ) = v minus E[c(v)]Seller must find it optimal to price at v and trade with probability one including whentrade generates negative surplus So Buyerrsquos payoff is E[v]minus v = limλrarrinfin Sλ(Γ)λ and totalsurplus is E[v minus c(v)] which includes the maximum amount of inefficient trade On theother hand an information structure that publicly reveals (only) whether trade is efficient(ie whether v ge c(v)) can implement efficient trade with all the surplus accruing to SellerIn this case Seller obtains S1(Γ) while Buyer obtains zero The proposition establishes thatin between S1(Γ) and πs(Γ) information structures and equilibria that lower Sellerrsquos profitmust reduce total surplus as the Pareto frontier Sλ(Γ) is a concave function with maximalslope minus1

The function capturing the Pareto frontier Sλ(Γ) is the maximal weighted sum of Buyerand Seller payoff assuming that Seller will trade at the low price p = v whenever it createspositive weighted total payoff Since we place higher weight on buyer (λ ge 1) S1(Γ) isthe upper bound for the weighted total payoff In the proof of Proposition 1 we showthat these bounds are tight mdash any payoff pair on the frontier can be implemented by someinformation structure The key observation is that whenever v minus c(v) + λ(v minus v) lt 0 itfollows that vminus c(v) lt (1minusλ)(vminus v) le 0 That is it is incentive compatible for Seller to notsell when it is common knowledge that v creates negative weighted total payoff Thereforewe can simply release a public signal indicating the sign of weighted total payoff and ifpositive induce Seller to sell at v which implies all bounds Sλ(Γ)

43 Linear Cost

When Sellerrsquos cost function c(v) is linear we can explicitly characterize an informationstructure that implements Sellerrsquos minimum implementable payoff πus

s (Γ) when Seller isuninformed (or per Theorem 2 when Buyer is better informed) subject to price-independentbeliefs We note that given the discussion in Subsection 41 a linear c(v) subsumes richerenvironments in which conditional expectations are linear including Gaussian environ-ments (cf Bar-Isaac et al 2018)

Condition 1 c(v) = λv + γ for some λ isin R

Let F (v) be the cumulative distribution function (CDF) corresponding to the prior mea-sure micro Let D(micro) be the set of all distributions whose CDF G satisfies

983133

V

vdG(v) =

983133

V

vdF (v) and983133 v

v

G(s)ds le983133 v

v

F (s)ds forallv isin V

20

That is D(micro) contains all distributions that are mean-preserving-contractions (MPC) of microIt is well known that D(micro) characterizes the set of distributions of Buyer posterior meansthat can be generated by any (uninformed-Seller) information structure

Definition 2 G is an incentive compatible distribution (ICD) if

983133

sgep

(pminus c(s))G(ds) = constant ge 0 forallp isin Supp(G)

That is when Buyerrsquos (mean) valuation is distributed according to an ICD G Seller isindifferent among all prices in the support of G assuming Buyer buys when indifferent Wefocus on a special family of ICDs whose supports are intervals [vlowast v

lowast] Given Condition 1such ICDs have analytical an expression

G(v) =

983099983105983105983103

983105983105983101

1minus983061

(1minus λ)v minus γ

(1minus λ)vlowast minus γ

983062 1λminus1

if λ ∕= 1

1minus evminusvlowast

γ if λ = 1

(3)

Given any vlowast and the above G(v) an upper bound vlowast is pinned down by the conditionEG[v] = EF [v] So a family of ICDs is parametrized by a single parameter vlowast and hence anICD is denoted by CDF 983144Gvlowast(v) with density 983144gvlowast(v) It is not difficult to verify that 983144Gvlowast(v) ismonotonic in vlowast for v in the support Consequently all ICDs are ordered according to theMPC order For vlowast isin [vE[v]] when vlowast increases the ICD increases in the MPC order

Proposition 2 Suppose Condition 1 is satisfied and let plowast equiv min983153vlowast983055983055 983144Gvlowast isin D(micro)

983154 It holds that

πuss (Γ) = plowast minus Emicro[c(v)]

Proposition 2 states that an uninformed Sellerrsquos minimum payoff (with price-independentbeliefs) is characterized by the ICDs that are MPCs of the prior distribution micro By defini-tion of ICDs Sellerrsquos profit when facing ICD 983144Gvlowast is vlowast minus E[v] Sellerrsquos minimum payoff isimplemented by the most dispersed ICD that is a MPC of the prior distribution

In proving Proposition 2 the key step is to show that given the prior micro garbling Buyerrsquosinformation such that the posterior mean distribution is an ICD makes Seller weakly worseoff It is then without loss to only consider ICDs to implement πus

s (Γ) We can thus workwith a tractable one-dimensional problem Consider the set D(micro) Suppose we find G(v)

such that (i) G isin D(micro) (ii) G is an ICD and (iii) existp such that983125 p

vG(s)ds =

983125 p

vF (s)ds Such

a G is the most-dispersed ICD that is a mean preserving contraction of F Consider the

21

following two manipulations

λ

983133 v

p

F (s)ds =

983133 v

p

(pminus c(s))dF (s)minus (1minus F (v)) (pminus c(v)) + λ(v minus p) (4)

λ

983133 v

p

G(s)ds =

983133 v

p

(pminus c(s))dG(s)minus (1minusG(v)) (pminus c(v)) + λ(v minus p) (5)

First by property (iii) above the LHS of Equation 4 equals the LHS of Equation 5 Secondsince

983125 v

p(F (s) minus G(s))ds is minimized at v this implies F (p) le G(p) Therefore the first

term on RHS of Equation 4 must be larger than that of Equation 5 Notice that the firstterm on RHS is Sellerrsquos profit when offering price p This implies that the profit fromoffering p given Buyer mean-valuation distribution G(v) is lower than that given F (v)which is lower than Sellerrsquos maximum profit given F (v) On the other hand by the ICDproperty p is already Sellerrsquos optimal price given G(v) The optimal profit given valuationdistribution F must be no lower than that of G Therefore we have proved that to minimizeSellerrsquos profit it is without loss to consider only ICDs within the set D(micro) which is a onedimensional subspace The detailed proof that such ICDs exist is relegated to the appendix

The monopoly-pricing environment with c(middot) = v is covered by Condition 1 with λ = 0

and γ = v The distribution G in (3) then reduces to that identified by Roesler and Szentes(2017)

If the prior micro has binary support then the cost function c(v) is linear In this special casewe can solve for πus

s (Γ) explicitly

Corollary 1 Suppose micro has binary support V = v1 v2 Let λ = c(v2)minusc(v1)v2minusv1

and let p be theunique solution to

(pminus c(p))1

λminus1 (pminus E[c(v)]) = (v2 minus c(v2))λ

λminus1 (6)

Then πuss (Γ) = max p v2minus E[c(v)]

References

AKERLOF G A (1970) ldquoThe Market for rdquoLemonsrdquo Quality Uncertainty and the MarketMechanismrdquo Quarterly Journal of Economics 84 488ndash500

BAGWELL K AND M H RIORDAN (1991) ldquoHigh and Declining Prices Signal ProductQualityrdquo American Economic Review 81 224ndash239

BAR-ISAAC H I JEWITT AND C LEAVER (2018) ldquoAdverse Selection Efficiency and theStructure of Informationrdquo Unpublished

22

BERGEMANN D B BROOKS AND S MORRIS (2015) ldquoThe Limits of Price Discrimina-tionrdquo American Economic Review 105 921ndash57

BERGEMANN D AND S MORRIS (2016) ldquoBayes Correlated Equilibrium and the Compar-ison of Information Structures in Gamesrdquo Theoretical Economics 11 487ndash522

BLACKWELL D (1953) ldquoEquivalent Comparisons of Experimentsrdquo Annals of MathematicalStatistics 24 265ndash272

CHO I-K AND D KREPS (1987) ldquoSignaling Games and Stable Equilibriardquo Quarterly Jour-nal of Economics 102 179ndash221

DU S (2018) ldquoRobust Mechanisms Under Common Valuationrdquo Econometrica 86 1569ndash1588

DURRETT R (1995) Probability Theory and Examples Duxbury Press 2nd ed

EINAV L AND A FINKELSTEIN (2011) ldquoSelection in Insurance Markets Theory and Em-pirics in Picturesrdquo Journal of Economic Perspectives 25 115ndash38

FUDENBERG D AND J TIROLE (1991) ldquoPerfect Bayesian Equilibrium and Sequential Equi-libriumrdquo Journal of Economic Theory 53 236ndash260

GARCIA D R TEPER AND M TSUR (2018) ldquoInformation Design in Insurance MarketsSelling Peaches in a Market for Lemonsrdquo Unpublished

GREENWALD B C (1986) ldquoAdverse Selection in the Labour Marketrdquo Review of EconomicStudies 53 325ndash347

JOVANOVIC B (1982) ldquoFavorable Selection with Asymmetric Informationrdquo QuarterlyJournal of Economics 97 535ndash539

KESSLER A S (2001) ldquoRevisiting the Lemons Marketrdquo International Economic Review 4225ndash41

KREPS D AND R WILSON (1982) ldquoSequential Equilibriardquo Econometrica 50 863ndash894

LEHMANN E L (1988) ldquoComparing Location Experimentsrdquo The Annals of Statistics 16521ndash533

LEVIN J (2001) ldquoInformation and the Market for Lemonsrdquo RAND Journal of Economics32 657ndash666

23

LIBGOBER J AND X MU (2019) ldquoInformational Robustness in Intertemporal PricingrdquoWorking paper

MAKRIS M AND L RENOU (2018) ldquoInformation Design in Multi-Stage Gamesrdquo Workingpaper

MILGROM P R AND R J WEBER (1985) ldquoDistributional Strategies for Games with In-complete Informationrdquo Mathematics of Operations Research 10 619ndash632

MUSSA M AND S ROSEN (1978) ldquoMonopoly and Product Qualityrdquo Journal of EconomicTheory 18 301ndash317

RAVID D A-K ROESLER AND B SZENTES (2019) ldquoLearning Before Trading On theInefficiency of Ignoring Free Informationrdquo Working paper

ROESLER A-K AND B SZENTES (2017) ldquoBuyer-Optimal Learning and Monopoly Pric-ingrdquo American Economic Review 107 2072ndash80

TERSTIEGE S AND C WASSER (2019) ldquoBuyer-optimal Extensionproof InformationrdquoWorking paper

Appendices omitted from this version but are available on request

24

Faced with a fully informed Buyer and any sequentially rational Buyer strategy an un-informed Seller can guarantee a profit level

