Competitive Information Disclosure in Search Markets

⇤

Simon Board

†and Jay Lu

‡

Current Version: July 14, 2017

First Version: May 2015

Abstract

Buyers often search across sellers to learn which product best fits their needs. We study how

sellers manage these search incentives through their disclosure strategies (e.g. product trials,

reviews and recommendations), and ask how competition affects information provision. If sellers

can observe the beliefs of buyers or can coordinate their strategies, then there is an equilibrium

in which sellers provide the “monopoly level” of information. In contrast, if buyers’ beliefs are

private, then there is an equilibrium in which sellers provide full information as search costs

vanish. Anonymity and coordination thus play important roles in understanding how advice

markets work.

1 Introduction

Starting with Stigler (1961), there has been a large literature examining the incentives of buyers tosearch for better prices. This paper studies the incentives of buyers to search for better information,and asks how sellers manage these incentives through their disclosure strategies. We consider asetting in which all sellers offer the same products and choose what information to disclose. Buyersthen engage in costly random search across sellers, trying to learn which product they prefer.

There are many environments that share these features. When shopping online for a TV, manywebsites provide product information and customer reviews, and also provide links to Amazon, earn-ing a commission for each sale.1 The websites therefore wish to steer customers towards expensive

⇤ We thank the editor and three anonymous referees for many excellent comments. We are grateful to John Asker,Alessandro Bonatti, Emir Kamenica, Moritz Meyer-ter-Vehn, Marek Pycia and Asher Wolinsky for comments, aswell as seminar audiences at Berkeley, ESSET (Gerzensee), SED (Warsaw), SITE, UNSW, UT Austin, UT Sydney,U Queensland, Munich, Bonn, World Congress (Montreal), Harvard-MIT, Microsoft, Duke-UNC, UCSD, Chicago,NEGT (USC), UCR, ASU, Miami, Columbia/Duke/MIT/Northwestern IO Conference and WUSTL. Keywords:Information Disclosure, Advertising, Learning, Search. JEL: D40, D83, L13.

† Department of Economics, UCLA. http://www.econ.ucla.edu/sboard/‡ Department of Economics, UCLA. http://www.econ.ucla.edu/jay/1 E.g. techradar.com, 4k.com, cnet.com and rtings.com.

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TVs, but must also provide enough information to prevent customers going elsewhere. Similarly,when a customer is looking to invest his savings, a financial adviser gathers information about hisfinancial needs and provides advice, but may stop soliciting information if the customer expressesa preference for high-fee mutual funds.2 And, when a patient is looking into an elective medicalprocedure, a doctor can order tests that both inform the patient and potentially guide him towardsmore lucrative procedures. In all these settings, a buyer can receive advice from many potentialsellers, and one would hope that competition enables buyers to become fully informed. Yet, this isoften not the case. For example, Clemens and Gottlieb (2014) find that small changes in Medicarefees lead to large changes in patients’ decisions, implying that patients’ information depends on theincentives of their doctors.

In this paper, we study the role of competition in information provision and show that itseffectiveness depends on the information structure of the market. We first show that competitionis ineffectual if sellers can perfectly observe buyers’ beliefs or perfectly correlate their disclosuresvia a coordination device. In either case, there is an equilibrium in which all sellers choose themonopoly disclosure strategy, manipulating buyers to purchase the most profitable, rather than themost suitable product. However, if sellers cannot fine-tune their disclosure strategies, then there isan equilibrium in which they fully reveal all the information as search costs vanish. Intuitively, abuyer can visit many sellers and accumulate more and more information, forcing any given seller tomatch the market and provide full information.

In the model, there are a large number of sellers (“she”) who sell the same set of products. Alarge number of prospective buyers (“he”) approach sellers in order to learn which product best suitstheir needs. Each seller chooses how much information to disclose by releasing a signal that mayincrease or decrease a buyer’s assessment of the product, as in Kamenica and Gentzkow (2011).3

After receiving the signal and updating his belief, the buyer chooses whether to buy a product, exit,or pay a search cost and randomly sample another seller. Different products yield different profitsfor a seller, so she tries to steer the buyer towards products that are more profitable, taking intoaccount that the buyer can always move on to a competitor.

We study how the disclosure by sellers depends on their information, as captured by noiseparameters (↵,�). When a seller is approached by a buyer, she observes a signal equal to thebuyer’s belief with noise scaled by ↵. She also has access to a coordination device with probability1� � or a noisy permutation with probability �. The ↵-parameter captures the degree to which aseller can condition her disclosure on the buyer’s belief. For example, this represents the quality ofpast browsing data on a customer’s cookies, or the information a financial adviser can deduce from

2 Mullainathan, Noeth, and Schoar (2012) document that when a customer holds a high-fee “returns-chasingportfolio, advisers were significantly more supportive than for either the company stock or the index portfolio”.

3 More formally, a seller can choose any distribution of posteriors that average to the prior. Kamenica andGentzkow show that a monopolist’s optimal solution corresponds to the concavification of her profit function.

2

an investor’s portfolio. The �-parameter captures the degree to which a seller can condition herdisclosure on a coordination device. For example, websites recommending TVs may use commonmarketing material provided by Amazon or third-party rating agencies (e.g. Energy Star), or maycollect their own independent customer reviews.

In Section 3, we show that if sellers can either perfectly observe beliefs (↵ = 0) or perfectlycoordinate their disclosures (� = 0), then the monopoly disclosure strategy is always an equilibrium.In either case, if all sellers use the monopoly strategy, then no buyer receives additional informationfrom searching a second time. The buyer thus has no incentive to continue searching, and since thestrategy maximizes profits, no seller will defect. This monopoly strategy may be very undesirablefor buyers: if there is a single product, then it provides the buyer with no valuable information.

When sellers can observe buyers’ beliefs (↵ = 0), this monopoly equilibrium is also unique ina wide-range of environments. For example, if there is a single product, the search cost impliesthat any one seller can provide a little less information than the market, iteratively pushing theequilibrium towards the monopoly outcome. The crucial condition is that, given any non-monopolystrategy, there are a series of local deviations that lead to monopoly. More precisely, we show thatthe monopoly strategy is the unique equilibrium if every non-maximal belief (i.e. where profit differsfrom the maximum over all possible states) is improvable (i.e. a monopolist wishes to release someinformation). In addition to the single-product example, this also covers cases when there are twohorizontally differentiated products or many “niche” products.

In Section 4, we show that if sellers observe a noisy signal of buyers’ beliefs (↵ > 0), thenthere exists a sequence of equilibria that converges to full-information as search costs vanish (i.e.full-information is a limit equilibrium). In equilibrium, buyers purchase from the first seller theyvisit but have the option to continue searching and mimic a new buyer. If all other sellers providesome information and search costs are small, then a buyer can threaten to visit many of them andaccumulate a large amount of information. This forces the current seller to provide (almost) fullinformation. Anonymity is therefore a powerful force in enabling buyers to become well-informed.

We then study the uniqueness of equilibria when ↵ > 0, focusing in equilibria where buyerspurchase from the first seller.4 If all sellers have access to a perfect coordination device (� = 0),then both monopoly and full-information are limit equilibria. However, if coordination is noisy(� > 0), full-information is the unique limit equilibrium if every conditional belief (i.e. the marginalof the prior on a subset of states) is fully improvable (i.e. a monopolist prefers full information tono information). Table 1 summarizes our existence results. In the case of our Example 1 (whichfeatures a single product and two states), these equilibria are unique if the buyer is initially skeptical,preferring not to buy in the absence of information.

4 Since all sellers sell identical products, it is natural to look at such equilibria. We discuss this assumption inSection 2.

3

Table 1: Summary of Limit Equilibria

Belief Observability Noise

↵ = 0 ↵ > 0

Coo

rdin

atio

nN

oise

� = 0

Monopoly

Monopoly &

Full Information

� > 0 Full Information

This table describes our existence results for limit equilibria, given the two noise parameters (↵,�). In the case ofExample 1, these equilibria are unique if the buyer prefers not to buy in the absence of information.

Comparing these cases, the key is whether sellers can choose strategies that discriminate be-tween new and old buyers. When sellers can observe buyers’ beliefs, it is optimal for them to provideless information to old buyers who are already informed, undercutting the incentive to search andundermining competition. Similarly, when sellers can coordinate, they can use a disclosure strategythat is useful for a new buyer but uninformative for an old one. In contrast, when sellers cannotdiscriminate, the option of going to a competitor forces sellers to provide all the pertinent infor-mation as search costs vanish. This finding is important for applications. Our results provide atheoretical foundation for the notion that tracking consumers can make them “exploitable”,5 andsuggest that common marketing material or industry training courses can help financial advisersimplicitly collude to the detriment of investors.

Finally, in Section 5, we explore several extensions. First, we argue that our results are robustto heterogenous buyers. Second, we consider a case in which sellers can observe buyer shoppinghistory but not beliefs, and show that since sellers can still discriminate between new and old buyers,equilibria are monopolistic. Third, we argue that buyers create a positive externality when theybecome anonymous (e.g. by deleting cookies), thus providing a rationale for regulating trackingprograms.

1.1 Literature

The paper contributes to the literature on search by allowing sellers to choose information disclosurestrategies. In the benchmark model where sellers choose prices, Diamond (1971) shows that allsellers charge the monopoly price. Intuitively, the search cost allows any one seller to raise her priceslightly above the prices set by others without losing customers.6 We show that when sellers choose

5 For example, see “Facebook Tries to Explain Its Privacy Settings but Advertising Still Rules,” New York Times

(November 13, 2014) or “Google to Pay $17 Million to Settle Privacy Case,” New York Times (November, 18 2013).6 This result continues to hold if sellers make multiple offers to each buyer (Board and Pycia (2014)). However,

it can break down if sellers sell heterogeneous products (Wolinsky (1986)), buyers have multi-unit demand (Zhou

4

disclosure strategies, the analysis depends on the information structure: when sellers can perfectlyobserve buyers’ beliefs, the logic is analogous to Diamond; however, when their observations areimperfect, competition induces sellers to fully reveal the information. In contrast to the Diamondmodel, when a seller provides information to a buyer, this changes his beliefs and changes how heacts at subsequent sellers.

The paper also contributes to a growing literature on information disclosure with competitionbased on the Kamenica-Gentzkow framework.7 Gentzkow and Kamenica (2017b) consider a generalmodel in which multiple firms simultaneously release signals and provide conditions under whichcompeting firms release more information than colluding firms. Gentzkow and Kamenica (2017a)study senders who release “coordinated” signals, characterize the equilibrium and show that com-petition increases information. Li and Norman (2017) show that the result may fail if sellers movesequentially, use independent signals or mixed strategies. Hoffmann, Inderst, and Ottaviani (2014)suppose heterogeneous sellers simultaneously release information to win over a customer and showthat competition increases information disclosure. In all these models, a buyer receives the marketinformation and then chooses his action; in our model, the buyer must pay a search cost in orderto acquire more information.8,9

We study sellers who choose disclosure strategies and obtain payoffs that depend on buyers’actions. For example, a website receives a commission if a buyer purchases a TV from Amazon, afinancial adviser receives a fee if she sells a mutual fund, and a doctor is reimbursed for proceduresby Medicare. In contrast, Anderson and Renault (2006) study a monopolist selling a single goodwho can choose both information and prices.10 They show that the seller should reveal whetherthe buyer’s value exceeds the cost and charge a price equal to the buyer’s expected value; thisimplements the efficient allocation and fully extracts from the buyer. More generally (e.g. if thereare two goods) the monopoly solution will be inefficient, and the forces we study will impact theeffectiveness of search in enhancing efficiency.

Finally, there is a large literature that studies costly acquisition of information. For example,Wald (1947) and Moscarini and Smith (2001) consider a decision maker who can purchase inde-pendent signals at a constant marginal cost. In contrast, the information provided to our buyer is

(2014), Rhodes (2015)), or buyers pre-commit to a number of searches (Burdett and Judd (1983)).7 See also Aumann and Maschler (1995) and Rayo and Segal (2010).8 There are other related papers. Forand (2013) studies a model of information revelation with directed search.

Au (2015) and Ely (2017) study dynamic information disclosure by a monopolistic seller.9 In addition, there are a variety of papers concerning the revelation of information when the senders are informed.

One literature considers verifiable information (“persuasion games”). Here, Milgrom and Roberts (1986) find condi-tions under which competition leads to full revelation, and Bhattacharya and Mukherjee (2013) consider a model withmultiple senders. A second literature supposes information is unverifiable (“cheap talk”). Here, Crawford and Sobel(1982) characterize the structure of communication, and Battaglini (2002) shows how multiple senders can increasethe information provided when senders have opposed preferences. Also related, Inderst and Ottaviani (2012) considera model with a single seller and study the choice of commission rates chosen by upstream sellers.

10 See also Lewis and Sappington (1994) and Johnson and Myatt (2006).

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chosen endogenously by strategic sellers.