πfbs (Γ) equiv sup

p

983133 v

p

(pminus c(v))micro(dv)

regardless of her information Plainly πfbs (Γ) ge πus

s (Γ) In monopoly pricing with c(middot) = vRoesler and Szentes (2017) have shown that πfb

s gt πuss if there is no trade due to adverse

selection when Buyer is fully informed and Seller is uniformed then πfbs = πus

s = 0 Belowwe establish that when Buyer is fully informed πfb

s is the only additional constraint onequilibrium payoffs Let Bε(πb πs) denote an ε-ball around (πb πs)

Theorem 3 Consider a fully informed Buyer and price-independent beliefs

1 Πlowastfb(Γ) = (πb πs) isin Π(Γ) πs ge πfb

s (Γ)

2 For any (πb πs) isin Πlowastfb(Γ) and any ε gt 0 there is τ isin Tfb with Π(Γ τ) sub Bε(πb πs)

The payoff set characterized in Theorem 3 corresponds to the triangle ABE in Figure 1Part 2 of the theorem establishes ldquoapproximately unique implementationrdquo for any point inthe triangle there is an information structure under which all equilibria yield payoff pairsin a neighborhood of that point

Here is the idea behind part 1 of Theorem 3 When Buyer is fully informed an informa-tion structure can be viewed as dividing vrsquos prior distribution micro into a set of microi that aver-age to micro with Seller informed of which microi she faces Theorem 3 is proven by establishingthat we can divide micro suitably so that against each microi Seller is indifferent between pricingat all prices in the support of microi including the price corresponding to πfb

s in that environ-ment We call such a microi a (Seller-price) incentive compatible distribution or ICD (Bergemannet al (2015) call their analogous construct an extreme market) The Appendix provides aldquogreedyrdquo algorithm to compute ICDs19

We can sketch how the algorithm works and construct a set of ICDs that average to theprior Suppose V = v1 v2 vK with vi lt vi+1 for i isin 1 K minus 1 and c(v) lt v forall v Given any small-enough mass of vK there is a unique mass of type vKminus1 that makesSeller indifferent between charging price vK and vKminus1 (If the mass is too low Seller prefersvK if it is too high she prefers vKminus1) Iterating down to keep Seller indifferent betweenall prices pins down an ICD Choose the maximum mass of type vK for which this worksRemove that ICD and repeat

19 The algorithm is defined for finite V and we take limits to handle the infinite case

17

Crucially whenever an ICD is removed the price corresponding to πfbs (Γ) remains op-

timal in the remaining ldquomarketrdquo this follows from the ICDrsquos defining property of Sellerindifference and an accounting identity Therefore Sellerrsquos profit in this segmentation ofICDs remains πfb

s (Γ) Moreover it is also optimal for Seller to always (ie for each microi) priceso that there is full trade or no trade Hence Buyerrsquos expected payoff can be either 0 or theentire surplus less πfb

s (Γ) It follows that the fully-informed Buyer information structuredefined by this set of ICDs implements point B and C in Figure 1 The entire triangle ABC

can then be implemented by convexification randomizing over this information structure(and two equilibria) and full information (where Seller obtains all the surplus)

The above logic of incentive compatible distributions reduces to that of Bergemann et al(2015) in the context of monopoly pricing Even in the monopoly pricing case Theorem 3part 2 strengthens the conclusions of Bergemann et al (2015) by ensuring approximatelyunique implementation

Remark 4 Theorems 1ndash3 imply that fully-informed-Buyer information structures imple-ment all implementable payoff pairs if and only if πfb

s (Γ) = πs(Γ) In that case trianglesAFG and ABC coincide in Figure 1 It follows that πfb

s (Γ) = πs(Γ) only when a fully-informed Buyer and uninformed Seller can result in full trade (v ge E[c(v)] and Sellerprices at v) or no trade (v le E[c(v)] and Seller prices at some p ge v) Interestinglywhen πfb

s (Γ) gt πs(Γ) fully-informed-Buyer information structures cannot even implementall payoff pairs implementable by uninformed-Seller information structures ie trianglesABC and ADE in Figure 1 are distinct That is when πfb

s (Γ) gt πs(Γ) there is an uninformed-Seller information structure that implements some πs lt πfb

s (Γ)20 Theorems 2ndash3 further im-ply in this case that Buyer can obtain a strictly higher payoff when he is not fully informed(and Seller is uninformed)

4 Discussion

41 Multidimensionality

Suppose Buyer and Sellerrsquos cost and valuation pair (c v) is a two-dimensional randomvariable distributed according to joint distribution micro with a compact support The ex-tension of our maintained assumption of commonly known gains from trade is for all

20 Pick any pprime gt v such that pprimeminusE[c(v)] lt πfbs (Γ) Following the construction described after Theorem 2 we

can mix all valuations v le pprime with a fraction λ gt 0 of valuations v gt pprime so that the mixture has posterior meanexactly pprime The remaining fraction 1 minus λ of valuations above pprime are revealed to Buyer With this uninformed-Seller information structure consider any equilibrium in which Buyer purchases when indifferent (Suchan equilibrium with price-independent belief exists) Sellerrsquos profit is at most (1 minus λ)πfb

s (Γ) from any pricep gt pprime and pprime minus E[c(v)] from price p = pprime Hence Sellerrsquos profit is strictly less than πfb

s (Γ)

18

(c v) isin Supp(micro) v ge c and E[v minus c] gt 0 An information structure is now a joint distribu-tion P (tb ts v c) whose marginal distribution on (v c) is micro

We claim that the content of our key results Theorem 1Theorem 1lowast Theorem 2 andTheorem 3 still hold To see why let v be the lowest valuation in the support of micro Sellerrsquosindividual rationality constraint is now maxv minus E[c] 0 as she can guarantee this by set-ting either a sufficiently high price or a price (arbitrarily close to) v regardless of her signalWe can define a cost function c(vprime) equiv Emicro[c|v = vprime] ge vprime This results in an environment sat-isfying all the maintained assumptions of our baseline model except that c(middot) may not becontinuous Recall that such continuity is not needed for proving Theorem 1 Theorem 1lowastBoth Theorem 2 and Theorem 3 use continuity of c(middot) to guarantee that Eν [c(v)] is a contin-uous function of ν isin ∆(V ) for a key convergence result However in the two-dimensionaltype environment Eν [c] is still a continuous function of ν isin ∆(V times C) Note that Theo-rem 3 uses upper semi-continuity of Sellerrsquos profit in price boundedness of c and c le v issufficient for such upper semi-continuity

42 Negative Trading Surplus

Returning to our baseline model we next discuss cases in which trade sometimes gen-erates negative surplus That is we drop the assumption that c(v) le v we do not requireE[v minus c(v)] gt 0 either Define Sλ(Γ) for λ isin [1infin) as

Sλ(Γ) equiv983133 v

v

[v minus c(v) + λ(v minus v)]+ micro(dv)

where [middot]+ equiv max middot 0 It is readily verified that S1(Γ) = E [[v minus c(v)]+] and limλrarr0

Sλ(Γ)λ =

E[v] minus v Allowing negative trading surplus does not affect our definition of wPBE Sothe notation Π(Γ) and πs(Γ) still have the same meanings as before The next propositionshows that Π(Γ) is now characterized by three constraints as before the two individualrationality constraints πs ge πs(Γ) and πb ge 0 and different now a Pareto frontier definedby all Sλ(Γ)

Proposition 1 Consider all information structures and equilibria when trade can generate negativesurplus

Π(Γ) =

983099983105983103

983105983101

πb ge 0

(πb πs) πs ge πs(Γ)

λπb + πs le Sλ(Γ) forallλ ge 1

983100983105983104

983105983102

Proposition 1 shows that when the trade can be inefficient it may be impossible to lower

19

Sellerrsquos profit without losing total surplus (ie πb + πs) This result is intuitive at the ex-tremes Suppose for simplicity v gt E[c(v)] If πs = πs(Γ) then since πs(Γ) = v minus E[c(v)]Seller must find it optimal to price at v and trade with probability one including whentrade generates negative surplus So Buyerrsquos payoff is E[v]minus v = limλrarrinfin Sλ(Γ)λ and totalsurplus is E[v minus c(v)] which includes the maximum amount of inefficient trade On theother hand an information structure that publicly reveals (only) whether trade is efficient(ie whether v ge c(v)) can implement efficient trade with all the surplus accruing to SellerIn this case Seller obtains S1(Γ) while Buyer obtains zero The proposition establishes thatin between S1(Γ) and πs(Γ) information structures and equilibria that lower Sellerrsquos profitmust reduce total surplus as the Pareto frontier Sλ(Γ) is a concave function with maximalslope minus1

The function capturing the Pareto frontier Sλ(Γ) is the maximal weighted sum of Buyerand Seller payoff assuming that Seller will trade at the low price p = v whenever it createspositive weighted total payoff Since we place higher weight on buyer (λ ge 1) S1(Γ) isthe upper bound for the weighted total payoff In the proof of Proposition 1 we showthat these bounds are tight mdash any payoff pair on the frontier can be implemented by someinformation structure The key observation is that whenever v minus c(v) + λ(v minus v) lt 0 itfollows that vminus c(v) lt (1minusλ)(vminus v) le 0 That is it is incentive compatible for Seller to notsell when it is common knowledge that v creates negative weighted total payoff Thereforewe can simply release a public signal indicating the sign of weighted total payoff and ifpositive induce Seller to sell at v which implies all bounds Sλ(Γ)

43 Linear Cost

When Sellerrsquos cost function c(v) is linear we can explicitly characterize an informationstructure that implements Sellerrsquos minimum implementable payoff πus

s (Γ) when Seller isuninformed (or per Theorem 2 when Buyer is better informed) subject to price-independentbeliefs We note that given the discussion in Subsection 41 a linear c(v) subsumes richerenvironments in which conditional expectations are linear including Gaussian environ-ments (cf Bar-Isaac et al 2018)

Condition 1 c(v) = λv + γ for some λ isin R

Let F (v) be the cumulative distribution function (CDF) corresponding to the prior mea-sure micro Let D(micro) be the set of all distributions whose CDF G satisfies

983133

V

vdG(v) =

983133

V

vdF (v) and983133 v

v

G(s)ds le983133 v

v

F (s)ds forallv isin V

20

That is D(micro) contains all distributions that are mean-preserving-contractions (MPC) of microIt is well known that D(micro) characterizes the set of distributions of Buyer posterior meansthat can be generated by any (uninformed-Seller) information structure

Definition 2 G is an incentive compatible distribution (ICD) if

983133

sgep

(pminus c(s))G(ds) = constant ge 0 forallp isin Supp(G)

That is when Buyerrsquos (mean) valuation is distributed according to an ICD G Seller isindifferent among all prices in the support of G assuming Buyer buys when indifferent Wefocus on a special family of ICDs whose supports are intervals [vlowast v

lowast] Given Condition 1such ICDs have analytical an expression

G(v) =

983099983105983105983103

983105983105983101

1minus983061

(1minus λ)v minus γ

(1minus λ)vlowast minus γ

983062 1λminus1

if λ ∕= 1

1minus evminusvlowast

γ if λ = 1

(3)

Given any vlowast and the above G(v) an upper bound vlowast is pinned down by the conditionEG[v] = EF [v] So a family of ICDs is parametrized by a single parameter vlowast and hence anICD is denoted by CDF 983144Gvlowast(v) with density 983144gvlowast(v) It is not difficult to verify that 983144Gvlowast(v) ismonotonic in vlowast for v in the support Consequently all ICDs are ordered according to theMPC order For vlowast isin [vE[v]] when vlowast increases the ICD increases in the MPC order

Proposition 2 Suppose Condition 1 is satisfied and let plowast equiv min983153vlowast983055983055 983144Gvlowast isin D(micro)

983154 It holds that

πuss (Γ) = plowast minus Emicro[c(v)]

Proposition 2 states that an uninformed Sellerrsquos minimum payoff (with price-independentbeliefs) is characterized by the ICDs that are MPCs of the prior distribution micro By defini-tion of ICDs Sellerrsquos profit when facing ICD 983144Gvlowast is vlowast minus E[v] Sellerrsquos minimum payoff isimplemented by the most dispersed ICD that is a MPC of the prior distribution

In proving Proposition 2 the key step is to show that given the prior micro garbling Buyerrsquosinformation such that the posterior mean distribution is an ICD makes Seller weakly worseoff It is then without loss to only consider ICDs to implement πus

s (Γ) We can thus workwith a tractable one-dimensional problem Consider the set D(micro) Suppose we find G(v)

such that (i) G isin D(micro) (ii) G is an ICD and (iii) existp such that983125 p

vG(s)ds =

983125 p

vF (s)ds Such

a G is the most-dispersed ICD that is a mean preserving contraction of F Consider the

21

following two manipulations

λ

983133 v

p

F (s)ds =

983133 v

p

(pminus c(s))dF (s)minus (1minus F (v)) (pminus c(v)) + λ(v minus p) (4)

λ

983133 v

p

G(s)ds =

983133 v

p

(pminus c(s))dG(s)minus (1minusG(v)) (pminus c(v)) + λ(v minus p) (5)

First by property (iii) above the LHS of Equation 4 equals the LHS of Equation 5 Secondsince

983125 v

p(F (s) minus G(s))ds is minimized at v this implies F (p) le G(p) Therefore the first

term on RHS of Equation 4 must be larger than that of Equation 5 Notice that the firstterm on RHS is Sellerrsquos profit when offering price p This implies that the profit fromoffering p given Buyer mean-valuation distribution G(v) is lower than that given F (v)which is lower than Sellerrsquos maximum profit given F (v) On the other hand by the ICDproperty p is already Sellerrsquos optimal price given G(v) The optimal profit given valuationdistribution F must be no lower than that of G Therefore we have proved that to minimizeSellerrsquos profit it is without loss to consider only ICDs within the set D(micro) which is a onedimensional subspace The detailed proof that such ICDs exist is relegated to the appendix

The monopoly-pricing environment with c(middot) = v is covered by Condition 1 with λ = 0

and γ = v The distribution G in (3) then reduces to that identified by Roesler and Szentes(2017)

If the prior micro has binary support then the cost function c(v) is linear In this special casewe can solve for πus

s (Γ) explicitly

Corollary 1 Suppose micro has binary support V = v1 v2 Let λ = c(v2)minusc(v1)v2minusv1

and let p be theunique solution to

(pminus c(p))1

λminus1 (pminus E[c(v)]) = (v2 minus c(v2))λ

λminus1 (6)

Then πuss (Γ) = max p v2minus E[c(v)]

References

AKERLOF G A (1970) ldquoThe Market for rdquoLemonsrdquo Quality Uncertainty and the MarketMechanismrdquo Quarterly Journal of Economics 84 488ndash500

BAGWELL K AND M H RIORDAN (1991) ldquoHigh and Declining Prices Signal ProductQualityrdquo American Economic Review 81 224ndash239

BAR-ISAAC H I JEWITT AND C LEAVER (2018) ldquoAdverse Selection Efficiency and theStructure of Informationrdquo Unpublished

22

BERGEMANN D B BROOKS AND S MORRIS (2015) ldquoThe Limits of Price Discrimina-tionrdquo American Economic Review 105 921ndash57

BERGEMANN D AND S MORRIS (2016) ldquoBayes Correlated Equilibrium and the Compar-ison of Information Structures in Gamesrdquo Theoretical Economics 11 487ndash522

BLACKWELL D (1953) ldquoEquivalent Comparisons of Experimentsrdquo Annals of MathematicalStatistics 24 265ndash272

CHO I-K AND D KREPS (1987) ldquoSignaling Games and Stable Equilibriardquo Quarterly Jour-nal of Economics 102 179ndash221

DU S (2018) ldquoRobust Mechanisms Under Common Valuationrdquo Econometrica 86 1569ndash1588

DURRETT R (1995) Probability Theory and Examples Duxbury Press 2nd ed

EINAV L AND A FINKELSTEIN (2011) ldquoSelection in Insurance Markets Theory and Em-pirics in Picturesrdquo Journal of Economic Perspectives 25 115ndash38

FUDENBERG D AND J TIROLE (1991) ldquoPerfect Bayesian Equilibrium and Sequential Equi-libriumrdquo Journal of Economic Theory 53 236ndash260

GARCIA D R TEPER AND M TSUR (2018) ldquoInformation Design in Insurance MarketsSelling Peaches in a Market for Lemonsrdquo Unpublished

GREENWALD B C (1986) ldquoAdverse Selection in the Labour Marketrdquo Review of EconomicStudies 53 325ndash347

JOVANOVIC B (1982) ldquoFavorable Selection with Asymmetric Informationrdquo QuarterlyJournal of Economics 97 535ndash539

KESSLER A S (2001) ldquoRevisiting the Lemons Marketrdquo International Economic Review 4225ndash41

KREPS D AND R WILSON (1982) ldquoSequential Equilibriardquo Econometrica 50 863ndash894

LEHMANN E L (1988) ldquoComparing Location Experimentsrdquo The Annals of Statistics 16521ndash533

LEVIN J (2001) ldquoInformation and the Market for Lemonsrdquo RAND Journal of Economics32 657ndash666

23

LIBGOBER J AND X MU (2019) ldquoInformational Robustness in Intertemporal PricingrdquoWorking paper

MAKRIS M AND L RENOU (2018) ldquoInformation Design in Multi-Stage Gamesrdquo Workingpaper

MILGROM P R AND R J WEBER (1985) ldquoDistributional Strategies for Games with In-complete Informationrdquo Mathematics of Operations Research 10 619ndash632

MUSSA M AND S ROSEN (1978) ldquoMonopoly and Product Qualityrdquo Journal of EconomicTheory 18 301ndash317

RAVID D A-K ROESLER AND B SZENTES (2019) ldquoLearning Before Trading On theInefficiency of Ignoring Free Informationrdquo Working paper

ROESLER A-K AND B SZENTES (2017) ldquoBuyer-Optimal Learning and Monopoly Pric-ingrdquo American Economic Review 107 2072ndash80

TERSTIEGE S AND C WASSER (2019) ldquoBuyer-optimal Extensionproof InformationrdquoWorking paper

Appendices omitted from this version but are available on request

24

Crucially whenever an ICD is removed the price corresponding to πfbs (Γ) remains op-

timal in the remaining ldquomarketrdquo this follows from the ICDrsquos defining property of Sellerindifference and an accounting identity Therefore Sellerrsquos profit in this segmentation ofICDs remains πfb

s (Γ) Moreover it is also optimal for Seller to always (ie for each microi) priceso that there is full trade or no trade Hence Buyerrsquos expected payoff can be either 0 or theentire surplus less πfb

s (Γ) It follows that the fully-informed Buyer information structuredefined by this set of ICDs implements point B and C in Figure 1 The entire triangle ABC

can then be implemented by convexification randomizing over this information structure(and two equilibria) and full information (where Seller obtains all the surplus)

The above logic of incentive compatible distributions reduces to that of Bergemann et al(2015) in the context of monopoly pricing Even in the monopoly pricing case Theorem 3part 2 strengthens the conclusions of Bergemann et al (2015) by ensuring approximatelyunique implementation

Remark 4 Theorems 1ndash3 imply that fully-informed-Buyer information structures imple-ment all implementable payoff pairs if and only if πfb

s (Γ) = πs(Γ) In that case trianglesAFG and ABC coincide in Figure 1 It follows that πfb

s (Γ) = πs(Γ) only when a fully-informed Buyer and uninformed Seller can result in full trade (v ge E[c(v)] and Sellerprices at v) or no trade (v le E[c(v)] and Seller prices at some p ge v) Interestinglywhen πfb

s (Γ) gt πs(Γ) fully-informed-Buyer information structures cannot even implementall payoff pairs implementable by uninformed-Seller information structures ie trianglesABC and ADE in Figure 1 are distinct That is when πfb

s (Γ) gt πs(Γ) there is an uninformed-Seller information structure that implements some πs lt πfb

s (Γ)20 Theorems 2ndash3 further im-ply in this case that Buyer can obtain a strictly higher payoff when he is not fully informed(and Seller is uninformed)

4 Discussion

41 Multidimensionality

Suppose Buyer and Sellerrsquos cost and valuation pair (c v) is a two-dimensional randomvariable distributed according to joint distribution micro with a compact support The ex-tension of our maintained assumption of commonly known gains from trade is for all

20 Pick any pprime gt v such that pprimeminusE[c(v)] lt πfbs (Γ) Following the construction described after Theorem 2 we

can mix all valuations v le pprime with a fraction λ gt 0 of valuations v gt pprime so that the mixture has posterior meanexactly pprime The remaining fraction 1 minus λ of valuations above pprime are revealed to Buyer With this uninformed-Seller information structure consider any equilibrium in which Buyer purchases when indifferent (Suchan equilibrium with price-independent belief exists) Sellerrsquos profit is at most (1 minus λ)πfb

s (Γ) from any pricep gt pprime and pprime minus E[c(v)] from price p = pprime Hence Sellerrsquos profit is strictly less than πfb

s (Γ)

18

(c v) isin Supp(micro) v ge c and E[v minus c] gt 0 An information structure is now a joint distribu-tion P (tb ts v c) whose marginal distribution on (v c) is micro