2 Model Setup

Overview. There is a unit mass of sellers offering the same set of products. Time is discrete,with a unit mass of buyers entering each period; each buyer searches for one product. When abuyer approaches a seller, the seller observes a signal about the buyer’s belief and decides how muchinformation to disclose. The buyer then chooses whether to buy, exit or continue searching. Ourresults will depend on two parameters: ↵, which determines the accuracy of the sellers’ signals, and�, which determines the sellers’ ability to coordinate their information disclosures.

Information. A buyer is initially uncertain about a finite set of payoff states S that are relevantfor his decision. There is also an independent coordination device uniformly distributed on [0, 1]

that sellers can use to coordinate their disclosures.11 Let ⌦ := S ⇥ [0, 1] be the overall state space,and �⌦ be the set of all possible beliefs for the buyer. Buyers start with an initial belief p0 2 �⌦,where the payoff belief p0 2 �S is the marginal of p0 on S and puts positive probability on everys 2 S. Denote a generic prior by p and a generic payoff prior by p. When there are only new buyersin the market, we will use p0 and p synonymously.

Buyers are randomly matched with sellers. When a buyer approaches a seller, the seller will havea prior ⌘0 about the buyer’s belief.12 The seller then observes a private signal about the buyer’sbelief, y = (1� ↵)p + ↵⇠, where the random variable ⇠ 2 R⌦ has full support. The parameter↵ 2 [0, 1] captures the noise of the signal, so ↵ = 0 corresponds to the case where sellers can perfectlyobserve buyers’ beliefs. Let Y be the space of private signals and ⌘y be her inference about thebuyer’s belief after receiving the signal y 2 Y .

Sellers’ strategy. A seller chooses how much information to disclose to a buyer based on theprivate signal y. With probability 1� �, the seller chooses a general signal structure �g

y : ⌦ ! �⇥

over a signal space ⇥ that allows her to coordinate with other sellers. With probability �, the sellerdoes not have access to the coordination device and can only send an independent signal structure

�iy : S ! �⇥. We denote the collection of signal structures by �g

= (�gy)y2Y and �i

=

�

�iy

�

y2Y ,and call them disclosure policies. A seller’s strategy is then � =

�

�g,�i�

.After receiving a signal ✓ 2 ⇥ from the seller, a buyer updates his belief to form posterior

q 2 �⌦. Given a buyer with belief p, a signal structure �gy induces a posterior distribution K

�gy

p

for the buyer such that the average posterior coincides with the prior,R

�⌦ q K�gy

p (dq) = p. Givena seller strategy � and a signal realization y, a buyer with belief p faces a distribution of posteriors

11 This formulation draws on Green and Stokey (1978) and Gentzkow and Kamenica (2017a).12 In equilibrium, the seller’s prior ⌘0 will be consistent with the strategies of other sellers in the market.

6

K�yp := (1� �)K

�gy

p +�K�iy

p . When sellers observe buyers’ beliefs (↵ = 0), we can drop the subscripty = p on the strategy and use the condensed notation K�

p .

Buyers’ decisions. After a buyer updates his belief to form a posterior q 2 �⌦, he chooseswhether to (i) buy a product from the seller, (ii) exit the market or (iii) pay a search cost c > 0

and pick a new seller at random. Each seller offers the same set of products {0, 1, . . . , I}. Producti � 1 delivers utility ui (s) to the buyer in state s 2 S and profit ⇡i > 0 to the seller.13 We assumethat different products yield different profits, that is, ⇡i 6= ⇡j for all i 6= j. Product 0 is the exitoption normalized so that u0 is the zero vector and ⇡0 = 0.

If a buyer with belief q 2 �⌦ decides to stop shopping, then let i⇤ (q) = argmaxiq · ui denotehis optimal product.14 We can define the indirect utility vector of the buyer by u (q) := ui⇤(q), andthe profit of the seller by ⇡ (q) := ⇡i⇤(q). If a buyer decides to continue shopping, then he pays thesearch cost and arrives at a new seller. The game then proceeds as above.

Value functions. Sellers choose their strategies to maximize profit. All sellers have identicaloptimization problems with the buyer’s belief p being the only relevant state variable. We look foran equilibrium in symmetric Markov strategies.15 If all other sellers choose a strategy �, then thecontinuation value of a buyer with belief q is

Vc (�,q) =� c+ Eq

Z

�⌦max {r · u (r) , Vc (�, r)}K�y

p (dr)

�

where r · u (r) is the buyer’s expected utility when stopping and the expectation Eq is taken overall realizations of y. A buyer purchases or exits when his belief falls in a stopping set given by

Qc (�) = {q 2 �⌦ | q · u (q) � Vc (�,q)}

We will also let Qc (�) ⇢ �S denote the corresponding stopping payoff beliefs.

Best-responses. Suppose other sellers in the market use strategy �. The current seller begins withbelief ⌘0 about the buyer’s prior p, and after receiving the realization y 2 Y , updates her belief toform a posterior ⌘y. She then chooses a strategy � that solves

max

�

Z

�⌦

Z

Qc(�)⇡ (q)K

�yp (dq) ⌘y (dp)

Geometrically, given p, the seller’s optimal profits are given by the concavification of ⇡ over beliefsin the stopping set Qc (�). If � solves this problem, then we say it is optimal given �.

13 As in Rayo and Segal (2010), a seller’s profit is pinned down by the product chosen by the buyer.14 In equilibrium, ties are resolved in the seller’s favor. Similarly, a buyer purchases when indifferent between

stopping and continuing.15 That is, the strategy depends neither on calendar time nor the identity of the seller. This assumption makes the

analysis simpler, but is not required for our motivating example (see footnote 20).

7

We will usually be interested in the case where sellers have degenerate beliefs about the distri-bution of buyers they face. This is either because they can observe buyers’ beliefs or because theybelieve all buyers are new. Letting � be the Dirac measure, then:

Lemma 1. If a seller knows a buyer’s belief, in that ⌘y = �p for all y, then it is optimal for her to

sell with probability one.

Proof. See Appendix A.1.

Intuitively, suppose a seller sends a buyer to a belief outside his stopping set with positiveprobability. Since she has access to arbitrary signals, she can increase her profits by disclosing fullinformation to this buyer whenever he would have waited, while holding constant the informationshe sends in all other situations.

If ↵ = 0, a seller observes buyers’ beliefs, meaning that ⌘y = �p. Lemma 1 then implies thatshe sells with probability one. If ↵ > 0, a seller observes a noisy signal of the true belief. If thecurrent seller believes that she only faces new buyers, then her prior is degenerate, ⌘0 = �p0 . Sincethe noise ⇠ has full support, her prior is consistent with any realization of the signal y, and Bayes’rule implies that her posterior coincides with the prior, ⌘y = �p0 , for any realization of y. Lemma1 thus implies that if the seller is correct about facing a new buyer, then she sells with probabilityone.

Equilibrium. In a symmetric Markov perfect equilibrium � of this game, (i) a seller’s strategyis optimal given that other sellers also use this strategy, and (ii) sellers’ priors ⌘0 are consistentwith buyers’ behavior. Given the discussion above, we consider a particular type of Markov perfectequilibrium in which buyers purchase from the first seller; when clear, we refer to these immediate

purchase equilibria simply as equilibria. When beliefs are observable (↵ = 0), this assumption iswithout loss. When beliefs are imperfectly observable (↵ > 0) then, on path, it is optimal for aseller to sell immediately if all other sellers sell immediately.

When ↵ > 0, it is natural to ask whether equilibria with delay also exist. In the SupplementaryAppendix S.2, we show that if there is no coordination, then selling immediately is a weak bestresponse to any Markov equilibrium strategy; moreover, under some conditions (e.g. normal noise,⇠), it is a strict best response, meaning that equilibria with delay cannot exist. Unfortunately,lack of tractability makes it hard to say more when there is partial coordination. Formally, thisrestriction does not affect Theorems 1-3, but does mean that the uniqueness result in Theorem 4only applies to immediate purchase equilibria.

The discussion highlights the critical difference between perfectly observable beliefs (↵ = 0) andimperfectly observable beliefs (↵ > 0). In the former case, sellers can see buyers beliefs and arethus correct both on- and off-path, ⌘y = �p. In the latter case, sellers think they are facing new

8

buyers, ⌘y = �p0 , which is correct on-path but not off-path if a buyer chooses to search. As we willsee, this distinction generates the disparity between the “monopoly” results in Section 3 and the“full-information” results in Section 4.

Remarks. As in Kamenica and Gentzkow (2011), we assume the seller has no private informationabout the payoff state. For example, when a website provides information about TVs, it does notknow the buyer’s preferences. When a financial adviser tries to help an investor, she learns abouthis needs and has a fiduciary obligation to give the correct advice conditional on this information.And, when a doctor orders a test for a patient, she does not know the outcome before it arrives.16

In addition, the signal structure is assumed to be observed by both parties (e.g. the number ofreviews on a website, the information acquired by the financial adviser, or the tests ordered by thedoctor).17

Sometimes we will consider equilibria without seller coordination. When sellers choose policies�g

= �i that are completely independent, we call the strategy � simple and equate the strategywith the policy. If a strategy delivers monopoly profits to the seller, then we call it a monopoly

strategy ; the set of monopoly strategies coincides with the optimal disclosure solution in Kamenicaand Gentzkow (2011). Since there is a single seller, we can ignore the coordination device, meaninga monopoly strategy is automatically a simple strategy on payoff beliefs. An equilibrium where allsellers use a monopoly strategy is called a monopoly equilibrium.

3 Monopoly Equilibria

In this section we show that the monopoly strategy is an equilibrium if sellers can perfectly observebuyers’ beliefs (↵ = 0) or can perfectly coordinate (� = 0). For the former case, we then provideconditions under which monopoly is the unique equilibrium. We first illustrate the main forcesthrough a simple single-product example.

3.1 Motivating Example

Example 1 (Single product). A seller has a single product to sell, there are two payoff states{L,H}, and the buyer wishes to buy the product if state H is more likely. For example, a consumerconsiders upgrading to a new TV, but does not know whether he will value the new technology. A

16 Gentzkow and Kamenica (2017c) provide a different interpretation: they show that the outcome of the test canbe privately observed by the seller if sellers’ messages are verifiable.

17 If the buyer observes signal structures before visiting, then he can direct his search towards the most informative,leading to Bertrand competition and full information revelation. While some companies develop reputations forrevealing information (e.g. Best and Quigley (2016)), it is generally hard to describe an entire signal structure to acustomer before they have visited (e.g. when compared to describing a price).

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0 1

Buyer

Seller

c>0

(a) Monopoly Equilibrium Policy

1

2

Qc(σ∗)

0 11

2

Figure 1: Single Product (Monopoly Equilibrium)

Buyer

Seller

c>0

(b) Non-Equilibrium Policy

b

b

π (q)

q⋅u(q)

V c(σ ,q)

b(1−c)b(1−c)

∫ r⋅u(r )K q

σ (dr)

∫ r⋅u(r )K q

σ∗

(dr )= q⋅u(q)

V c(σ∗, q)

π (q)

∫Qc(σ

∗)π (r )K q

σ∗

(dr) ∫Qc(σ )

π (r )K q

σ(dr)

0 11

2

0 11

2Qc(σ) Qc(σ)

This figure illustrates Example 1 when sellers observe buyers’ beliefs (↵ = 0). Figure 1(a) shows that the monopolystrategy is an equilibrium. The top panel shows the buyer’s payoffs: the bold line is the payoff from stopping q ·u (q),the shaded line is the expected payoff

R�S

r ·u (r)K�⇤q (dr) under the monopoly strategy �⇤, and the dotted line is the

buyer’s value function Vc (�⇤, q). The bottom panel shows the seller’s payoffs: the bold line is the profit function ⇡ (q)

and the shaded line is the expected profitRQc(�⇤) ⇡ (r)K�⇤

q (dr). Figure 1(b) shows that a more informative strategycannot be an equilibrium since a seller can raise her profits by providing a little less information, as illustrated bythe dashed line.

sale generates profits ⇡ = 1 for the seller. Payoffs to the buyer are u (L) = �1 and u (H) = 1, sothe buyer prefers to buy if p = Pr (H) is greater than 1

2 .If there were a single seller, then the monopoly strategy from Kamenica and Gentzkow (2011)

provides just enough information to persuade the buyer to buy. In other words, the monopolystrategy �⇤ is given by18

K�⇤

p =

8

<

:

�p if p 2⇥

12 , 1⇤

(1� 2p) �{0} + 2p �{ 12} if p 2

⇥

0, 12�

This is illustrated in Figure 1(a). If the buyer starts with a prior p � 12 , then the seller provides no

information and the buyer buys; if p < 12 , then the seller sends the buyer to posteriors 0 and 1

2 andthe buyer buys with probability 2p. For shorthand, we will sometimes denote this binary signal bythe condensed notation p !