We claim that the content of our key results Theorem 1Theorem 1lowast Theorem 2 andTheorem 3 still hold To see why let v be the lowest valuation in the support of micro Sellerrsquosindividual rationality constraint is now maxv minus E[c] 0 as she can guarantee this by set-ting either a sufficiently high price or a price (arbitrarily close to) v regardless of her signalWe can define a cost function c(vprime) equiv Emicro[c|v = vprime] ge vprime This results in an environment sat-isfying all the maintained assumptions of our baseline model except that c(middot) may not becontinuous Recall that such continuity is not needed for proving Theorem 1 Theorem 1lowastBoth Theorem 2 and Theorem 3 use continuity of c(middot) to guarantee that Eν [c(v)] is a contin-uous function of ν isin ∆(V ) for a key convergence result However in the two-dimensionaltype environment Eν [c] is still a continuous function of ν isin ∆(V times C) Note that Theo-rem 3 uses upper semi-continuity of Sellerrsquos profit in price boundedness of c and c le v issufficient for such upper semi-continuity

42 Negative Trading Surplus

Returning to our baseline model we next discuss cases in which trade sometimes gen-erates negative surplus That is we drop the assumption that c(v) le v we do not requireE[v minus c(v)] gt 0 either Define Sλ(Γ) for λ isin [1infin) as

Sλ(Γ) equiv983133 v

v

[v minus c(v) + λ(v minus v)]+ micro(dv)

where [middot]+ equiv max middot 0 It is readily verified that S1(Γ) = E [[v minus c(v)]+] and limλrarr0

Sλ(Γ)λ =

E[v] minus v Allowing negative trading surplus does not affect our definition of wPBE Sothe notation Π(Γ) and πs(Γ) still have the same meanings as before The next propositionshows that Π(Γ) is now characterized by three constraints as before the two individualrationality constraints πs ge πs(Γ) and πb ge 0 and different now a Pareto frontier definedby all Sλ(Γ)

Proposition 1 Consider all information structures and equilibria when trade can generate negativesurplus

Π(Γ) =

983099983105983103

983105983101

πb ge 0

(πb πs) πs ge πs(Γ)

λπb + πs le Sλ(Γ) forallλ ge 1

983100983105983104

983105983102

Proposition 1 shows that when the trade can be inefficient it may be impossible to lower

19

Sellerrsquos profit without losing total surplus (ie πb + πs) This result is intuitive at the ex-tremes Suppose for simplicity v gt E[c(v)] If πs = πs(Γ) then since πs(Γ) = v minus E[c(v)]Seller must find it optimal to price at v and trade with probability one including whentrade generates negative surplus So Buyerrsquos payoff is E[v]minus v = limλrarrinfin Sλ(Γ)λ and totalsurplus is E[v minus c(v)] which includes the maximum amount of inefficient trade On theother hand an information structure that publicly reveals (only) whether trade is efficient(ie whether v ge c(v)) can implement efficient trade with all the surplus accruing to SellerIn this case Seller obtains S1(Γ) while Buyer obtains zero The proposition establishes thatin between S1(Γ) and πs(Γ) information structures and equilibria that lower Sellerrsquos profitmust reduce total surplus as the Pareto frontier Sλ(Γ) is a concave function with maximalslope minus1

The function capturing the Pareto frontier Sλ(Γ) is the maximal weighted sum of Buyerand Seller payoff assuming that Seller will trade at the low price p = v whenever it createspositive weighted total payoff Since we place higher weight on buyer (λ ge 1) S1(Γ) isthe upper bound for the weighted total payoff In the proof of Proposition 1 we showthat these bounds are tight mdash any payoff pair on the frontier can be implemented by someinformation structure The key observation is that whenever v minus c(v) + λ(v minus v) lt 0 itfollows that vminus c(v) lt (1minusλ)(vminus v) le 0 That is it is incentive compatible for Seller to notsell when it is common knowledge that v creates negative weighted total payoff Thereforewe can simply release a public signal indicating the sign of weighted total payoff and ifpositive induce Seller to sell at v which implies all bounds Sλ(Γ)

43 Linear Cost

When Sellerrsquos cost function c(v) is linear we can explicitly characterize an informationstructure that implements Sellerrsquos minimum implementable payoff πus

s (Γ) when Seller isuninformed (or per Theorem 2 when Buyer is better informed) subject to price-independentbeliefs We note that given the discussion in Subsection 41 a linear c(v) subsumes richerenvironments in which conditional expectations are linear including Gaussian environ-ments (cf Bar-Isaac et al 2018)

Condition 1 c(v) = λv + γ for some λ isin R

Let F (v) be the cumulative distribution function (CDF) corresponding to the prior mea-sure micro Let D(micro) be the set of all distributions whose CDF G satisfies

983133

V

vdG(v) =

983133

V

vdF (v) and983133 v

v

G(s)ds le983133 v

v

F (s)ds forallv isin V

20

That is D(micro) contains all distributions that are mean-preserving-contractions (MPC) of microIt is well known that D(micro) characterizes the set of distributions of Buyer posterior meansthat can be generated by any (uninformed-Seller) information structure

Definition 2 G is an incentive compatible distribution (ICD) if

983133

sgep

(pminus c(s))G(ds) = constant ge 0 forallp isin Supp(G)

That is when Buyerrsquos (mean) valuation is distributed according to an ICD G Seller isindifferent among all prices in the support of G assuming Buyer buys when indifferent Wefocus on a special family of ICDs whose supports are intervals [vlowast v

lowast] Given Condition 1such ICDs have analytical an expression

G(v) =

983099983105983105983103

983105983105983101

1minus983061

(1minus λ)v minus γ

(1minus λ)vlowast minus γ

983062 1λminus1

if λ ∕= 1

1minus evminusvlowast

γ if λ = 1

(3)

Given any vlowast and the above G(v) an upper bound vlowast is pinned down by the conditionEG[v] = EF [v] So a family of ICDs is parametrized by a single parameter vlowast and hence anICD is denoted by CDF 983144Gvlowast(v) with density 983144gvlowast(v) It is not difficult to verify that 983144Gvlowast(v) ismonotonic in vlowast for v in the support Consequently all ICDs are ordered according to theMPC order For vlowast isin [vE[v]] when vlowast increases the ICD increases in the MPC order

Proposition 2 Suppose Condition 1 is satisfied and let plowast equiv min983153vlowast983055983055 983144Gvlowast isin D(micro)

983154 It holds that

πuss (Γ) = plowast minus Emicro[c(v)]

Proposition 2 states that an uninformed Sellerrsquos minimum payoff (with price-independentbeliefs) is characterized by the ICDs that are MPCs of the prior distribution micro By defini-tion of ICDs Sellerrsquos profit when facing ICD 983144Gvlowast is vlowast minus E[v] Sellerrsquos minimum payoff isimplemented by the most dispersed ICD that is a MPC of the prior distribution

In proving Proposition 2 the key step is to show that given the prior micro garbling Buyerrsquosinformation such that the posterior mean distribution is an ICD makes Seller weakly worseoff It is then without loss to only consider ICDs to implement πus

s (Γ) We can thus workwith a tractable one-dimensional problem Consider the set D(micro) Suppose we find G(v)

such that (i) G isin D(micro) (ii) G is an ICD and (iii) existp such that983125 p

vG(s)ds =

983125 p

vF (s)ds Such

a G is the most-dispersed ICD that is a mean preserving contraction of F Consider the

21

following two manipulations

λ

983133 v

p

F (s)ds =

983133 v

p

(pminus c(s))dF (s)minus (1minus F (v)) (pminus c(v)) + λ(v minus p) (4)

λ

983133 v

p

G(s)ds =

983133 v

p

(pminus c(s))dG(s)minus (1minusG(v)) (pminus c(v)) + λ(v minus p) (5)

First by property (iii) above the LHS of Equation 4 equals the LHS of Equation 5 Secondsince

983125 v

p(F (s) minus G(s))ds is minimized at v this implies F (p) le G(p) Therefore the first

term on RHS of Equation 4 must be larger than that of Equation 5 Notice that the firstterm on RHS is Sellerrsquos profit when offering price p This implies that the profit fromoffering p given Buyer mean-valuation distribution G(v) is lower than that given F (v)which is lower than Sellerrsquos maximum profit given F (v) On the other hand by the ICDproperty p is already Sellerrsquos optimal price given G(v) The optimal profit given valuationdistribution F must be no lower than that of G Therefore we have proved that to minimizeSellerrsquos profit it is without loss to consider only ICDs within the set D(micro) which is a onedimensional subspace The detailed proof that such ICDs exist is relegated to the appendix

The monopoly-pricing environment with c(middot) = v is covered by Condition 1 with λ = 0

and γ = v The distribution G in (3) then reduces to that identified by Roesler and Szentes(2017)

If the prior micro has binary support then the cost function c(v) is linear In this special casewe can solve for πus

s (Γ) explicitly

Corollary 1 Suppose micro has binary support V = v1 v2 Let λ = c(v2)minusc(v1)v2minusv1

and let p be theunique solution to

(pminus c(p))1

λminus1 (pminus E[c(v)]) = (v2 minus c(v2))λ

λminus1 (6)

Then πuss (Γ) = max p v2minus E[c(v)]

References

AKERLOF G A (1970) ldquoThe Market for rdquoLemonsrdquo Quality Uncertainty and the MarketMechanismrdquo Quarterly Journal of Economics 84 488ndash500

BAGWELL K AND M H RIORDAN (1991) ldquoHigh and Declining Prices Signal ProductQualityrdquo American Economic Review 81 224ndash239

BAR-ISAAC H I JEWITT AND C LEAVER (2018) ldquoAdverse Selection Efficiency and theStructure of Informationrdquo Unpublished

22

BERGEMANN D B BROOKS AND S MORRIS (2015) ldquoThe Limits of Price Discrimina-tionrdquo American Economic Review 105 921ndash57

BERGEMANN D AND S MORRIS (2016) ldquoBayes Correlated Equilibrium and the Compar-ison of Information Structures in Gamesrdquo Theoretical Economics 11 487ndash522

BLACKWELL D (1953) ldquoEquivalent Comparisons of Experimentsrdquo Annals of MathematicalStatistics 24 265ndash272

CHO I-K AND D KREPS (1987) ldquoSignaling Games and Stable Equilibriardquo Quarterly Jour-nal of Economics 102 179ndash221

DU S (2018) ldquoRobust Mechanisms Under Common Valuationrdquo Econometrica 86 1569ndash1588

DURRETT R (1995) Probability Theory and Examples Duxbury Press 2nd ed

EINAV L AND A FINKELSTEIN (2011) ldquoSelection in Insurance Markets Theory and Em-pirics in Picturesrdquo Journal of Economic Perspectives 25 115ndash38

FUDENBERG D AND J TIROLE (1991) ldquoPerfect Bayesian Equilibrium and Sequential Equi-libriumrdquo Journal of Economic Theory 53 236ndash260

GARCIA D R TEPER AND M TSUR (2018) ldquoInformation Design in Insurance MarketsSelling Peaches in a Market for Lemonsrdquo Unpublished

GREENWALD B C (1986) ldquoAdverse Selection in the Labour Marketrdquo Review of EconomicStudies 53 325ndash347

JOVANOVIC B (1982) ldquoFavorable Selection with Asymmetric Informationrdquo QuarterlyJournal of Economics 97 535ndash539

KESSLER A S (2001) ldquoRevisiting the Lemons Marketrdquo International Economic Review 4225ndash41

KREPS D AND R WILSON (1982) ldquoSequential Equilibriardquo Econometrica 50 863ndash894

LEHMANN E L (1988) ldquoComparing Location Experimentsrdquo The Annals of Statistics 16521ndash533

LEVIN J (2001) ldquoInformation and the Market for Lemonsrdquo RAND Journal of Economics32 657ndash666

23

LIBGOBER J AND X MU (2019) ldquoInformational Robustness in Intertemporal PricingrdquoWorking paper

MAKRIS M AND L RENOU (2018) ldquoInformation Design in Multi-Stage Gamesrdquo Workingpaper

MILGROM P R AND R J WEBER (1985) ldquoDistributional Strategies for Games with In-complete Informationrdquo Mathematics of Operations Research 10 619ndash632

MUSSA M AND S ROSEN (1978) ldquoMonopoly and Product Qualityrdquo Journal of EconomicTheory 18 301ndash317

RAVID D A-K ROESLER AND B SZENTES (2019) ldquoLearning Before Trading On theInefficiency of Ignoring Free Informationrdquo Working paper

ROESLER A-K AND B SZENTES (2017) ldquoBuyer-Optimal Learning and Monopoly Pric-ingrdquo American Economic Review 107 2072ndash80

TERSTIEGE S AND C WASSER (2019) ldquoBuyer-optimal Extensionproof InformationrdquoWorking paper

Appendices omitted from this version but are available on request

24

(c v) isin Supp(micro) v ge c and E[v minus c] gt 0 An information structure is now a joint distribu-tion P (tb ts v c) whose marginal distribution on (v c) is micro

We claim that the content of our key results Theorem 1Theorem 1lowast Theorem 2 andTheorem 3 still hold To see why let v be the lowest valuation in the support of micro Sellerrsquosindividual rationality constraint is now maxv minus E[c] 0 as she can guarantee this by set-ting either a sufficiently high price or a price (arbitrarily close to) v regardless of her signalWe can define a cost function c(vprime) equiv Emicro[c|v = vprime] ge vprime This results in an environment sat-isfying all the maintained assumptions of our baseline model except that c(middot) may not becontinuous Recall that such continuity is not needed for proving Theorem 1 Theorem 1lowastBoth Theorem 2 and Theorem 3 use continuity of c(middot) to guarantee that Eν [c(v)] is a contin-uous function of ν isin ∆(V ) for a key convergence result However in the two-dimensionaltype environment Eν [c] is still a continuous function of ν isin ∆(V times C) Note that Theo-rem 3 uses upper semi-continuity of Sellerrsquos profit in price boundedness of c and c le v issufficient for such upper semi-continuity

42 Negative Trading Surplus

Returning to our baseline model we next discuss cases in which trade sometimes gen-erates negative surplus That is we drop the assumption that c(v) le v we do not requireE[v minus c(v)] gt 0 either Define Sλ(Γ) for λ isin [1infin) as

Sλ(Γ) equiv983133 v

v

[v minus c(v) + λ(v minus v)]+ micro(dv)

where [middot]+ equiv max middot 0 It is readily verified that S1(Γ) = E [[v minus c(v)]+] and limλrarr0

Sλ(Γ)λ =

E[v] minus v Allowing negative trading surplus does not affect our definition of wPBE Sothe notation Π(Γ) and πs(Γ) still have the same meanings as before The next propositionshows that Π(Γ) is now characterized by three constraints as before the two individualrationality constraints πs ge πs(Γ) and πb ge 0 and different now a Pareto frontier definedby all Sλ(Γ)

Proposition 1 Consider all information structures and equilibria when trade can generate negativesurplus

Π(Γ) =

983099983105983103

983105983101

πb ge 0

(πb πs) πs ge πs(Γ)

λπb + πs le Sλ(Γ) forallλ ge 1

983100983105983104

983105983102

Proposition 1 shows that when the trade can be inefficient it may be impossible to lower

19

Sellerrsquos profit without losing total surplus (ie πb + πs) This result is intuitive at the ex-tremes Suppose for simplicity v gt E[c(v)] If πs = πs(Γ) then since πs(Γ) = v minus E[c(v)]Seller must find it optimal to price at v and trade with probability one including whentrade generates negative surplus So Buyerrsquos payoff is E[v]minus v = limλrarrinfin Sλ(Γ)λ and totalsurplus is E[v minus c(v)] which includes the maximum amount of inefficient trade On theother hand an information structure that publicly reveals (only) whether trade is efficient(ie whether v ge c(v)) can implement efficient trade with all the surplus accruing to SellerIn this case Seller obtains S1(Γ) while Buyer obtains zero The proposition establishes thatin between S1(Γ) and πs(Γ) information structures and equilibria that lower Sellerrsquos profitmust reduce total surplus as the Pareto frontier Sλ(Γ) is a concave function with maximalslope minus1

The function capturing the Pareto frontier Sλ(Γ) is the maximal weighted sum of Buyerand Seller payoff assuming that Seller will trade at the low price p = v whenever it createspositive weighted total payoff Since we place higher weight on buyer (λ ge 1) S1(Γ) isthe upper bound for the weighted total payoff In the proof of Proposition 1 we showthat these bounds are tight mdash any payoff pair on the frontier can be implemented by someinformation structure The key observation is that whenever v minus c(v) + λ(v minus v) lt 0 itfollows that vminus c(v) lt (1minusλ)(vminus v) le 0 That is it is incentive compatible for Seller to notsell when it is common knowledge that v creates negative weighted total payoff Thereforewe can simply release a public signal indicating the sign of weighted total payoff and ifpositive induce Seller to sell at v which implies all bounds Sλ(Γ)

43 Linear Cost

When Sellerrsquos cost function c(v) is linear we can explicitly characterize an informationstructure that implements Sellerrsquos minimum implementable payoff πus

s (Γ) when Seller isuninformed (or per Theorem 2 when Buyer is better informed) subject to price-independentbeliefs We note that given the discussion in Subsection 41 a linear c(v) subsumes richerenvironments in which conditional expectations are linear including Gaussian environ-ments (cf Bar-Isaac et al 2018)

Condition 1 c(v) = λv + γ for some λ isin R

Let F (v) be the cumulative distribution function (CDF) corresponding to the prior mea-sure micro Let D(micro) be the set of all distributions whose CDF G satisfies

983133

V

vdG(v) =

983133

V

vdF (v) and983133 v

v

G(s)ds le983133 v

v

F (s)ds forallv isin V

20

That is D(micro) contains all distributions that are mean-preserving-contractions (MPC) of microIt is well known that D(micro) characterizes the set of distributions of Buyer posterior meansthat can be generated by any (uninformed-Seller) information structure

Definition 2 G is an incentive compatible distribution (ICD) if

983133

sgep

(pminus c(s))G(ds) = constant ge 0 forallp isin Supp(G)

That is when Buyerrsquos (mean) valuation is distributed according to an ICD G Seller isindifferent among all prices in the support of G assuming Buyer buys when indifferent Wefocus on a special family of ICDs whose supports are intervals [vlowast v

lowast] Given Condition 1such ICDs have analytical an expression

G(v) =

983099983105983105983103

983105983105983101

1minus983061

(1minus λ)v minus γ

(1minus λ)vlowast minus γ

983062 1λminus1

if λ ∕= 1

1minus evminusvlowast

γ if λ = 1

(3)

Given any vlowast and the above G(v) an upper bound vlowast is pinned down by the conditionEG[v] = EF [v] So a family of ICDs is parametrized by a single parameter vlowast and hence anICD is denoted by CDF 983144Gvlowast(v) with density 983144gvlowast(v) It is not difficult to verify that 983144Gvlowast(v) ismonotonic in vlowast for v in the support Consequently all ICDs are ordered according to theMPC order For vlowast isin [vE[v]] when vlowast increases the ICD increases in the MPC order

Proposition 2 Suppose Condition 1 is satisfied and let plowast equiv min983153vlowast983055983055 983144Gvlowast isin D(micro)

983154 It holds that

πuss (Γ) = plowast minus Emicro[c(v)]

Proposition 2 states that an uninformed Sellerrsquos minimum payoff (with price-independentbeliefs) is characterized by the ICDs that are MPCs of the prior distribution micro By defini-tion of ICDs Sellerrsquos profit when facing ICD 983144Gvlowast is vlowast minus E[v] Sellerrsquos minimum payoff isimplemented by the most dispersed ICD that is a MPC of the prior distribution

In proving Proposition 2 the key step is to show that given the prior micro garbling Buyerrsquosinformation such that the posterior mean distribution is an ICD makes Seller weakly worseoff It is then without loss to only consider ICDs to implement πus

s (Γ) We can thus workwith a tractable one-dimensional problem Consider the set D(micro) Suppose we find G(v)

such that (i) G isin D(micro) (ii) G is an ICD and (iii) existp such that983125 p

vG(s)ds =

983125 p

vF (s)ds Such

a G is the most-dispersed ICD that is a mean preserving contraction of F Consider the

21

following two manipulations

λ

983133 v

p

F (s)ds =

983133 v

p

(pminus c(s))dF (s)minus (1minus F (v)) (pminus c(v)) + λ(v minus p) (4)

λ

983133 v

p

G(s)ds =

983133 v

p

(pminus c(s))dG(s)minus (1minusG(v)) (pminus c(v)) + λ(v minus p) (5)

First by property (iii) above the LHS of Equation 4 equals the LHS of Equation 5 Secondsince

983125 v

p(F (s) minus G(s))ds is minimized at v this implies F (p) le G(p) Therefore the first

term on RHS of Equation 4 must be larger than that of Equation 5 Notice that the firstterm on RHS is Sellerrsquos profit when offering price p This implies that the profit fromoffering p given Buyer mean-valuation distribution G(v) is lower than that given F (v)which is lower than Sellerrsquos maximum profit given F (v) On the other hand by the ICDproperty p is already Sellerrsquos optimal price given G(v) The optimal profit given valuationdistribution F must be no lower than that of G Therefore we have proved that to minimizeSellerrsquos profit it is without loss to consider only ICDs within the set D(micro) which is a onedimensional subspace The detailed proof that such ICDs exist is relegated to the appendix

The monopoly-pricing environment with c(middot) = v is covered by Condition 1 with λ = 0

and γ = v The distribution G in (3) then reduces to that identified by Roesler and Szentes(2017)