�

0, 12

.18 Technically, there is a set of monopoly strategies that deliver monopoly profits to the seller. Here, �⇤ is the

“one-shot” monopoly strategy defined in Section 3.2.

10

If sellers perfectly observe buyers’ beliefs (↵ = 0), then the monopoly strategy is an equilibrium.If all sellers choose this strategy, then a buyer learns nothing from a second seller after leavingthe first seller and thus, given the search cost, buys immediately. Since the seller makes monopolyprofits, she has no incentive to deviate.

More surprising, the monopoly strategy is the the unique equilibrium of the game. To gain someintuition, fix b > 1

2 and suppose that all sellers provide a simple binary signal p ! {0, b} as illustratedin Figure 1(b). Call this strategy � and observe that it induces a stopping set Qc (�) =

h

0, b2b�1c

i

[[b (1� c) , 1] as shown in the lower panel.19 Now, consider a seller who deviates and uses the lessinformative simple strategy p ! {0, b (1� c)}. Since there is a strictly positive search cost, a buyerwho receives this signal will not subsequently search and this deviation strictly raises the seller’sprofits. As we prove below, this logic implies that whenever sellers provide more information thanthe monopoly strategy, there is a profitable deviation, analogous to the classic result of Diamond(1971).20

Similarly, if sellers can perfectly coordinate (� = 0), then the coordinated monopoly strategy isan equilibrium. Under such coordination, all sellers provide the same signal, so a buyer will purchaseafter visiting the first seller. Since this strategy yields monopoly profits, no seller has any incentiveto deviate. In contrast to the case of observed beliefs (↵ = 0), this equilibrium is not unique: if allother sellers fail to use the coordination device, then the current seller has no incentive to use iteither. 4

3.2 Monopoly is an Equilibrium

We now return to the general model, with multiple states and multiple products. We wish to showthat if sellers can perfectly observe buyers’ beliefs (↵ = 0) or perfectly coordinate (� = 0), thenmonopoly is always an equilibrium.

First, we need a preliminary result. Consider a monopolist who uses a strategy �, and letA� ⇢ �S be the set of absorbing beliefs where no information is provided, i.e. K�

p = �p for allp 2 A�. We say such a strategy is one-shot if it only takes the buyer to absorbing beliefs, that is,K�

p (A�) = 1 for all p 2 �S. This means the buyer gets no more information if he were to return

to the monopolist under �.

Lemma 2. There exists a one-shot monopoly strategy.

Proof. Kamenica and Gentzkow (2011) show that a monopolist’s profit, which we denote by ⇧

⇤,equals the concavification of the profit function ⇡. Let R be the set of beliefs where the two

19 Note that if b (1� c) 12 , then Qc (�) = [0, 1] and the seller can trivially deviate to the monopoly strategy.

20 In fact, in this example we can show an even stronger result: the monopoly strategy is the only rationalizablestrategy. If we start with an stopping set Q0 = {0}[ {1} then, no matter what other sellers do, a buyer purchases ifhis belief falls within Q1 = [0, c][ [1� c, 1]. Iterating n � � log (2) / log (1� c) rounds,

⇥12 , 1

⇤⇢ Qn and the monopoly

strategy is the only undominated strategy remaining.

11

functions coincide, that is ⇧

⇤(q) = ⇡ (q). Consider the following strategy �⇤: the seller uses any

optimal strategy if p 62 R and releases no information if p 2 R. First observe that �⇤ takes thebuyer to posteriors in R. This is because the optimal strategy at any prior p must send the buyerto beliefs where no information is optimal; otherwise, the seller should incorporate this informationin her initial signal. Second, using the definition of �⇤, beliefs in R are absorbing so �⇤ is one-shot.Since �⇤ yields monopoly profits, it is thus a one-shot monopoly strategy.

We now present our first main result: monopoly is always an equilibrium. Note that this alsoimplies the existence of an equilibrium.

Theorem 1. Suppose sellers can perfectly observe beliefs (↵ = 0) or perfectly coordinate (� = 0).

There exists a monopoly equilibrium.

Proof. First, suppose that ↵ = 0 and all sellers use a simple one-shot monopoly strategy �⇤, whichexists by Lemma 2. By definition, this strategy sends buyers to absorbing beliefs, i.e. K�⇤

p

�

A�⇤�

= 1.Since buyers receive no information at absorbing beliefs, they stop, i.e. A�⇤ ⇢ Qc (�

⇤). Hence

the seller achieves her monopoly profit and has no incentive to deviate, meaning that �⇤ is anequilibrium.

Now suppose � = 0 and let �⇤⇤ be the perfectly-coordinated strategy corresponding to �⇤. If abuyer were to return to the market, then he would receive no information, i.e. K�⇤⇤

q = �q for all qin the support of K�⇤⇤

p for any p 2 �⌦. Hence the buyer prefers to stop rather than continue, i.e.q 2 Q�

c (�⇤⇤) for all q in the support of K�⇤⇤

p for any p 2 �⌦. Since the seller cannot do betterthan achieving monopoly profits, she will not deviate, meaning that �⇤⇤ is an equilibrium.

Intuitively, if all sellers use a one-shot monopoly strategy, a buyer only receives one round ofinformation and therefore purchases from the first seller. Since all sellers make monopoly profits,they have no incentive to deviate. This logic relies on sellers’ ability to give information to newbuyers but not to old buyers, either because sellers can observe buyers’ beliefs or because they canperfectly coordinate their disclosure strategies.

3.3 Uniqueness of the Monopoly Equilibrium

When sellers can perfectly coordinate (� = 0), there is a monopoly equilibrium even if buyers’beliefs are imperfectly observed (↵ > 0). However, if all other sellers use independent signals, thenno one seller can coordinate by herself. As we show in Section 4, this means that full-informationequilibria also exist as search costs vanish.

In contrast, when buyers’ beliefs are perfectly observed (↵ = 0), the monopoly equilibrium is“often” unique. Before establishing this result, we first provide an example of a non-monopoly equi-librium. This motivates the sufficient condition for the uniqueness result. For the remainder of this

12

section, we thus assume that beliefs are perfectly observed (↵ = 0).21

Example 2 (Vertical differentiation). Suppose there are two vertically differentiated prod-ucts and two payoff states L and H representing the buyer’s taste for quality (e.g. Kamenica (2008,Section IIA)), with p = Pr (H). The first product is a cheap, low-quality TV yielding utilityu1 =

�

�13 ,

23

�

; the second is an expensive, high-quality TV, yielding utility u2 = (�1, 1). Thus, thebuyer prefers no TV if p 2

⇥

0, 13�

, the cheap TV if p 2⇥

13 ,

23

�

and the expensive TV if p 2⇥

23 , 1⇤

.Assume that the expensive TV is more profitable for the seller, i.e. ⇡1 = 1 and ⇡2 =

32 . The

monopoly strategy �⇤ is given by

K�⇤

p =

8

>

>

>

>

<

>

>

>

>

:

(1� 3p) �{0} + 3p�{ 13} if p 2

⇥

0, 13�

(2� 3p) �{ 13} + (3p� 1) �{ 2

3} if p 2⇥

13 ,

23

�

�p if p 2⇥

23 , 1⇤

By Theorem 1, this monopoly strategy is an equilibrium, as illustrated in Figure 2(a). One can in-terpret this strategy as the seller recommending the high-quality TV for high beliefs, recommendingeither the high- or low-quality TV for intermediate beliefs, and recommending either the low-qualityTV or no TV for low beliefs.

When c > 0 is small enough, there is also a second simple equilibrium � where sellers providemore information,

K�p =

8

<

:

2�3p2 �{0} +

3p2 �{ 2

3} if p 2⇥

0, 23�

�p if p 2⇥

23 , 1⇤

as illustrated in Figure 2(b). This gives rise to the stopping set Qc (�) = [0, "c][⇥

23 � "c, 1

⇤

for some"c > 0 and "c > 0. This strategy differs from the monopoly strategy by never recommending thelow-quality TV. As shown in Figure 2(b), the presence of the search cost allows the seller to providea little less information than her competitors. However, there is no local deviation that is profitable:if a seller sends the buyer to p !

�

0, 23 � "c

, then the seller’s profit would drop. There is alsono profitable global deviation: if the seller sends the buyer to p !

�

0, 13

as under the monopolystrategy, then the buyer with belief 1

3 would refuse to buy, correctly anticipating that other sellerswill provide significantly more information. 4

We now provide a sufficient condition for monopoly to be the unique equilibrium. We say abelief p 2 �⌦ is maximal if ⇡ (p) = maxs2Sp

⇡ (1s) where Sp ⇢ S is the support of the payoff belief21 When the search cost c is sufficiently high, monopoly is trivially the unique equilibrium. All of the results in

this section apply to any positive search cost, including those arbitrarily small. As one might expect, the set ofequilibria is decreasing in the search cost; indeed, we show in the Supplementary Appendix S.1 that given any simpleequilibrium for c > 0, there is an equilibrium for all c0 < c with the same profits.

13

(b) Vertical Differentiation: Non-monopoly

ɛ cɛc

(c) Horizontal Differentiation: Monopoly

Figure 2: Vertical and Horizontal Differentiation

(a) Vertical Differentiation: Monopoly

2

3

0 1

Buyer

Seller

c>0

1

3

2

3

0 11

3

2

3

0 11

3

2

3

0 11

3

2

3

0 11

3

2

3

0 11

3

Buyer

Seller

Buyer

Seller

c>0

c>0

Qc(σ∗) Qc(σ) Qc(σ

∗) Qc(σ∗)Qc(σ)

Figures 2(a) and (b) show the two equilibria that arise with vertical differentiated products. Figure 2(c) showsthe unique equilibrium that arises with horizontally differentiated products; this coincides with the monopoly strategy.For a description of the different lines, see the caption for Figure 1.

p. We say p is non-maximal if the equality fails. A belief p is improvable if a monopolist wouldlike to provide some non-trivial information, i.e. ⇧⇤

(p) > ⇡ (p) where ⇧

⇤(p) is the monopoly profit.

We first present a preliminary lemma.

Lemma 3. Suppose ↵ = 0. If any non-maximal belief is improvable, then in any equilibrium, a

buyer stops at all maximal beliefs.

Proof. See Appendix A.2.

To illustrate what it means for non-maximal beliefs to be improvable, consider Figure 3(a) wherethere are three products {1, 2, 3} with ⇡1 > ⇡2 > ⇡3. The set Hi ⇢ �S denotes the set of payoffbeliefs where the buyer favors product i. Note that 1si 2 Hi for all i 2 {1, 2, 3} so the most profitableproduct is favored in state s1, the second most profitable in state s2, and the third most profitablein state s3. The shaded dark regions represent the set of maximal beliefs. This example satisfies thecondition in Lemma 3: given any belief p that is not maximal, a monopolist would choose to releasesome information (i.e. p is improvable). In particular, the monopoly strategy can be decomposed intwo steps. First, it releases a binary news signal about s1, taking the prior p ! {q1, q23} where q1

just persuades the buyer to buy the first product and q23 is on the edge between s2 and s3. Second,the strategy releases a binary signal about s2 such that q23 ! {q2, 1s3}, where q2 just persuades the

14

(a) Non-Maximal Improvable

s1

Figure 3: Uniqueness of Monopoly Equilibrium

H1

p

q1

q2

H3

H2

s2s

3

s1

p

s2s

3

q1

H1

H2

q23

(b) Non-Maximal Improvable

Figure 3(a) shows the monopoly strategy when all non-maximal beliefs are improvable. The shadeddark regions (H1, 1s3 and the segment from q2 to 1s2) represent the set of maximal beliefs. In thiscase, maximal beliefs are in the stopping set of any equilibrium. Figure 3(b) shows that otherequilibria can exist when the sufficient condition fails. The belief p is non-maximal and isunimprovable when the second product is sufficiently profitable. For small c, there is a non-monopolyequilibrium in which all sellers “skip over” product 2 and send the buyer p ! {q1, 1s3}.

buyer to purchase product 2.22

To understand Lemma 3, suppose by contradiction that there is a maximal payoff belief in H1

that is not in the stopping set Qc (�) of some equilibrium strategy �. A buyer with a belief on theboundary of Qc (�) \H1 gets no information since he purchases the most profitable product. By acontinuity argument, we can find a belief close enough to the boundary but not in the stopping setsuch that the buyer gets very little information. Since the search cost is strictly positive, this smallamount of information is not enough to incentivize the buyer to shop, so the buyer stops at thatbelief yielding a contradiction. This shows that the buyer stops at any belief in H1, and the prooffollows by applying the same argument to all maximal beliefs.

Given this lemma, we have the following uniqueness result:

Theorem 2. Suppose ↵ = 0. Any equilibrium is monopoly if every non-maximal belief is improvable.