If the prior micro has binary support then the cost function c(v) is linear In this special casewe can solve for πus

s (Γ) explicitly

Corollary 1 Suppose micro has binary support V = v1 v2 Let λ = c(v2)minusc(v1)v2minusv1

and let p be theunique solution to

(pminus c(p))1

λminus1 (pminus E[c(v)]) = (v2 minus c(v2))λ

λminus1 (6)

Then πuss (Γ) = max p v2minus E[c(v)]

References

AKERLOF G A (1970) ldquoThe Market for rdquoLemonsrdquo Quality Uncertainty and the MarketMechanismrdquo Quarterly Journal of Economics 84 488ndash500

BAGWELL K AND M H RIORDAN (1991) ldquoHigh and Declining Prices Signal ProductQualityrdquo American Economic Review 81 224ndash239

BAR-ISAAC H I JEWITT AND C LEAVER (2018) ldquoAdverse Selection Efficiency and theStructure of Informationrdquo Unpublished

22

BERGEMANN D B BROOKS AND S MORRIS (2015) ldquoThe Limits of Price Discrimina-tionrdquo American Economic Review 105 921ndash57

BERGEMANN D AND S MORRIS (2016) ldquoBayes Correlated Equilibrium and the Compar-ison of Information Structures in Gamesrdquo Theoretical Economics 11 487ndash522

BLACKWELL D (1953) ldquoEquivalent Comparisons of Experimentsrdquo Annals of MathematicalStatistics 24 265ndash272

CHO I-K AND D KREPS (1987) ldquoSignaling Games and Stable Equilibriardquo Quarterly Jour-nal of Economics 102 179ndash221

DU S (2018) ldquoRobust Mechanisms Under Common Valuationrdquo Econometrica 86 1569ndash1588

DURRETT R (1995) Probability Theory and Examples Duxbury Press 2nd ed

EINAV L AND A FINKELSTEIN (2011) ldquoSelection in Insurance Markets Theory and Em-pirics in Picturesrdquo Journal of Economic Perspectives 25 115ndash38

FUDENBERG D AND J TIROLE (1991) ldquoPerfect Bayesian Equilibrium and Sequential Equi-libriumrdquo Journal of Economic Theory 53 236ndash260

GARCIA D R TEPER AND M TSUR (2018) ldquoInformation Design in Insurance MarketsSelling Peaches in a Market for Lemonsrdquo Unpublished

GREENWALD B C (1986) ldquoAdverse Selection in the Labour Marketrdquo Review of EconomicStudies 53 325ndash347

JOVANOVIC B (1982) ldquoFavorable Selection with Asymmetric Informationrdquo QuarterlyJournal of Economics 97 535ndash539

KESSLER A S (2001) ldquoRevisiting the Lemons Marketrdquo International Economic Review 4225ndash41

KREPS D AND R WILSON (1982) ldquoSequential Equilibriardquo Econometrica 50 863ndash894

LEHMANN E L (1988) ldquoComparing Location Experimentsrdquo The Annals of Statistics 16521ndash533

LEVIN J (2001) ldquoInformation and the Market for Lemonsrdquo RAND Journal of Economics32 657ndash666

23

LIBGOBER J AND X MU (2019) ldquoInformational Robustness in Intertemporal PricingrdquoWorking paper

MAKRIS M AND L RENOU (2018) ldquoInformation Design in Multi-Stage Gamesrdquo Workingpaper

MILGROM P R AND R J WEBER (1985) ldquoDistributional Strategies for Games with In-complete Informationrdquo Mathematics of Operations Research 10 619ndash632

MUSSA M AND S ROSEN (1978) ldquoMonopoly and Product Qualityrdquo Journal of EconomicTheory 18 301ndash317

RAVID D A-K ROESLER AND B SZENTES (2019) ldquoLearning Before Trading On theInefficiency of Ignoring Free Informationrdquo Working paper

ROESLER A-K AND B SZENTES (2017) ldquoBuyer-Optimal Learning and Monopoly Pric-ingrdquo American Economic Review 107 2072ndash80

TERSTIEGE S AND C WASSER (2019) ldquoBuyer-optimal Extensionproof InformationrdquoWorking paper

Appendices omitted from this version but are available on request

24

Sellerrsquos profit without losing total surplus (ie πb + πs) This result is intuitive at the ex-tremes Suppose for simplicity v gt E[c(v)] If πs = πs(Γ) then since πs(Γ) = v minus E[c(v)]Seller must find it optimal to price at v and trade with probability one including whentrade generates negative surplus So Buyerrsquos payoff is E[v]minus v = limλrarrinfin Sλ(Γ)λ and totalsurplus is E[v minus c(v)] which includes the maximum amount of inefficient trade On theother hand an information structure that publicly reveals (only) whether trade is efficient(ie whether v ge c(v)) can implement efficient trade with all the surplus accruing to SellerIn this case Seller obtains S1(Γ) while Buyer obtains zero The proposition establishes thatin between S1(Γ) and πs(Γ) information structures and equilibria that lower Sellerrsquos profitmust reduce total surplus as the Pareto frontier Sλ(Γ) is a concave function with maximalslope minus1

The function capturing the Pareto frontier Sλ(Γ) is the maximal weighted sum of Buyerand Seller payoff assuming that Seller will trade at the low price p = v whenever it createspositive weighted total payoff Since we place higher weight on buyer (λ ge 1) S1(Γ) isthe upper bound for the weighted total payoff In the proof of Proposition 1 we showthat these bounds are tight mdash any payoff pair on the frontier can be implemented by someinformation structure The key observation is that whenever v minus c(v) + λ(v minus v) lt 0 itfollows that vminus c(v) lt (1minusλ)(vminus v) le 0 That is it is incentive compatible for Seller to notsell when it is common knowledge that v creates negative weighted total payoff Thereforewe can simply release a public signal indicating the sign of weighted total payoff and ifpositive induce Seller to sell at v which implies all bounds Sλ(Γ)

43 Linear Cost

When Sellerrsquos cost function c(v) is linear we can explicitly characterize an informationstructure that implements Sellerrsquos minimum implementable payoff πus

s (Γ) when Seller isuninformed (or per Theorem 2 when Buyer is better informed) subject to price-independentbeliefs We note that given the discussion in Subsection 41 a linear c(v) subsumes richerenvironments in which conditional expectations are linear including Gaussian environ-ments (cf Bar-Isaac et al 2018)

Condition 1 c(v) = λv + γ for some λ isin R

Let F (v) be the cumulative distribution function (CDF) corresponding to the prior mea-sure micro Let D(micro) be the set of all distributions whose CDF G satisfies

983133

V

vdG(v) =

983133

V

vdF (v) and983133 v

v

G(s)ds le983133 v

v

F (s)ds forallv isin V

20

That is D(micro) contains all distributions that are mean-preserving-contractions (MPC) of microIt is well known that D(micro) characterizes the set of distributions of Buyer posterior meansthat can be generated by any (uninformed-Seller) information structure

Definition 2 G is an incentive compatible distribution (ICD) if

983133

sgep

(pminus c(s))G(ds) = constant ge 0 forallp isin Supp(G)

That is when Buyerrsquos (mean) valuation is distributed according to an ICD G Seller isindifferent among all prices in the support of G assuming Buyer buys when indifferent Wefocus on a special family of ICDs whose supports are intervals [vlowast v

lowast] Given Condition 1such ICDs have analytical an expression

G(v) =

983099983105983105983103

983105983105983101

1minus983061

(1minus λ)v minus γ

(1minus λ)vlowast minus γ

983062 1λminus1

if λ ∕= 1

1minus evminusvlowast

γ if λ = 1

(3)

Given any vlowast and the above G(v) an upper bound vlowast is pinned down by the conditionEG[v] = EF [v] So a family of ICDs is parametrized by a single parameter vlowast and hence anICD is denoted by CDF 983144Gvlowast(v) with density 983144gvlowast(v) It is not difficult to verify that 983144Gvlowast(v) ismonotonic in vlowast for v in the support Consequently all ICDs are ordered according to theMPC order For vlowast isin [vE[v]] when vlowast increases the ICD increases in the MPC order

Proposition 2 Suppose Condition 1 is satisfied and let plowast equiv min983153vlowast983055983055 983144Gvlowast isin D(micro)

983154 It holds that

πuss (Γ) = plowast minus Emicro[c(v)]

Proposition 2 states that an uninformed Sellerrsquos minimum payoff (with price-independentbeliefs) is characterized by the ICDs that are MPCs of the prior distribution micro By defini-tion of ICDs Sellerrsquos profit when facing ICD 983144Gvlowast is vlowast minus E[v] Sellerrsquos minimum payoff isimplemented by the most dispersed ICD that is a MPC of the prior distribution

In proving Proposition 2 the key step is to show that given the prior micro garbling Buyerrsquosinformation such that the posterior mean distribution is an ICD makes Seller weakly worseoff It is then without loss to only consider ICDs to implement πus

s (Γ) We can thus workwith a tractable one-dimensional problem Consider the set D(micro) Suppose we find G(v)

such that (i) G isin D(micro) (ii) G is an ICD and (iii) existp such that983125 p

vG(s)ds =

983125 p

vF (s)ds Such

a G is the most-dispersed ICD that is a mean preserving contraction of F Consider the

21

following two manipulations

λ

983133 v

p

F (s)ds =

983133 v

p

(pminus c(s))dF (s)minus (1minus F (v)) (pminus c(v)) + λ(v minus p) (4)

λ

983133 v

p

G(s)ds =

983133 v

p

(pminus c(s))dG(s)minus (1minusG(v)) (pminus c(v)) + λ(v minus p) (5)

First by property (iii) above the LHS of Equation 4 equals the LHS of Equation 5 Secondsince

983125 v

p(F (s) minus G(s))ds is minimized at v this implies F (p) le G(p) Therefore the first

term on RHS of Equation 4 must be larger than that of Equation 5 Notice that the firstterm on RHS is Sellerrsquos profit when offering price p This implies that the profit fromoffering p given Buyer mean-valuation distribution G(v) is lower than that given F (v)which is lower than Sellerrsquos maximum profit given F (v) On the other hand by the ICDproperty p is already Sellerrsquos optimal price given G(v) The optimal profit given valuationdistribution F must be no lower than that of G Therefore we have proved that to minimizeSellerrsquos profit it is without loss to consider only ICDs within the set D(micro) which is a onedimensional subspace The detailed proof that such ICDs exist is relegated to the appendix

The monopoly-pricing environment with c(middot) = v is covered by Condition 1 with λ = 0

and γ = v The distribution G in (3) then reduces to that identified by Roesler and Szentes(2017)

If the prior micro has binary support then the cost function c(v) is linear In this special casewe can solve for πus

s (Γ) explicitly

Corollary 1 Suppose micro has binary support V = v1 v2 Let λ = c(v2)minusc(v1)v2minusv1

and let p be theunique solution to

(pminus c(p))1

λminus1 (pminus E[c(v)]) = (v2 minus c(v2))λ

λminus1 (6)