Proof. If other sellers use some equilibrium strategy �, the current seller can send the buyer to anybelief in the stopping set Qc (�). Suppose she uses the one-shot monopoly strategy �⇤ which existsby Lemma 2. This sends the buyer to beliefs in its absorbing set A�⇤ which are unimprovable, andthus by the premise, maximal. By Lemma 3, these beliefs must be in the stopping set Qc (�). Hencethe seller can attain her monopoly profits, as required.

22 If p is maximal, then there is some freedom about the monopoly strategy, but since the buyer’s decision is thesame (he buys the product), such differences are payoff-irrelevant.

15

One can understand the uniqueness result in terms of “local deviations” as in Example 1 or Figure3(a). Whenever the monopolist only sends the buyer to beliefs that are maximal, then no matterwhat the other sellers do, one seller can always provide a little less information and creep closer tothe monopoly outcome. This logic does not apply when a monopolist wishes to use products in themiddle of the probability space as in Example 2 or Figure 3(b).

The following corollary illustrates a wide range of cases where the sufficient condition in Theorem2 holds.

Corollary 1. Suppose ↵ = 0. All equilibria are monopoly if:

(i) There is one product.

(ii) There are two “horizontally differentiated” products in that u1 (s) � 0 implies u2 (s) 0.

(iii) Products are “niche” in that p · ui � 0 implies p · uj 0 for any j 6= i.

Proof. See Appendix A.3.

Corollary 1 provides sufficient conditions for uniqueness. Part (i) considers the case of a singleproduct, extending Example 1 to any finite number of states. Part (ii) considers two “horizontallydifferentiated” products as illustrated in Figure 2(c), where in each state, buyers only want oneproduct or the other, although for intermediate beliefs they may want both. Finally part (iii)considers a large number of “niche” products as illustrated in Figure 3(a), where buyers only wantone product at any given belief.

It is notable that the improvability condition in Theorem 2 only involves the monopoly strategy,and is therefore stricter than required. For example, consider a reversed version of Example 2 wherethe low-quality product is more profitable than the high-quality product, and p 2

�

0, 13�

. While thisdoes not satisfy the sufficient condition, monopoly is still the unique equilibrium since any seller canraise her profits by providing slightly less information than the market. More generally, one couldshow uniqueness by checking whether profits drop along each path of local deviations; this dependson how buyer-optimal products are ordered in belief space.

4 Full-Information Equilibria

In this section, we suppose that sellers imperfectly observe buyers’ beliefs (↵ > 0). Our first resultis that, as search costs vanish, there is a sequence of equilibria that converges to full-information.We then derive conditions under which all equilibria reveal full information as search costs vanish,in stark contrast to the results in the last section.

Given there are only new buyers on path, sellers have a degenerate prior ⌘0 = �p0 , ignore thenoisy signal y and thus have degenerate beliefs ⌘y = �p0 . We can therefore drop the subscript y

16

and let �g and �i denote the general and independent signal structures respectively, which we willrefer to as policies without loss of generality.

We now describe how buyers update their payoff beliefs when they receive independent signalsfrom a seller. Let r (✓) be the posterior of a buyer with prior p after observing a signal ✓ 2 ⇥, andlet rq (✓) denote the posterior of a buyer with prior q who observes the same signal. As shown byAlonso and Camara (2016), the posteriors of the two buyers are related as follows

rq (✓) = �q (r (✓))

where �q : �S ! �S is a mapping from the posteriors of buyer p to the posteriors of buyer q

satisfying Bayes’ rule,

[�q (r)] (s) =q (s) r(s)

p(s)P

s0 q (s0)

r(s0)p(s0)

. (1)

Intuitively, for any two buyers, the proportionality of their likelihood ratios for any two statesremains constant when we update via Bayes’ rule. That is, if buyer q starts off twice as optimisticas buyer p, then buyer q will remain twice as optimistic as p after any signal realization.

This section will be interested in convergence results as search costs get small. A limit equilibrium

is a sequence of equilibria �n and associated search costs cn ! 0 such that the buyer’s equilibriumpayoffs converge. We say a strategy is fully-informative if it has full support on all degenerateposteriors, and a limit equilibrium is full-information if the buyer’s equilibrium payoffs converge tothat of a fully-informative strategy.

4.1 Motivating Example

Example 1 (cont). As before there are two payoff states {L,H}, and the buyer’s initial prior isp = Pr (H). We first abstract from coordination and assume that sellers use simple strategies. Weclaim that when p < 1

2 , the only equilibrium strategy provides full information as c ! 0. Intuitively,since the monopoly strategy provides some information, as the cost vanishes, a buyer can obtain alarge number signals at low cost and become (almost) fully-informed. Hence the current seller mustprovide (almost) full information in order to beat this outside option and make a sale.

Since sellers use independent policies that do not depend on y, the stopping sets are of the formQc (�) = [0, a][ [b, 1], where a 1

2 b. This is because information has no value at the boundariesand a buyer’s value function is convex and increasing in the posterior q. As a result, the seller’soptimal strategy is a binary signal p ! {0, b} as shown in Figure 4(a). Moreover, a buyer with prior

17

(a) Non-monopoly Equilibrium

Figure 4: Single Product (Full-Information Equilibrium)

(b) Full-information Limit Equilibrium

1

2bc

0 1

Buyer

Seller

Qc(σ)

c>0 c→0

p

π (q)

∫Qc(σ )

π (r )K q

σ(dr)

q⋅u(q)

V c(σ ,q)

ac bc→1

Buyer

Seller

1

2

0 1 1

2

01

1

2

0p

p pQc(σ) Qc(σ)→←Qc(σ )

∫ r⋅u(r )K q

σ (dr)

This figure illustrates Example 1 when sellers receiver noisy signals about buyers’ beliefs (↵ > 0) anduse simple strategies. Figure 4(a) shows the unique equilibrium, which provides more informationthan a monopolist. Figure 4(b) shows that as the search cost vanishes, i.e. c ! 0, this equilibriumconverges to full-information.

b will have posteriors b ! {�b (0) ,�b (b)}. Using equation (1), we get

�b (b) =b bp

b bp+(1�b) 1�b

1�p

with probability b�b(b)

�b (0) = 0 with probability 1� b�b(b)

By the definition of the stopping set Qc (�), the buyer with belief b must be indifferent betweenstopping and searching again, that is

2b� 1 =

b

�b (b)(2�b (b)� 1)� c

Rearranging, this becomes

b2 � [1 + p� c (1� p)] b+ p = 0. (2)

Since p < 12 , for small search cost c, only the larger root b = ¯bc of this quadratic is greater than 1

2

and leads to a sale. Hence, the unique equilibrium is that sellers provide a signal p !�

0,¯bc

whichinduces a stopping set Qc (�) = [0, ac][

⇥

¯bc, 1⇤

. Moreover, as c ! 0, the value of searching increasesand ¯bc ! 1, as shown in Figure 4(b). This means that the seller provides full information.

18

When p � 12 , a monopolist would provide no information and there are two limit equilibria

as search costs vanish: full information and no information. First observe that “no information”is an equilibrium for any search cost: if sellers provide no information, the buyer will not search,and since “no information” is the monopoly strategy, no seller will defect. There are two furtherequilibria which correspond to the two roots of the quadratic in equation (2), which are denoted by�

bc,¯bc

⇢⇥

12 , 1⇤

. As c ! 0, bc ! p, which corresponds to no information, whereas ¯bc ! 1, whichcorresponds to full-information. In summary, full-information is always a limit equilibrium and isthe unique limit equilibrium if the monopolist provides any information.

So far, we have abstracted from coordination. No matter the level of �, there is always an equi-librium where sellers ignore the coordination device, inducing the full-information limit equilibriumabove. Moreover, if coordination is imperfect (� > 0) and the monopolist provides information(p < 1

2), then full-information is the only limit equilibrium. Intuitively, no matter what informationthe coordinating policy provides, there is at least proportion � of potential sellers who providenon-trivial independent information. Therefore, as the search cost vanishes, a buyer can obtain alarge number signals at low cost and become almost fully-informed. 4

This example shows that when sellers imperfectly observe buyers’ beliefs, equilibria can be muchmore informative than the monopoly strategy. Crucially, when sellers can perfectly observe buyers’beliefs, a buyer who receives information from one seller obtains no useful information if he were togo back to the market. However, when the observation of beliefs is imperfect, the buyer can pretendto be uninformed and receive more information from another seller in the market. The possibilityof taking this outside option enables competition to function.

4.2 Full-Information is a Limit Equilibrium

We now return to the general model with multiple states and multiple products and show that therealways exists a full-information limit equilibrium.

Theorem 3. Suppose ↵ > 0. There exists a full-information limit equilibrium.

Proof. See Appendix B.2.

The basic intuition is as follows. Suppose sellers ignore the coordination device and use simplestrategies. Although a buyer only visits one seller on the equilibrium path, he always has the optionof visiting other sellers and obtaining new information. As the search cost vanishes, the buyer canthus threaten to visit a growing number of sellers at a shrinking total cost. This means that if eachseller provides some information about each state, the buyer will become fully informed. Hence thecurrent seller has to match the market and also provide full information.23

23 Note that Theorem 3 does not say that the seller must provide full information but that the limit equilibrium

19

We now provide an overview of the proof. Fix a search cost c > 0 and suppose all other sellersuse a simple strategy �. This generates a value function Vc (�, ·) for the buyer and a correspondingstopping set Qc (�). Faced with this stopping set, the current seller has a set of optimal simplestrategies which we denote by 'c (�). In other words, the mapping � ◆ 'c (�) is the best-responsecorrespondence. The set of all simple strategies is a convex compact space. If the best-responsecorrespondence 'c is nonempty, convex-valued and has a closed graph, then a direct application ofthe Kakutani-Fan-Glicksberg fixed point theorem implies that an equilibrium exists. Unfortunately,'c may not have a closed graph: as �n changes, the set of stopping beliefs Qc (�n) can increase,meaning that there are products available to the seller in the limit that are not be available alongthe sequence resulting in profits that may jump up discontinuously.24

To construct an equilibrium, we consider strategies that provide a lot of information. Forexample, consider Figure 5(a) where the D" regions represent payoff beliefs such that the buyer’sstopping payoff is "-away from that of full-information. The first step is to observe that if allsellers use strategies with support in D", then when search costs are small, the best-response alsohas support in D". Intuitively, if other sellers provide a lot of information, then buyers have highcontinuation values and the stopping sets are inside D". Second, we show that 'c has a closed graph.When " is small, D" only intersects with beliefs where products are chosen under full-information.This means that as other sellers change their strategies, this does not affect the set of products thatare available to the current seller. Together, these properties imply that an equilibrium exists witha payoff that is "-close to that of full-information. We can then take " ! 0 as c ! 0 as required.

4.3 Uniqueness of the Full-Information Limit Equilibrium

Theorem 3 establishes that there always exists a full-information limit equilibrium. In this section,we characterize the set of limit equilibria and provide a sufficient condition under which the full-information limit equilibrium is unique.25

To illustrate our results, consider the example illustrated in Figures 5(b) and (c) where there isa single product and p is the buyer’s initial prior over the three payoff states S = {s1, s2, s3}. Forany event E ⇢ S, let pE 2 �S denote its conditional belief given E, that is, pE (s) = p (s) /p (E)

for s 2 E (and pE (s) = 0 for s /2 E). Figure 5(b) shows a limit equilibrium that is partitionaland provides less than full information. Here, sellers provide independent information about s1

only, so a buyer’s posterior lies on the line connecting the conditional beliefs p{s1} and p{s2,s3}.

strategy must give the buyer his full-information payoff. Intuitively, a buyer who makes the same decision in twodifferent states will not pay a search cost to distinguish them.

24 For example, suppose that in Example 2 there is a sequence of simple strategies �n such that Qc (�n) =⇥0, 1

3 � "n⇤[⇥23 , 1

⇤. For all "n > 0, the optimal strategy is p !

�0, 2

3

, whereas the optimal strategy in the limit is

no information.25 Theorem 1 shows that if there is perfect coordination (� = 0), then monopoly is an equilibrium; our discussion

thus assumes that � > 0.

20

(a) Full-information limit equilibrium

Figure 5: Limit Equilibria

s1

H1

s2s

3

H2

H3

H4

Dɛ

Dɛ

(c) p{s2,s3}

is fully improvable (d) Fully unimprovable partition not a limit equilibrium

s1

H2

s2s

3

H1

s1

p

p{s2, s3}

s2s

3

H1

p{s1}

p

p{s2, s3}

q r

(b) p{s2,s3}

is not fully improvable

s1

p

p{s2, s3}

s2s

3

H1

p{s1}

p

Figure 5(a) shows the construction of a full-information limit equilibrium in the proof of Theorem 3.It also shows that this limit is unique as every non-trivial conditional belief is fully improvable.Figure 5(b) shows a limit equilibrium where the buyer only learns about s1. Note that p{s2,s3} is notfully improvable since the buyer purchases the most profitable product possible at that conditionalbelief. Figure 5(c) shows the case where learning only about s1 is not a limit equilibrium as p{s2,s3}is fully improvable. By Theorem 4, the unique limit equilibrium is full-information. Figure 5(d)shows an example where p{s2,s3} is not fully improvable, but learning only about s1 is still not a limitequilibrium.

21

From the current seller’s perspective, a buyer near p{s2,s3} purchases her product, so the sellerhas no incentive to provide information about s2 versus s3. As search costs go to zero, buyersthus learn for sure whether s1 is true or not, but learn nothing about s2 versus s3; this limitequilibrium is thus characterized by the partition {{s1} , {s2, s3}} and yields the buyer lower payoffsthan full-information. In contrast, in Figure 5(c), we will see that full-information is the only limitequilibrium. In particular, the partition {{s1} , {s2, s3}} is no longer an equilibrium. Intuitively,a buyer with belief close to p{s2,s3} does not buy, so the seller can strictly increase her profitsby providing information about s2 versus s3; as long as sellers provide some information in thisdimension, then as search costs vanish, a buyer can visit many sellers and fully learn the state.Hence, a single seller must match the market and the unique limit equilibrium is full-information.

We now formally characterize limit equilibria in terms of partitions. We call a strategy partitional

if it corresponds to revealing to the buyer a partition E of the payoff state space S. Given anyindependent policy �, we say a partition E� is induced by � if for every event E 2 E�, s, s0 2 E

iff the policy does not distinguish between them, i.e. � (s) = � (s0). Finally, observe that since thespace of independent policies is compact,26 we can consider convergent sequences, �n ! �, wherewe call � its limit.

Lemma 4. Suppose ↵ > 0 and � > 0. In any limit equilibrium, buyers’ payoffs exceed those from

the partition induced by the limit of the independent policies. Moreover, payoffs are equal if � = 1.

Proof. See Appendix B.3.

When sellers cannot coordinate (� = 1), Lemma 4 says that all limit equilibria are partitional.Intuitively, if sellers use independent policies that are informative about state s versus s0, thenas the search cost vanishes, a buyer can visit a large number of sellers, receive a large number ofindependent signals and perfectly distinguish between the two states in the limit. In order to makea sale, any one seller must therefore match the market by providing this information immediatelyto the buyer, meaning that any limit equilibrium must be partitional.

However, there are a couple of subtleties with this intuition. First, as the search cost vanishes,it may be the case that the information provided by �n shrinks so that the partitions induced byeach �n are richer than the partition induced by the limit policy �. Even though a buyer can obtainmore and more signals for a given search expenditure, Lemma 4 says that his payoff is determinedby the coarser partition corresponding to �. Intuitively, sellers provide enough information so thatthe buyer only searches once on path, and his limit payoff is determined by the limit policy �.27

26 See Lemma 9 in the Appendix.27 As an illustration, consider a reversed version of Example 2 in which the low-quality good is more profitable

than the high-quality good. Suppose p > 23 and sellers provide vanishing amounts of information as cn ! 0, so that

Qcn (�n) \⇥13 ,

23

⇤= ;. For any positive search cost, E�n = {{L} , {H}}, whereas in the limit E� = {{L,H}}.

22

Second, when sellers can coordinate (� < 1), limit equilibria may not be partitional. This occurswhen sellers’ independent policies do not distinguish between states s and s0, while the coordinatingsignal does, so the buyer only learns about the two states from one draw of the coordinating signal.This can be an equilibrium because if a seller defected and provided independent information abouts versus s0, then the buyer would continue searching for the coordinated signal.28 In such as case,Lemma 4 says that buyers still earn at least as much as from learning the partition induced by thelimit of sellers’ independent policies.

We now provide a sufficient condition for full-information to be the unique limit equilibrium.We say a payoff belief p is non-trivial if there are at least two products that a buyer with belief pcould potentially buy, i.e. ⇡ (1s) is not constant for all s 2 Sp. Recall from Section 3 that a beliefis improvable if a monopolist prefers to release some information at that belief. We say a payoffbelief p is fully improvable if a monopolist prefers to release full information than no information,i.e. ¯

⇧ (p) > ⇡ (p), where ¯

⇧ (p) :=P

s p (s)⇡ (1s) is the full-information profit.

Theorem 4. Suppose ↵ > 0 and � > 0. Any limit equilibrium is full-information if every non-trivial

conditional belief is fully improvable.

Proof. See Appendix B.4.

Intuitively, if a single seller would like to release a little information along some dimension,a buyer can visit many sellers and accumulate this information thus forcing a seller to release fullinformation to make a sale. To prove this result, note that Lemma 4 implies that buyers’ payoffs arebounded below by the limit of the independent policies. We would thus like to take an equilibriumwhere sellers provide no independent information about state s versus s0, and show that any currentseller would like to deviate and provide some information. If the stopping set for the current sellerincludes all beliefs on the segment connecting s and s0, then the result would only require that amonopolist would want to release some information at the conditional belief p{s,s0} (i.e. p{s,s0} isimprovable). However, this is complicated by the two “subtleties” discussed after Lemma 4. First,the partitions induced by the independent policies can be discontinuous in the limit, so even thoughthe limit stopping set Q0 (�) may contain the entire segment of beliefs, each Qcn (�n) may not (seefootnote 27). Second, with coordination, limit equilibria may not even be partitional so Q0 (�)

itself may not include the segment (see footnote 28). Nevertheless, a seller can always provide fullinformation at p{s,s0}. Thus, Theorem 4 states that if every conditional belief is fully improvable,then any limit equilibrium is full-information.

28 Again, consider the reversed version of Example 2 from footnote 27. When p > 23 and the cost is small, there is a

non-partitional limit equilibrium in which the coordinating policy is a monopoly strategy (i.e. p !�

23 , 1

), while the

independent policy reveals no information. A buyer purchases after receiving either of these signals; in both cases, asubsequent signal provides no additional value.

23

In terms of the examples, Example 1 (with p < 12) and Figures 5(a,c) satisfy full-improvability

so full-information is the only limit equilibrium. On the other hand, Figure 5(b) does not satisfyfull-improvability and full-information is not the unique limit equilibrium.29

It is notable that the full-improvability condition in Theorem 4 only considers a monopolist’sincentives at conditional beliefs, and is therefore stricter than required. Indeed, a partitional strategythat provides less than full information can be ruled out in other ways. First, as suggested by theabove intuition, the partition may contain a conditional belief that is improvable (rather than fullyimprovable).30 Second, even if no conditional belief is improvable, there may be no sequence ofequilibria that converges to the strategy.31

Taken together, Theorems 1-4 show how market outcomes depend on whether sellers can ob-serve buyers’ beliefs or coordinate their disclosures. Under perfect observation of beliefs or perfectcoordination, monopoly is an equilibrium. Intuitively, sellers can discriminate between new and oldbuyers, allowing them to implicitly collude. However, under imperfect observation and imperfectcoordination, full-information is an equilibrium and, if conditional beliefs are fully improvable, theonly equilibrium. Even though buyers purchase from the first seller on-path, the option to anony-mously mimic an uninformed buyer and receive more information forces sellers to compete againsteach other and provide much more information to buyers.

5 Extensions

In this section we discuss a number of extensions and applications. For simplicity, we discuss allthe issues in the context of the two-state, single-product example (Example 1) and abstract fromthe possibility of coordination, i.e. we set � = 1.

The model assumes that buyers have a common prior p0. We first argue that the spirit of ourresults is not affected if priors were heterogenous. To see this, suppose that the prior is p0 2 {pL, pH}with probabilities {fL, fH}. If sellers can perfectly observe buyers’ beliefs, they can condition theirdisclosure strategy on the buyer’s type. Hence, monopoly is the unique equilibrium for any search

29 One can also ask whether monopoly is still an equilibrium with ↵ > 0. When there is no coordination (� = 1),monopoly is generically a limit equilibrium iff the monopolist provides no information. This follows from Lemma4 and the fact that a monopolist generically does not provide full information along any dimension. When thereis coordination (� < 1), footnote 28 illustrates that there can still be equilibria whereby the coordinated policy ismonopoly. However, in order for this to be an equilibrium, Lemma 4 requires that the independent policy releasesno information.

30 For example, consider Example 2 with p 2�13 ,

23

�and no coordination (� = 1). Even though p is not fully

improvable, the no information limit equilibrium can be ruled out because a seller can always earn higher profits bysending the buyer to p ! {p� ", 1� "} for small " > 0.

31 For example, consider Figure 5(d) with three payoff states and two goods. Given the partition E =

{{s1} , {s2, s3}}, the seller has no incentive to provide information about s2 versus s3 since p{s2,s3} is unimprov-able. However, away from the limit, the seller wishes to provide information about s2 versus s3. In particular, theseller would like to send the buyer to posterior r rather than q, meaning that there is no limit equilibrium thatcorresponds to E .

24

cost exactly as in Section 3.1. In contrast, if sellers imperfectly observe buyers’ beliefs, then full-information is a limit equilibrium as in Section 4.1. This is easiest to see if buyers’ beliefs areprivate (↵ = 1), so disclosure policies are independent of y and buyers’ acceptance sets are of theform Qc (�) = [0, a] [ [b, 1] for a 1

2 b. If other sellers provide a lot of information, then b ⇡ 1

and it is optimal for the current seller to use a strategy that sells to both types, i.e. pL ! {0, b}.32

Replacing p with pL in Section 4.1, this means that a buyer can become fully informed as c ! 0

and the equilibrium converges to full-information. Moreover, when pH < 12 , full-information is the

only limit equilibrium. Intuitively, if there are lots of pL types, then the seller always sells to bothtypes, i.e. pL ! {0, b} as above. If there are lots of pH types, then the low types may stay in themarket for multiple periods; nevertheless, the seller always sells to pH types immediately, and sothe signal is at least as informative as pH ! {0, b}.33 In either case, buyers become fully informedas search costs vanish and sellers provide full information in the limit.

The model also assumes that sellers observe signals of buyers’ beliefs, leading to the dichotomybetween the cases with perfect observability (↵ = 0) and imperfect observability (↵ > 0). Wenow claim that the monopoly results of Section 3.1 carry over to a model in which sellers observewhich sellers a buyer has visited in the past and their signal structures but not the resulting belief.First, one can see that there is an equilibrium in which the first seller provides the monopolylevel of information and subsequent sellers provide no information. Given such strategies, a buyerpurchases from the first seller who makes monopoly profits. If a buyer deviates and searches, thenany subsequent seller knows that the buyer’s belief is already in the acceptance set and thus providesno more information. Second, under the assumptions of Example 1, this monopoly strategy is theunique equilibrium. Given an acceptance set of the form Qc (�) = [0, a1] [ [b1, 1], the first selleruses a binary signal, p ! {0, b1}. If the buyer approaches seller 2, then she knows the buyer’sbelief q lies in {0, b1}. If q = 0, seller 2 cannot affect the buyer’s posterior, so she should assumethat q = b1. Given an acceptance set Qc (�) = [0, a2] [ [b2, 1], seller 2 thus uses a binary signalb1 ! {0, b2}. Crucially, seller 2 would use exactly the same strategy if she could observe the beliefof the buyer. The same logic applies to subsequent sellers, so the equilibria of the game coincidewith those of the game in Section 3.1. Given that the monopoly strategy is the only rationalizableoutcome (see footnote 20), it is also the unique equilibrium here. Intuitively, the logic behind themonopoly outcome relies on the seller being able to discriminate between new and old buyers; seeingthe history of the buyer’s shopping behavior is sufficient for this.

As a final extension, one might think that buyers can affect whether their beliefs are observed by32 If the seller sells to both types, then its profits are bounded below by fLpL + fHpH which can be attained by

releasing full information. If the seller sells to pH only, then its profits are fHpH/b which is smaller when b ⇡ 1.33 To see this, the first step is to show that the seller uses a binary signal of the form pH ! {0, r}; this follows

from the fact that “bad news” that lowers beliefs never leads to a sale, so the seller should send the buyer to posteriorq = 0. The second step is to show the seller seller to pH immediately; if this did not happen, then a seller can provide“two rounds” of information and raise her profits.

25

sellers. This may mean browsing the internet anonymously or being tight-lipped when a financialadviser asks about their portfolio. By deleting their cookies, anonymous buyers exert a positiveexternality on others. Intuitively, when faced with an anonymous buyer, a seller provides moreinformation, anticipating that he can return to the market and pose as a new buyer. This extrainformation benefits both new and anonymous buyers. However, if there is a small cost to becominganonymous, then there is no equilibrium where all buyers choose to do so. If this were the case,then sellers would reveal enough information to prevent an anonymous buyer from searching again;hence any anonymous buyer obtains the same payoff as one who can be tracked, while the formerhas to pay the fixed cost. This externality may provide a rationale for regulating online tracking,encouraging sellers to provide more information to their customers.