Then πuss (Γ) = max p v2minus E[c(v)]

References

AKERLOF G A (1970) ldquoThe Market for rdquoLemonsrdquo Quality Uncertainty and the MarketMechanismrdquo Quarterly Journal of Economics 84 488ndash500

BAGWELL K AND M H RIORDAN (1991) ldquoHigh and Declining Prices Signal ProductQualityrdquo American Economic Review 81 224ndash239

BAR-ISAAC H I JEWITT AND C LEAVER (2018) ldquoAdverse Selection Efficiency and theStructure of Informationrdquo Unpublished

22

BERGEMANN D B BROOKS AND S MORRIS (2015) ldquoThe Limits of Price Discrimina-tionrdquo American Economic Review 105 921ndash57

BERGEMANN D AND S MORRIS (2016) ldquoBayes Correlated Equilibrium and the Compar-ison of Information Structures in Gamesrdquo Theoretical Economics 11 487ndash522

BLACKWELL D (1953) ldquoEquivalent Comparisons of Experimentsrdquo Annals of MathematicalStatistics 24 265ndash272

CHO I-K AND D KREPS (1987) ldquoSignaling Games and Stable Equilibriardquo Quarterly Jour-nal of Economics 102 179ndash221

DU S (2018) ldquoRobust Mechanisms Under Common Valuationrdquo Econometrica 86 1569ndash1588

DURRETT R (1995) Probability Theory and Examples Duxbury Press 2nd ed

EINAV L AND A FINKELSTEIN (2011) ldquoSelection in Insurance Markets Theory and Em-pirics in Picturesrdquo Journal of Economic Perspectives 25 115ndash38

FUDENBERG D AND J TIROLE (1991) ldquoPerfect Bayesian Equilibrium and Sequential Equi-libriumrdquo Journal of Economic Theory 53 236ndash260

GARCIA D R TEPER AND M TSUR (2018) ldquoInformation Design in Insurance MarketsSelling Peaches in a Market for Lemonsrdquo Unpublished

GREENWALD B C (1986) ldquoAdverse Selection in the Labour Marketrdquo Review of EconomicStudies 53 325ndash347

JOVANOVIC B (1982) ldquoFavorable Selection with Asymmetric Informationrdquo QuarterlyJournal of Economics 97 535ndash539

KESSLER A S (2001) ldquoRevisiting the Lemons Marketrdquo International Economic Review 4225ndash41

KREPS D AND R WILSON (1982) ldquoSequential Equilibriardquo Econometrica 50 863ndash894

LEHMANN E L (1988) ldquoComparing Location Experimentsrdquo The Annals of Statistics 16521ndash533

LEVIN J (2001) ldquoInformation and the Market for Lemonsrdquo RAND Journal of Economics32 657ndash666

23

LIBGOBER J AND X MU (2019) ldquoInformational Robustness in Intertemporal PricingrdquoWorking paper

MAKRIS M AND L RENOU (2018) ldquoInformation Design in Multi-Stage Gamesrdquo Workingpaper

MILGROM P R AND R J WEBER (1985) ldquoDistributional Strategies for Games with In-complete Informationrdquo Mathematics of Operations Research 10 619ndash632

MUSSA M AND S ROSEN (1978) ldquoMonopoly and Product Qualityrdquo Journal of EconomicTheory 18 301ndash317

RAVID D A-K ROESLER AND B SZENTES (2019) ldquoLearning Before Trading On theInefficiency of Ignoring Free Informationrdquo Working paper

ROESLER A-K AND B SZENTES (2017) ldquoBuyer-Optimal Learning and Monopoly Pric-ingrdquo American Economic Review 107 2072ndash80

TERSTIEGE S AND C WASSER (2019) ldquoBuyer-optimal Extensionproof InformationrdquoWorking paper

Appendices omitted from this version but are available on request

24

That is D(micro) contains all distributions that are mean-preserving-contractions (MPC) of microIt is well known that D(micro) characterizes the set of distributions of Buyer posterior meansthat can be generated by any (uninformed-Seller) information structure

Definition 2 G is an incentive compatible distribution (ICD) if

983133

sgep

(pminus c(s))G(ds) = constant ge 0 forallp isin Supp(G)

That is when Buyerrsquos (mean) valuation is distributed according to an ICD G Seller isindifferent among all prices in the support of G assuming Buyer buys when indifferent Wefocus on a special family of ICDs whose supports are intervals [vlowast v

lowast] Given Condition 1such ICDs have analytical an expression

G(v) =

983099983105983105983103

983105983105983101

1minus983061

(1minus λ)v minus γ

(1minus λ)vlowast minus γ

983062 1λminus1

if λ ∕= 1

1minus evminusvlowast

γ if λ = 1

(3)

Given any vlowast and the above G(v) an upper bound vlowast is pinned down by the conditionEG[v] = EF [v] So a family of ICDs is parametrized by a single parameter vlowast and hence anICD is denoted by CDF 983144Gvlowast(v) with density 983144gvlowast(v) It is not difficult to verify that 983144Gvlowast(v) ismonotonic in vlowast for v in the support Consequently all ICDs are ordered according to theMPC order For vlowast isin [vE[v]] when vlowast increases the ICD increases in the MPC order

Proposition 2 Suppose Condition 1 is satisfied and let plowast equiv min983153vlowast983055983055 983144Gvlowast isin D(micro)

983154 It holds that

πuss (Γ) = plowast minus Emicro[c(v)]

Proposition 2 states that an uninformed Sellerrsquos minimum payoff (with price-independentbeliefs) is characterized by the ICDs that are MPCs of the prior distribution micro By defini-tion of ICDs Sellerrsquos profit when facing ICD 983144Gvlowast is vlowast minus E[v] Sellerrsquos minimum payoff isimplemented by the most dispersed ICD that is a MPC of the prior distribution

In proving Proposition 2 the key step is to show that given the prior micro garbling Buyerrsquosinformation such that the posterior mean distribution is an ICD makes Seller weakly worseoff It is then without loss to only consider ICDs to implement πus

s (Γ) We can thus workwith a tractable one-dimensional problem Consider the set D(micro) Suppose we find G(v)

such that (i) G isin D(micro) (ii) G is an ICD and (iii) existp such that983125 p

vG(s)ds =

983125 p

vF (s)ds Such

a G is the most-dispersed ICD that is a mean preserving contraction of F Consider the

21

following two manipulations

λ

983133 v

p

F (s)ds =

983133 v

p

(pminus c(s))dF (s)minus (1minus F (v)) (pminus c(v)) + λ(v minus p) (4)

λ

983133 v

p

G(s)ds =

983133 v

p

(pminus c(s))dG(s)minus (1minusG(v)) (pminus c(v)) + λ(v minus p) (5)

First by property (iii) above the LHS of Equation 4 equals the LHS of Equation 5 Secondsince

983125 v

p(F (s) minus G(s))ds is minimized at v this implies F (p) le G(p) Therefore the first

term on RHS of Equation 4 must be larger than that of Equation 5 Notice that the firstterm on RHS is Sellerrsquos profit when offering price p This implies that the profit fromoffering p given Buyer mean-valuation distribution G(v) is lower than that given F (v)which is lower than Sellerrsquos maximum profit given F (v) On the other hand by the ICDproperty p is already Sellerrsquos optimal price given G(v) The optimal profit given valuationdistribution F must be no lower than that of G Therefore we have proved that to minimizeSellerrsquos profit it is without loss to consider only ICDs within the set D(micro) which is a onedimensional subspace The detailed proof that such ICDs exist is relegated to the appendix

The monopoly-pricing environment with c(middot) = v is covered by Condition 1 with λ = 0

and γ = v The distribution G in (3) then reduces to that identified by Roesler and Szentes(2017)

If the prior micro has binary support then the cost function c(v) is linear In this special casewe can solve for πus

s (Γ) explicitly

Corollary 1 Suppose micro has binary support V = v1 v2 Let λ = c(v2)minusc(v1)v2minusv1

and let p be theunique solution to

(pminus c(p))1

λminus1 (pminus E[c(v)]) = (v2 minus c(v2))λ

λminus1 (6)

Then πuss (Γ) = max p v2minus E[c(v)]

References

AKERLOF G A (1970) ldquoThe Market for rdquoLemonsrdquo Quality Uncertainty and the MarketMechanismrdquo Quarterly Journal of Economics 84 488ndash500

BAGWELL K AND M H RIORDAN (1991) ldquoHigh and Declining Prices Signal ProductQualityrdquo American Economic Review 81 224ndash239

BAR-ISAAC H I JEWITT AND C LEAVER (2018) ldquoAdverse Selection Efficiency and theStructure of Informationrdquo Unpublished

22

BERGEMANN D B BROOKS AND S MORRIS (2015) ldquoThe Limits of Price Discrimina-tionrdquo American Economic Review 105 921ndash57

BERGEMANN D AND S MORRIS (2016) ldquoBayes Correlated Equilibrium and the Compar-ison of Information Structures in Gamesrdquo Theoretical Economics 11 487ndash522

BLACKWELL D (1953) ldquoEquivalent Comparisons of Experimentsrdquo Annals of MathematicalStatistics 24 265ndash272

CHO I-K AND D KREPS (1987) ldquoSignaling Games and Stable Equilibriardquo Quarterly Jour-nal of Economics 102 179ndash221

DU S (2018) ldquoRobust Mechanisms Under Common Valuationrdquo Econometrica 86 1569ndash1588

DURRETT R (1995) Probability Theory and Examples Duxbury Press 2nd ed

EINAV L AND A FINKELSTEIN (2011) ldquoSelection in Insurance Markets Theory and Em-pirics in Picturesrdquo Journal of Economic Perspectives 25 115ndash38

FUDENBERG D AND J TIROLE (1991) ldquoPerfect Bayesian Equilibrium and Sequential Equi-libriumrdquo Journal of Economic Theory 53 236ndash260

GARCIA D R TEPER AND M TSUR (2018) ldquoInformation Design in Insurance MarketsSelling Peaches in a Market for Lemonsrdquo Unpublished

GREENWALD B C (1986) ldquoAdverse Selection in the Labour Marketrdquo Review of EconomicStudies 53 325ndash347

JOVANOVIC B (1982) ldquoFavorable Selection with Asymmetric Informationrdquo QuarterlyJournal of Economics 97 535ndash539

KESSLER A S (2001) ldquoRevisiting the Lemons Marketrdquo International Economic Review 4225ndash41

KREPS D AND R WILSON (1982) ldquoSequential Equilibriardquo Econometrica 50 863ndash894

LEHMANN E L (1988) ldquoComparing Location Experimentsrdquo The Annals of Statistics 16521ndash533

LEVIN J (2001) ldquoInformation and the Market for Lemonsrdquo RAND Journal of Economics32 657ndash666