6 Conclusion

This paper studies a market in which buyers search for better information about products andsellers choose how much information to disclose. This is particularly applicable to markets whereintermediaries provide advice and are compensated via commission, such as when websites recom-mend products, financial advisers advise on mutual funds, and doctors suggest different medicaltreatments. This also applies to areas such as organizational economics and political economy.For example, consider a CEO (“buyer”) who is interested in investing in a new innovation and se-quentially approaches potential project managers (“sellers”) to investigate the market potential andreport back. While the CEO wants to maximize company profits, the managers want the firm tomake the investment and be chosen to lead the project.

It has been widely recognized that if there is a single seller, then the advice she provides will bedistorted in her interest (e.g. high commission products). However, one might think that competitionis a substitute for regulation: intuitively, a buyer can visit many sellers and buy from the onewho gives the best advice. Since competition is quite extensive among websites, financial advisersand doctors, this would suggest that regulation would be unnecessary. Our analysis suggests thatthe impact of competition in advice markets depends on the information structure of the market.We show that even if there are many competing sellers, the monopoly disclosure strategy is anequilibrium if sellers can observe buyers’ beliefs or can successfully coordinate their disclosures. Ineither case, sellers can provide less information to old buyers than new ones, discouraging buyersfrom searching and undermining competition. On the other hand, when sellers cannot fine-tune theirdisclosures, there is an equilibrium in which sellers provide full information as search costs vanish.When buyers search for information, they can visit many sellers and become better informed. Thisforces any given seller to match the market and provide close to full information in order to makea sale.

26

On the normative side, our results suggest that the effectiveness of competition depends on thedetails of the market. Thus, a regulator might be able to enhance the role of competition withoutdirectly regulating commissions (e.g. the UK’s ban on commissions for financial advisers). For ex-ample, regulators can make it harder to condition advice on buyers’ beliefs by requiring financialadvisers to describe a strategy for clients before they see the customer’s current portfolio. Alterna-tively, regulators could make coordination harder by restricting the common marketing materialsprovided by mutual funds, increasing the dispersion of opinions. These forces are particularly pow-erful online, where sellers are increasingly able to calibrate information disclosure to the needs andexperiences of the customer. Our results highlight the role that anonymity can play in fosteringcompetition between recommender systems and illustrate how tracking technology can underminethe competitive mechanism and support collusive equilibria.

A Proofs for Section 3

A.1 Proof of Lemma 1

Lemma 1 says if a seller knows a buyer’s belief, i.e. ⌘y = �p, then it is optimal for her to sell withprobability one. In other words, the stopping set Qc (�) is almost-sure under the measure K

�yp for

any possible realization of y given p. We will prove a stronger version of this below which will beuseful for proving subsequent results.

Lemma 5. Suppose � is optimal given �. If ↵ = 0, then for all p 2 �⌦, K �p (Qc (�)) = 1.

Moreover, if � = � is an equilibrium, then

Vc (�,p) = �c+

Z

�⌦q · u (q)K�

p (dq) (3)

If ↵ > 0 and ⌘0 = �p0 , then the same holds for p = p0.

Proof. Let � is optimal given � and Q := Qc (�) be the corresponding stopping set. First, considerthe case where ↵ = 0 so the seller knows the buyer’s belief, i.e. ⌘y = �p for any buyer withbelief p = y. Let µ = K �

p be the distribution on posteriors induced by �, so profit is given byR

Q ⇡ (r)µ (dr). We will show that the buyer will always choose to purchase from the seller, that is,µ (Q) = 1.

Suppose otherwise so µ puts strictly positive weight outside Q. Consider the following deviation:provide an independent draw of full information to all buyers who would have ended up outside Q

under µ. Call this new signal � and let ⌫ = K �p be the corresponding posterior distribution. The

seller’s profit from using ⌫ isZ

Q⇡ (r) ⌫ (dr) =

Z

Q⇡ (r)µ (dr) +

Z

�⌦\Q¯

⇧ (r)µ (dr)

27

where ¯

⇧ (r) is the seller’s profit from providing full information for a buyer with payoff belief r. Wenow show that the second term on the RHS is strictly positive. For any r that is not in Q, thebuyer pays the search cost c > 0 in order to acquire some information. Hence, after being given fullinformation, he must purchase with strictly positive probability (else he should have just stoppedat r). Since all products have positive profits, this means the seller strictly benefits by providingfull information at r, that is ¯

⇧ (r) > 0. By assumption, µ puts strictly positive weight on beliefsoutside Q, so the expected profit from using � is strictly larger than from � contradicting the factthat � is optimal. Hence, µ must put all its weight in Q and the buyer stops with probability one.Moreover, if � = � is an equilibrium, then we have

Vc (�,p) = �c+

Z

�⌦max {q · u (q) , Vc (�,q)}K�

p (dq)

= �c+

Z

�⌦q · u (q)K�

p (dq)

as desired.Finally, consider the case of ↵ > 0 and ⌘0 = �p0 . Recall from the text that from the full-support

assumption, this implies that for the initial prior p0, ⌘y = �p0 for all realizations of y. Everythingnow follows as above but for p = p0.

A.2 Proof of Lemma 3

Let ↵ = 0 and suppose any non-maximal belief is improvable. Consider some equilibrium � and let

⇧ (p) :=

Z

Qc(�)⇡ (q)K�

p (dq)

be the corresponding profit from the equilibrium strategy. Fix some p that is maximal, i.e,

⇡ (p) = max

s02Sp

⇡ (1s0) = ⇡ (1s) = ⇡k

where ⇡k and s 2 Sp are the corresponding profit and state. We will now show that p is also astopping belief under �, that is, p 2 Qc (�).

Let Hk be the set of payoff beliefs where product k is chosen under no information. Considerbeliefs between p and some degenerate belief at s, that is,

p� := (1� �)p+ �1s

for � 2 [0, 1] and let p� denote the corresponding payoff belief. Note that since ⇧ is concave andupper semicontinuous,34

⇧ (p�) as a function of � 2 [0, 1] is also concave and upper semicontinuous.This implies that ⇧ (p�) as a function of � is continuous on [0, 1].

34 See Lemma 17.30 of Aliprantis and Border (2006).

28

We will now use a continuity argument as � ! 0 to show that the buyer stops at p. Bythe contrapositive of our assumption, every belief that is not improvable must be maximal. Thismeans that ⇡k is the highest attainable profit in �Sp.35 Note that the convexity of Hk implies thatp� 2 Hk as both p 2 Hk and 1s 2 Hk. Since ⇡k is the highest attainable profit in �Sp, ⇡k is alsothe monopoly profit at p� 2 Hk, that is ⇧

⇤(p�) = ⇡k. This implies that

⇧ (p�) ⇧

⇤(p�) = ⇡k

Finally, define¯� := inf {� 2 [0, 1] | ⇧ (p�) = ⇡k }

and ¯p := p�. Note that by the continuity of ⇧ (p�), ⇧ (

¯p) = ⇡k.We will now show that ¯� = 0 so ⇧ (p) = ⇡k. Suppose otherwise so we can find a increasing

sequence �n % ¯�. Let pn := p�nso by continuity

lim

n⇧ (pn) = ⇧ (

¯p) = ⇡k

Since ⇡k is the highest attainable profit, it must be that limnK�pn

(Hk) = 1. By the definition of ¯�,⇧ (pn) < ⇡k. Thus, pn cannot be in the stopping set Qc (�) since otherwise the seller can achievegreater profits by providing no information. By the definition of Qc (�), it must be that

pn · uk < Vc (�,pn) = �c+

Z

�⌦q · u (q)K�

pn(dq)

where the last the expression follows from equation (3). Rearranging terms yield

pn · uk �

Z

{q2�⌦ | q2Hk }q K�

pn(dq)

!

· uk < �c+

Z

{q2�⌦ | q 62Hk }q · u (q)K�

pn(dq)

Note that both the left expression and the second term in the right expression vanish as K�pn

(Hk) !1. Since c > 0, we can find some �n < ¯� such that the above inequality is not satisfied implyingpn 2 Qc (�), which yields a contradiction. Thus, ¯� = 0 so ⇧ (p) = ⇧ (

¯p) = ⇡k.Finally, since ⇡k is the most profitable product in �Sp and ⇧ (p) = ⇡k, it must be that K�

p (Hk) =

1. Hence, information has no effect on the buyer’s purchasing behavior so the buyer might as wellstop , i.e. p 2 Qc (�).

A.3 Proof of Corollary 1

We will show that in all three cases, non-maximal beliefs are improvable and then apply Theorem 2.Let ⇧

⇤ denote the monopoly profit function. Note that part (i) follows trivially from either (ii) or35 Suppose otherwise and let ⇡j > ⇡k be the highest attainable profit in �Sp. Let q 2 �Sp be such that ⇡ (q) = ⇡j .

Clearly q is not improvable so it must be maximal which contradicts the definition of ⇡k.

29

(iii). For part (ii), consider two “horizontally differentiated” products with ⇡1 > ⇡2. We will provethis by contradiction. Suppose there is a belief p that is neither maximal nor improvable, so

⇧

⇤(p) = ⇡2 < ⇡1 = ⇡ (1s)

for some s 2 Sp. We will show that it is a strictly profitable deviation for the seller to providea perfect signal about s. Since 1s 2 H1, u1 (s) � 0 and “horizontal differentiation” implies that1s · u2 = u2 (s) 0. Let r be the conditional belief knowing that s is not true, that is,

p = p (s)1s + (1� p (s)) r

where 1s is the conditional belief knowing that s is true. Note that 1s · u2 0 and p · u2 � 0

imply that r · u2 � 0. Hence, the buyer chooses some profitable product at r so ⇡ (r) � ⇡2. Thisimplies that the seller can earn a strictly higher payoff by providing a perfect signal about state s

contradicting the fact that ⇧

⇤(p) = ⇡2. Hence, all non-maximal beliefs are improvable.

The argument for part (iii) is analogous. We prove it again by contradiction. Suppose there isa belief p that is neither maximal nor improvable, so

⇧

⇤(p) = ⇡i < ⇡1 = ⇡ (1s)

where ⇡1 is the state-wise optimal product in Sp and s 2 Sp. Again, consider the deviation ofproviding a perfect signal about s and let r be the conditional belief knowing that s is not true, sop = p (s)1s + (1� p (s)) r. By “nicheness” at 1s, 1s · u1 � 0 implies 1s · ui 0. Since p · ui � 0, thisimplies that r · ui � 0 from the definition of r. By “nicheness” at r, this means that r · uj 0 forall j 6= i. Hence, the profit at r is at least that of ⇡i so by the same argument as before, the sellercan obtain a strictly higher payoff by providing a perfect signal about s yielding a contradiction.

B Proofs for Section 4

This section includes the proofs for Theorems 3 and 4. In subsection B.1, we first introduce somepreliminary notation and results that will be useful in subsections B.2 and B.3 when we prove thetheorems

B.1 Preliminaries

We first establish some useful preliminary notation and results. First, note that when ↵ > 0, thenall sellers ignore their private signal y. Thus, �g

y and �iy are independent of y so without loss

of generality, we let �g and �i denote the general and independent signal structures. The value

30

function of the buyer is now given by

Vc (�,q) =� c+

Z

�⌦max {r · u (r) , Vc (�, r)}

h

(1� �)K�g

q + �K�i

q

i

(dr)

Moreover, if a seller uses a simple strategy, then she only provides a single independent signalstructure which we will simply refer to as the strategy.

When sellers use a simple strategy �, we can only work with payoff beliefs and a buyer with theinitial prior p0 has posterior beliefs distributed according to K�

p0. Hence, we can associate a simple

strategy � with its posterior distribution µ = K�p0

. Henceforth, we will use the simple strategy �

and its associated posterior distribution µ on payoff beliefs in �S interchangeably. We will refer toµ as the seller’s strategy as well. Since signals are all independent, without loss we can define thecanonical signal space as the set of posterior beliefs of buyer p. In other words, each signal realizationcorresponds to a posterior belief r 2 �S of the buyer with prior p. The joint distribution of statesand signals from the perspective of buyer p is given by Bayes’ rule as

p (s)� (s, dr) = µ (dr) r (s) (4)

Now, a buyer with prior q has signal distribution µq and for each signal realization r, his posteriorbelief is �q (r), that is

q (s)� (s, dr) = µq (dr) [�q (r)] (s)

Combining this with equation (4) yields the following

µq (dr) = µ (dr)X

s2Sq (s)

r (s)

p (s)(5)

Also recall the following expression for �q (r) from equation (1)

[�q (r)] (s)

X

s2Sq (s)

r (s)

p (s)

!

= q (s)r (s)

p (s)

Consider a buyer with prior q0 who searches t times when all sellers are using strategy µ. Letqt := �qt�1 (rt) be his posterior belief in period t given the t-period signal realization rt 2 �S.Assuming zero search costs, his payoff at the end of the t periods is given by

W t(µ, q0) : =

Z

(�S)tmax

i(ui · qt)µq0 (dr1) · · ·µqt�2 (drt�1)µqt�1 (drt)

When t = 1, we just set W := W 1. We first show that W t has a simpler expression.