23

LIBGOBER J AND X MU (2019) ldquoInformational Robustness in Intertemporal PricingrdquoWorking paper

MAKRIS M AND L RENOU (2018) ldquoInformation Design in Multi-Stage Gamesrdquo Workingpaper

MILGROM P R AND R J WEBER (1985) ldquoDistributional Strategies for Games with In-complete Informationrdquo Mathematics of Operations Research 10 619ndash632

MUSSA M AND S ROSEN (1978) ldquoMonopoly and Product Qualityrdquo Journal of EconomicTheory 18 301ndash317

RAVID D A-K ROESLER AND B SZENTES (2019) ldquoLearning Before Trading On theInefficiency of Ignoring Free Informationrdquo Working paper

ROESLER A-K AND B SZENTES (2017) ldquoBuyer-Optimal Learning and Monopoly Pric-ingrdquo American Economic Review 107 2072ndash80

TERSTIEGE S AND C WASSER (2019) ldquoBuyer-optimal Extensionproof InformationrdquoWorking paper

Appendices omitted from this version but are available on request

24

following two manipulations

λ

983133 v

p

F (s)ds =

983133 v

p

(pminus c(s))dF (s)minus (1minus F (v)) (pminus c(v)) + λ(v minus p) (4)

λ

983133 v

p

G(s)ds =

983133 v

p

(pminus c(s))dG(s)minus (1minusG(v)) (pminus c(v)) + λ(v minus p) (5)

First by property (iii) above the LHS of Equation 4 equals the LHS of Equation 5 Secondsince

983125 v

p(F (s) minus G(s))ds is minimized at v this implies F (p) le G(p) Therefore the first

term on RHS of Equation 4 must be larger than that of Equation 5 Notice that the firstterm on RHS is Sellerrsquos profit when offering price p This implies that the profit fromoffering p given Buyer mean-valuation distribution G(v) is lower than that given F (v)which is lower than Sellerrsquos maximum profit given F (v) On the other hand by the ICDproperty p is already Sellerrsquos optimal price given G(v) The optimal profit given valuationdistribution F must be no lower than that of G Therefore we have proved that to minimizeSellerrsquos profit it is without loss to consider only ICDs within the set D(micro) which is a onedimensional subspace The detailed proof that such ICDs exist is relegated to the appendix

The monopoly-pricing environment with c(middot) = v is covered by Condition 1 with λ = 0

and γ = v The distribution G in (3) then reduces to that identified by Roesler and Szentes(2017)

If the prior micro has binary support then the cost function c(v) is linear In this special casewe can solve for πus

s (Γ) explicitly

Corollary 1 Suppose micro has binary support V = v1 v2 Let λ = c(v2)minusc(v1)v2minusv1

and let p be theunique solution to

(pminus c(p))1

λminus1 (pminus E[c(v)]) = (v2 minus c(v2))λ

λminus1 (6)

Then πuss (Γ) = max p v2minus E[c(v)]

References

AKERLOF G A (1970) ldquoThe Market for rdquoLemonsrdquo Quality Uncertainty and the MarketMechanismrdquo Quarterly Journal of Economics 84 488ndash500

BAGWELL K AND M H RIORDAN (1991) ldquoHigh and Declining Prices Signal ProductQualityrdquo American Economic Review 81 224ndash239

BAR-ISAAC H I JEWITT AND C LEAVER (2018) ldquoAdverse Selection Efficiency and theStructure of Informationrdquo Unpublished

22

BERGEMANN D B BROOKS AND S MORRIS (2015) ldquoThe Limits of Price Discrimina-tionrdquo American Economic Review 105 921ndash57

BERGEMANN D AND S MORRIS (2016) ldquoBayes Correlated Equilibrium and the Compar-ison of Information Structures in Gamesrdquo Theoretical Economics 11 487ndash522

BLACKWELL D (1953) ldquoEquivalent Comparisons of Experimentsrdquo Annals of MathematicalStatistics 24 265ndash272

CHO I-K AND D KREPS (1987) ldquoSignaling Games and Stable Equilibriardquo Quarterly Jour-nal of Economics 102 179ndash221

DU S (2018) ldquoRobust Mechanisms Under Common Valuationrdquo Econometrica 86 1569ndash1588

DURRETT R (1995) Probability Theory and Examples Duxbury Press 2nd ed

EINAV L AND A FINKELSTEIN (2011) ldquoSelection in Insurance Markets Theory and Em-pirics in Picturesrdquo Journal of Economic Perspectives 25 115ndash38

FUDENBERG D AND J TIROLE (1991) ldquoPerfect Bayesian Equilibrium and Sequential Equi-libriumrdquo Journal of Economic Theory 53 236ndash260

GARCIA D R TEPER AND M TSUR (2018) ldquoInformation Design in Insurance MarketsSelling Peaches in a Market for Lemonsrdquo Unpublished

GREENWALD B C (1986) ldquoAdverse Selection in the Labour Marketrdquo Review of EconomicStudies 53 325ndash347

JOVANOVIC B (1982) ldquoFavorable Selection with Asymmetric Informationrdquo QuarterlyJournal of Economics 97 535ndash539

KESSLER A S (2001) ldquoRevisiting the Lemons Marketrdquo International Economic Review 4225ndash41

KREPS D AND R WILSON (1982) ldquoSequential Equilibriardquo Econometrica 50 863ndash894

LEHMANN E L (1988) ldquoComparing Location Experimentsrdquo The Annals of Statistics 16521ndash533

LEVIN J (2001) ldquoInformation and the Market for Lemonsrdquo RAND Journal of Economics32 657ndash666

23

LIBGOBER J AND X MU (2019) ldquoInformational Robustness in Intertemporal PricingrdquoWorking paper

MAKRIS M AND L RENOU (2018) ldquoInformation Design in Multi-Stage Gamesrdquo Workingpaper

MILGROM P R AND R J WEBER (1985) ldquoDistributional Strategies for Games with In-complete Informationrdquo Mathematics of Operations Research 10 619ndash632

MUSSA M AND S ROSEN (1978) ldquoMonopoly and Product Qualityrdquo Journal of EconomicTheory 18 301ndash317

RAVID D A-K ROESLER AND B SZENTES (2019) ldquoLearning Before Trading On theInefficiency of Ignoring Free Informationrdquo Working paper

ROESLER A-K AND B SZENTES (2017) ldquoBuyer-Optimal Learning and Monopoly Pric-ingrdquo American Economic Review 107 2072ndash80

TERSTIEGE S AND C WASSER (2019) ldquoBuyer-optimal Extensionproof InformationrdquoWorking paper

Appendices omitted from this version but are available on request

24

BERGEMANN D B BROOKS AND S MORRIS (2015) ldquoThe Limits of Price Discrimina-tionrdquo American Economic Review 105 921ndash57

BERGEMANN D AND S MORRIS (2016) ldquoBayes Correlated Equilibrium and the Compar-ison of Information Structures in Gamesrdquo Theoretical Economics 11 487ndash522

BLACKWELL D (1953) ldquoEquivalent Comparisons of Experimentsrdquo Annals of MathematicalStatistics 24 265ndash272

CHO I-K AND D KREPS (1987) ldquoSignaling Games and Stable Equilibriardquo Quarterly Jour-nal of Economics 102 179ndash221

DU S (2018) ldquoRobust Mechanisms Under Common Valuationrdquo Econometrica 86 1569ndash1588

DURRETT R (1995) Probability Theory and Examples Duxbury Press 2nd ed

EINAV L AND A FINKELSTEIN (2011) ldquoSelection in Insurance Markets Theory and Em-pirics in Picturesrdquo Journal of Economic Perspectives 25 115ndash38

FUDENBERG D AND J TIROLE (1991) ldquoPerfect Bayesian Equilibrium and Sequential Equi-libriumrdquo Journal of Economic Theory 53 236ndash260

GARCIA D R TEPER AND M TSUR (2018) ldquoInformation Design in Insurance MarketsSelling Peaches in a Market for Lemonsrdquo Unpublished

GREENWALD B C (1986) ldquoAdverse Selection in the Labour Marketrdquo Review of EconomicStudies 53 325ndash347

JOVANOVIC B (1982) ldquoFavorable Selection with Asymmetric Informationrdquo QuarterlyJournal of Economics 97 535ndash539

KESSLER A S (2001) ldquoRevisiting the Lemons Marketrdquo International Economic Review 4225ndash41

KREPS D AND R WILSON (1982) ldquoSequential Equilibriardquo Econometrica 50 863ndash894

LEHMANN E L (1988) ldquoComparing Location Experimentsrdquo The Annals of Statistics 16521ndash533

LEVIN J (2001) ldquoInformation and the Market for Lemonsrdquo RAND Journal of Economics32 657ndash666

23

LIBGOBER J AND X MU (2019) ldquoInformational Robustness in Intertemporal PricingrdquoWorking paper

MAKRIS M AND L RENOU (2018) ldquoInformation Design in Multi-Stage Gamesrdquo Workingpaper

MILGROM P R AND R J WEBER (1985) ldquoDistributional Strategies for Games with In-complete Informationrdquo Mathematics of Operations Research 10 619ndash632

MUSSA M AND S ROSEN (1978) ldquoMonopoly and Product Qualityrdquo Journal of EconomicTheory 18 301ndash317

RAVID D A-K ROESLER AND B SZENTES (2019) ldquoLearning Before Trading On theInefficiency of Ignoring Free Informationrdquo Working paper

ROESLER A-K AND B SZENTES (2017) ldquoBuyer-Optimal Learning and Monopoly Pric-ingrdquo American Economic Review 107 2072ndash80

TERSTIEGE S AND C WASSER (2019) ldquoBuyer-optimal Extensionproof InformationrdquoWorking paper

Appendices omitted from this version but are available on request

24

LIBGOBER J AND X MU (2019) ldquoInformational Robustness in Intertemporal PricingrdquoWorking paper

MAKRIS M AND L RENOU (2018) ldquoInformation Design in Multi-Stage Gamesrdquo Workingpaper

MILGROM P R AND R J WEBER (1985) ldquoDistributional Strategies for Games with In-complete Informationrdquo Mathematics of Operations Research 10 619ndash632

MUSSA M AND S ROSEN (1978) ldquoMonopoly and Product Qualityrdquo Journal of EconomicTheory 18 301ndash317

RAVID D A-K ROESLER AND B SZENTES (2019) ldquoLearning Before Trading On theInefficiency of Ignoring Free Informationrdquo Working paper

ROESLER A-K AND B SZENTES (2017) ldquoBuyer-Optimal Learning and Monopoly Pric-ingrdquo American Economic Review 107 2072ndash80

TERSTIEGE S AND C WASSER (2019) ldquoBuyer-optimal Extensionproof InformationrdquoWorking paper

Appendices omitted from this version but are available on request

24