31

Lemma 6. For any strategy µ, belief q and time period t

W t(µ, q) =

Z

(�S)tmax

i

X

s

ui (s) q (s)r1 (s)

p (s)· · · rt (s)

p (s)

!

µ (dr1) · · ·µ (drt)

Proof. Using the expression for µqt from equation (5), we obtain

W t(µ, q0) =

Z

(�S)tmax

i(ui · qt)

X

s

qt�1 (s)rt (s)

p (s)

!

µq0 (dr1) · · ·µqt�2 (drt�1)µ (drt)

=

Z

(�S)tmax

i

X

s

ui (s) qt (s)

X

s

qt�1 (s)rt (s)

p (s)

!!

µq0 (dr1) · · ·µqt�2 (drt�1)µ (drt)

Since qt := �qt�1 (rt), from equation (1), we have

W t(µ, q0) =

Z

(�S)tmax

i

X

s

ui (s) qt�1 (s)rt (s)

p (s)

!

µq0 (dr1) · · ·µqt�2 (drt�1)µ (drt)

Repeating the above argument for each µq, we obtain the desired expression for W t

We now show that if the signal is informative about some event, then the buyer will eventuallylearn it. In order to prove this, recall some of the notation from Section 4.3. Recall that pE denotesthe conditional belief of event E ⇢ S under the prior p, that is, pE (s) = p (s) /p (E) for all s 2 E

and pE (s) = 0 otherwise. Given any strategy µ with corresponding signal structure �, let Eµ denotethe smallest such partition of S such that � (s) = � (s0) for every s, s0 2 E 2 Eµ. Note that fromequation 4, this implies that for s, s0 2 E 2 Eµ,

µ (dr)r (s)

r (s0)= p (s)� (s, dr)

1

r (s0)= �

�

s0, dr� p (s)

r (s0)= µ (dr)

p (s)

p (s0)

so r(s)r(s0) =

p(s)p(s0) µ-a.s.. This means that conditional posteriors are equal to the conditional priors,

that is, rE = pE µ-a.s. for all E 2 Eµ. Let µE denote the limit partitional strategy given µ, that is

µE :=

X

E2Eµ

p (E) �pE

We now show that the payoff of a buyer who searches indefinitely eventually converges to that ofits limit partitional strategy.

Lemma 7. For any strategy µ, W t(µ, q) ! W (µE , q).

Proof. Fix a strategy µ and let Eµ be its limit partition. Since r(s)p(s) =

r(E)p(E) for all s 2 E 2 Eµ, we

32

can rewrite W t from Lemma 6 as

W t(µ, q) =

Z

(�S)tmax

i

X

E

r1 (E)

p (E)

· · · rt (E)

p (E)

X

s2Eui (s) q (s)

!

µ (dr1) · · ·µ (drt)

=

Z

(�S)tmax

i

X

E

r1 (E)

p (E)

· · · rt (E)

p (E)

q (E) (ui · qE)!

µ (dr1) · · ·µ (drt)

Defining zt (E) :=

r1(E)p(E) · · · rt(E)

p(E) q (E), we can rewrite this as

W t(µ, q) =

Z

(�S)tmax

i

✓

P

E0 zt (E0) (ui · qE0

)

P

E0 zt (E0)

◆

X

E

zt (E)

!

µ (dr1) · · ·µ (drt)

=

X

E

q (E)

Z

(�S)tmax

i

✓

P

E0 zt (E0) (ui · qE0

)

P

E0 zt (E0)

◆

µE (dr1) · · ·µE (drt)

where µE (dr) := r(E)p(E)µ (dr).

Consider two events E,E0 2 Eµ such that E 6= E0. We want to show that the ratio of zt (E0) to

zt (E) goes to zero µE-a.s. as t ! 1. Now,

1

tlog

zt (E0)

zt (E)

=

1

tlog

✓

r1 (E0)

p (E0)

· · · rt (E0)

p (E0)

p (E)

r1 (E)

· · · p (E)

rt (E)

q (E0)

q (E)

◆

=

1

t

X

t0t

log

✓

rt0 (E0)

p (E0)

p (E)

rt0 (E)

◆

+

1

tlog

✓

q (E0)

q (E)

◆

By the law of large numbers, we have

1

tlog

zt (E0)

zt (E)

!Z

�Slog

✓

r (E0)

p (E0)

p (E)

r (E)

◆

µE (dr)

Since E 6= E0, by the definition of Eµ, it must be that r(E0)r(E) is different on some strictly positive

µ-measure. This is because if r(E0)r(E) is constant µ-a.s., then for all s 2 E and s0 2 E0,

r (s)

p (s)=

r (E)

p (E)

=

r (E0)

p (E0)

=

r (s0)

p (s0)

so E [ E0 2 Eµ which contradicts the fact that Eµ is the smallest such partition of S by definition.By Jensen’s inequality,

Z

�Slog

✓

r (E0)

p (E0)

p (E)

r (E)

◆

µE (dr) < log

✓

Z

�S

r (E0)

p (E0)

p (E)

r (E)

µE (dr)

◆

= log

✓

Z

�S

r (E0)

p (E0)

µ (dr)

◆

= log

✓

R

�S r (E0)µ (dr)

p (E0)

◆

= 0

where the strict inequality is due to the fact that r(E0)r(E) is non-constant µE-a.s.. Thus, 1

t logzt(E0)zt(E)

converges to something strictly less than zero. Hence, it must be that log zt(E0)zt(E) ! �1 or zt(E0)

zt(E) ! 0

33

for all E0 6= E. By dominated convergence, this implies that

lim

tW t

(µ, q) =X

E

q (E)

Z

(�S)tmax

i

✓

lim

t

P

E0 zt (E0) (ui · qE0

)

P

E0 zt (E0)

◆

µE (dr1) · · ·µE (drt)

=

X

E

q (E)

Z

(�S)tmax

i(ui · qE)µE (dr1) · · ·µE (drt) =

X

E

q (E)max

i(ui · qE)

From Lemma 6 and the definition of µE , we have

W (µE , q) =

Z

�Smax

i

X

s

ui (s) q (s)r (s)

p (s)

!

µE (dr) =X

E

p (E)max

i

X

s2Eui (s)

q (s)

p (E)

!

=

X

E

p (E)max

i

✓

q (E)

p (E)

ui · qE◆

=

X

E

q (E)max

i(ui · qE)

so limtWt(µ, q) = W (µE , q).

B.2 Proof of Theorem 3

We now prove that when ↵ > 0, there always exists a full-information limit equilibrium. We willonly be considering simple strategies, so without loss we will only consider payoff beliefs in �S. Wewill also be making use of the preliminary results in the previous section which uses some of thenotation from Section 4.3. Given any " > 0, let D" ⇢ �S be the set of beliefs such that the buyer’sstopping payoff is at most "-away from his full-information payoff. In other words,

D" := {q 2 �S | q · u (q) � W (µ, q)� "}

where µ is the fully-informative strategy. Let M" be the set of simple strategies that only put weightin D", that is, µ (D") = 1 for all µ 2 M". Denote ⇧c (µ, ⌫) as the profit of a seller for using simplestrategy ⌫ if all other sellers are using simple strategy µ, that is,

⇧c (µ, ⌫) :=

Z

Qc(µ)⇡ (q) ⌫ (dq)

and define the best-response correspondence 'c : M" ◆ M" such that ⌫ 2 'c (µ)

⇧c (µ, ⌫) � ⇧c

�

µ, ⌫ 0�

for any other simple strategy ⌫ 0.We first show that we can restrict ourselves to simple strategies and these best-response corre-

spondences without loss of generality.

Lemma 8. Suppose � =

�

�g, �i�

is optimal given a simple strategy �. Then the simple strategy �i

is also optimal given �.

34

Proof. Since � =

�

�g, �i�

is optimal given �, it solves

max

�

Z

Qc(�)⇡ (q)

h

(1� �)K �g

p + �K �i

p

i

(dr)

Note that since � is simple, we can work with payoff beliefs when buyers go back to the mar-ket. Define µg

:= K �g

p and µi:= K �i

p and supposeR

Qc(�)⇡ (r)µg

(dr) >R

Qc(�)⇡ (r)µi

(dr). Inthis case, if the seller were to use the strategy (�g, �) where � is the independent version of �g,then she would earn a strictly higher profit violating the optimality of �. Thus,

R

Qc(�)⇡ (r)µg

(dr) R

Qc(�)⇡ (r)µi

(dr). Moreover, since the seller can always set �g= �i, it must be that

R

Qc(�)⇡ (r)µg

(dr) =R

Qc(�)⇡ (r)µi

(dr). Thus, by using the simple strategy �i, the seller can achieve the same optimalprofit.

The main idea is that we will construct a sequence of equilibrium payoffs that converges tothe buyer’s full-information payoff as search costs vanish. For each ", we will find some smallenough search cost c such that 'c (µ) is non-empty for all µ 2 M". We will then use the Kaku-tani–Fan–Glicksberg (KFG) theorem to show that there exists an equilibrium strategy µ 2 M".Hence, as " ! 0, we can find a sequence of decreasing search costs and corresponding equilibriumstrategies µ 2 M" such that the buyer’s payoff converges to W (µ, p).

First, we show that the domain M" satisfies the usual conditions so that we can apply KFG.

Lemma 9. M" is a non-empty, compact and convex metric space.

Proof. To see why M" is non-empty, note that the fully-informative strategy µ puts weight onlyon degenerate beliefs so µ (D") = 1 and µ 2 M". The convexity of M" follows trivially. To seewhy M" is compact, note that since �S is a compact metric space, the set of all simple strategiesis also a compact metric space by Theorem 15.11 of Aliprantis and Border (2006). By Theorem3.28 of Aliprantis and Border (2006), it is also sequentially compact. We now show that M" is alsosequentially compact. Consider the sequence µm 2 M" so it has a convergent subsequence µn ! µ.Since µn are all strategies, it must be that

Z

�Sq µ (dq) = lim

n

Z

�Sq µn (dq) = p

so µ is also a simple strategy. Moreover, since µn (D") = 1 and D" is closed, by Theorem 15.3 ofAliprantis and Border (2006),

µ (D") � lim sup

nµn (D") = 1

so µ 2 M". Thus, M" is sequentially compact and therefore compact.

Next, we show that 'c is convex-valued. The proof follows directly from the fact that expectedprofits are linear in strategies.

35

Lemma 10. 'c (µ) is convex for all µ 2 M".

Proof. Let ⌫1, ⌫2 2 'c (µ) and consider ⌫ := a⌫1 + (1� a) ⌫2 for some a 2 [0, 1]. Note that ⌫ is alsoa strategy and

⇧c (µ, ⌫) = a⇧c (µ, ⌫1) + (1� a)⇧c (µ, ⌫2) � ⇧c

�

µ, ⌫ 0�

for any other strategy ⌫ 0. Hence, ⌫ 2 'c (µ) so 'c (µ) is convex.

As an intermediary step, we show that the buyer’s value function is jointly continuous in bothstrategies and posterior beliefs.

Lemma 11. The buyer value function Vc is continuous in (µ, q) and convex in q.

Proof. Fix some c > 0 and suppose all sellers are using the strategy µ. If the buyer searches once,then his payoff is

V 1c (µ, q) := �c+W (µ, q)

Using the expression for W from Lemma 6, we see that W is continuous by Corollary 15.7 ofAliprantis and Border (2006). Moreover, since support functions are convex, W is also convex inq. Hence, V 1

c is continuous and convex in q, and define a sequence of finite-period value functionsiteratively where

V t+1c (µ, q) := �c+

Z

�Smax

n

max

i�q (r) · ui, V t

c (µ,�q (r))o

µq (dr)

Recall from equation (5), that

µq (dr) = µ (dr)X

s

q (s)r (s)

p (s)

and also note that �q (r) is continuous in q. Hence, applying the same argument for V 1c inductively,

we conclude that V tc is continuous and convex in q. Since V t

c converges uniformly to the valuefunction Vc, the latter is also continuous and convex in q.

The next step is to show that the best-response correspondence 'c takes on non-empty valuesso we can apply KFG. While this may not be true in general, we will choose " small enoughsuch that this holds. Set " > 0 such that it is less than the minimal non-zero difference betweenthe buyer’s full and partitional information payoffs. In other words, choose " such that wheneverW (µ, p) > W (µE , p) for any limit partitional strategy µE , then

" < W (µ, p)�W (µE , p)

Note that such an " always exists as there are only a finite number of partitions of S.

36

We now show that for all " ", we can always find a search cost c small enough such that 'c

takes on non-empty values on M". The main idea is as follows. Recall that D" is the set of beliefssuch that the buyer’s payoff is "-close to his full-information payoff. Now, whenever µ only sends thebuyer to beliefs in D" for some " ", then the definition of " ensures that the buyer’s payoff underµ approaches that of full-information as search costs vanish. Hence, the buyer will only stop if hegets close to his full-information payoff so we can find a search cost c small enough such that thestopping set Qc (µ) is completely contained in D" for all µ 2 M". Since any optimal best responsestrategy ⌫ 2 'c (µ) will not send the buyer outside his stopping set, it must be that ⌫ also puts fullweight in D" so ⌫ 2 M" as well.

Lemma 12. If " ", then there exists a c" > 0 such that for all c c", 'c (µ) is non-empty for all

µ 2 M".

Proof. Let " ". We first show that for every µ 2 M", the value function Vc (µ, q) converges toW (µ, q) uniformly as c ! 0. For each search cost c, suppose the buyer searches tc times where tc isthe largest integer less than c�

12 so tcc c

12 . Since the buyer can always do this, his value function

is at least his payoff from visiting tc sellers. In other words,

Vc (µ, q) � �tcc+W tc(µ, q) � �c

12+W tc

(µ, q)

As c ! 0, tc ! 1 so lim infn Vcn (µ, q) � W (µE , q) from Lemma 7.We now show that W (µE , q) = W (µ, q). First note that since µ (D") = 1, the definition of D"

means that µ-a.s.max

ir · ui � W (µ, r)� "

Note that the left-hand expression is convex in r so if we let r =

P

E2Eµr (E) rE , then µ-a.s.

X

E2Eµ

r (E)max

i(rE · ui) � max

i

0

@

X

E2Eµ

r (E) rE

1

A · ui � W (µ, r)� "

Since rE = pE µ-a.s. for all E 2 Eµ, rearranging we have that µ-a.s.

" � W (µ, r)�X

E2Eµ

r (E)max

i(pE · ui)

Note that the buyer’s full-information payoff W (µ, r) is linear in r. Hence, taking expectationswith respect to µ yields

" � W (µ, p)�X

E2Eµ

p (E)max

i(pE · ui) = W (µ, p)�W (µE , p)

By the definition of ", this implies that W (µ, p) = W (µE , p). From Lemma 6, it is straightforward

37

to show that this implies W (µE , q) = W (µ, q).We thus have lim infn Vcn (µ, q) � W (µ, q). Since µ is fully-informative, this implies that

the buyer’s value function converges to that of full-information as search costs vanish, that is,Vcn (µ, q) ! W (µ, q). We will use Dini’s theorem (Theorem 2.66 of Aliprantis and Border (2006))to show that this convergence is uniform on the domain M" ⇥ �S. Note that M" ⇥ �S is com-pact by Lemma 9 and Vc is continuous by Lemma 11. Since Vcn is decreasing monotonically,Vcn (µ, q) ! W (µ, q) uniformly by Dini’s Theorem. Hence, we can find some c" > 0 such that

|Vc" (µ, q)�W (µ, q)| < " for all (µ, q) 2 M" ⇥�S

This implies that for all c c"

Qc (µ) ⇢ Qc" (µ) = {q 2 �S | q · u (q) � Vc" (µ, q)}

⇢ {q 2 �S | q · u (q) � W (µ, q)� "} = D"

If ⌫ is a best-response strategy to any µ 2 M", then by the same argument as in Lemma 5, ⌫ musthave full support in the stopping set, that is, ⌫ (Qc (µ)) = 1. Hence, ⌫ (D") � ⌫ (Qc (µ)) = 1 so⌫ 2 M" as desired.

Going forward, we assume " < ". The last step before we can apply KFG which is to show that'c has a closed graph. In general, this may not be true, but again we will find " small enough suchthat this holds. Let µn ! µ, ⌫n 2 'c (µn) and ⌫n ! ⌫. If we can show that ⌫ 2 'c (µ), then 'c

has a closed graph. Let ⌫⇤ 2 'c (µ) be a best-response strategy to µ. Recall that Hi is the setof beliefs where ui is chosen under no information, assuming ties are resolved in the seller’s favor.The first step is to find " small enough such that whenever ⌫⇤ recommends some product i, that is,⌫⇤ (Hi) > 0, then we can find some state s 2 S where 1s 2 Hi as well. For instance, in the verticaldifferentiation example of Section 3.3, this means that the optimal strategy never recommends thelow-quality TV. In Figure 5(a) of Section 4, this means that D" is close enough to the vertices suchthat the seller will never recommend product 4.

Let �u denote the smallest non-zero difference between buyer payoffs across all products at alldegenerate beliefs. In other words, �u |uj (s)� ui (s)| for all i, j and s 2 S where uj (s) 6= ui (s).Define " > 0 such that for every ⇡j > ⇡i,

" <

✓

⇡j � ⇡i⇡j

◆

�u

This is the smallest profitable profit deviation for the smallest difference in buyer payoffs. Note thatthis is well-defined as the number of products and states are both finite. Set "⇤ := min {", "}.

Lemma 13. Let " "⇤. Then ⌫⇤ (Hi) > 0 implies 1s 2 Hi for some s 2 S.

38

Proof. We will prove this by contradiction. Suppose ⌫⇤ (Hi) > 0 for some product i and 1s 62 Hi

for all s 2 S. Since " ", ⌫⇤ 2 M" so we know that ⌫⇤ (Hi \D") > 0. Now, consider a beliefq 2 Hi \D". We will show that providing full information at q will yield a strictly higher profit forthe seller which contradicts the fact that ⌫⇤ is a best-response strategy. Let Si ⇢ S denote the setof states such that product i is buyer-optimal, that is, ui (s) = u (1s) for all s 2 Si. Since 1s 62 Hi

for all s 2 S by assumption, this means that for every s 2 S, either s 62 Si or s 2 Si but there isanother buyer-optimal product that is more profitable for the seller, that is ⇡ (1s) � ⇡j > ⇡i forsome product j. Hence, by providing full information at q, the seller will get profit

X

s2Sq (s)⇡ (1s) �

X

s 62Si

q (s)⇡ (1s) +

X

s2Si

q (s)⇡j � ⇡jX

s2Si

q (s) (6)

as all profits have positive profits. Since q 2 Hi \D", we know that

q · ui = q · u (q) � W (µ, q)� "

" �X

s

q (s) (u (1s)� ui (s)) =X

s 62Si

q (s) (u (1s)� ui (s)) � (�u)X

s 62Si

q (s)

This implies thatX

s2Si

q (s) = 1�X

s 62Si

q (s) � 1� "

�u

Combining this with inequality (6), we get

X

s2Sq (s)⇡ (1s) � ⇡j

⇣

1� "

�u

⌘

= ⇡j � "⇡j�u

> ⇡i

where the last strict inequality follows from the definition of ". Since providing full information atq is more profitable for the seller, ⌫⇤ cannot be optimal yielding a contradiction.

The next step is to show that we can find a sequence of strategies ⌫⇤n that approximates theseller’s optimal payoff with arbitrary precision.

Lemma 14. Let " "⇤. Then there exists a sequence of strategies ⌫⇤n such that

⇧c (µn, ⌫⇤n) ! ⇧c (µ, ⌫

⇤)

Proof. We are going to partition the stopping set Qc (µ) into different regions corresponding to thedifferent products. In other words, define Qi

:= Qc (µ) \Hi so all the Qi together form a partitionof Qc (µ). For each product such that ⌫⇤

�

Qi�

> 0, let qi be the ⌫⇤-average belief in Qi, that is,

qi :=1

⌫⇤ (Qi)

Z

Qi

q ⌫⇤ (dq)

Note that qi 2 Qi since Qi is convex. Since ⌫⇤ is a strategy, the prior p must be the ⌫⇤-average

39

of the beliefs qi. Moreover, since each ⌫⇤�

Qi�

is strictly positive, p must be in the interior of theconvex hull of all qi.

Note that if every qi is in every stopping set Qc (µn), then a strategy with supports on qi willgive the seller exactly the optimal profit ⇧c (µ, ⌫

⇤) and we are done. Unfortunately, qi may not be

in Qc (µn) which means that the limit profit from such a strategy may be less than ⇧c (µ, ⌫⇤). To

overcome this, we will construct a sequence of beliefs qin 2 Qc (µn) such that qin ! qi and the profitsfrom a sequence of strategies with supports on qin converge to ⇧c (µ, ⌫

⇤). We do this as follows. For

each product i and signal µn, define �in = 1 if qi 2 Qi

n and

�in :=

c

c+ Vc (µn, qi)� qi · ui

otherwise. Note that in the case of the latter, Vc

�

µn, qi�

> qi · ui so �in < 1. Finally, define

qin := �inq

i+

�

1� �in

�

1si where 1si 2 Hi as guaranteed by Lemma 13. Since Vc is continuousby Lemma 11, Vc

�

µn, qi�

! Vc

�

µ, qi�

qi · ui where the inequality follows from the fact thatqi 2 Qc (µ). By the definition of �i

n, this implies that �in ! 1 and qin ! qi. Since p is in the interior

of the convex hull of the average beliefs qi, for large enough n, p is also in the convex hull of thebeliefs qin. Hence, for each n, we can define a strategy ⌫⇤n with finite support on each qin such that⌫⇤n��

qin �

! ⌫⇤�

Qi�

.Finally, since Vc (µn, ·) is convex,

Vc

�

µn, qin

�

�inVc

�

µn, qi�

+

�

1� �in

�

Vc (µn, 1si) = �inVc

�

µn, qi�

+

�

1� �in

�

(�c)

= �in

�

qi · ui�

qi · ui

Hence, each qin 2 Qc (µn) so

⇧c (µn, ⌫⇤n) =

X

i

⇡i⌫⇤n

��

qin �

!X

i

⇡i⌫⇤ �Qi

�

= ⇧c (µ, ⌫⇤)

as desired.

In this last step, we use the sequence of strategies ⌫⇤n that converges to the optimal strategy ⌫⇤

to show that the limit strategy ⌫ is also optimal. This proves that 'c has a closed graph.

Lemma 15. If " "⇤, then 'c has a closed graph.

Proof. Let µn ! µ, ⌫n 2 'c (µn) and ⌫n ! ⌫. We want to prove that ⌫ 2 'c (µ). First, we show that⌫ (Qc (µ)) = 1. Note that ⌫n (Qc (µn)) = 1 for all n as otherwise the seller can profitably deviateby providing information at any belief that is not stopping. In order to prove that ⌫ (Qc (µ)) = 1,we will show that the stopping set mapping Qc is upper hemi-continuous and takes on non-empty,closed values. We can then use Theorem 17.13 of Aliprantis and Border (2006) which says thatin this case, ⌫n (Qc (µn)) = 1 for all n implies that ⌫ (Qc (µ)) = 1. Note that the non-emptiness

40

and closedness of Qc are trivial (the latter follows from the continuity of the value function fromLemma 11). For upper hemi-continuity, suppose (µm, qm) ! (µ, q) where qm 2 Qc (µm). Thus,qm ·u (qm) � Vc (µm, qm) so by continuity again, q ·u (q) � Vc (µ, q) implying q 2 Qc (µ). Hence, Qc

is upper hemi-continuous, so applying the theorem yields ⌫ (Qc (µ)) = 1.Since ⌫ (Qc (µ)) = 1, the seller profits are ⇧c (µ, ⌫) =

R

�S ⇡ (q) ⌫ (dq). Since ⇡ is upper semicon-tinuous, the expected profit of the seller is also upper semicontinuous with respect to strategies (seeTheorem 15.5 of Aliprantis and Border (2006)). Using ⌫⇤n as defined from Lemma 14, we thus have

⇧c (µ, ⌫) =

Z

�S⇡ (q) ⌫ (dq) � lim supn

Z

�S⇡ (q) ⌫n (dq) = lim supn⇧ (µn, ⌫n)

� lim

n⇧c (µn, ⌫

⇤n) = ⇧c (µ, ⌫

⇤) � ⇧c (µ, ⌫)

In other words, since the payoffs from the optimal strategies ⌫n must be higher than those of theapproximate strategies ⌫⇤n and the profits of the latter converge to that of the optimal strategy ⌫⇤,so must the profits from ⌫n. Thus ⇧c (µ, ⌫) = ⇧c (µ, ⌫

⇤) so ⌫ 2 'c (µ) as desired.

Finally, we put everything together. Consider a sequence of "n ! 0 such that "n < "⇤ and let cnbe such that 'cn is non-empty as guaranteed by Lemma 12. Combined with Lemmas 9, 10 and 15,the preconditions of KFG (Corollary 17.55 of Aliprantis and Border (2006)) are satisfied so thereexists an equilibrium µn 2 M"n . Moreover, from Lemma 12, we know that |Vcn (µn, q)�W (µ, q)| <"n ! 0 so the buyer’s value function converges to his full-information payoff in the limit. We havethus constructed a full-information limit equilibrium as desired.

B.3 Proof of Lemma 4

Suppose ↵ > 0 and � > 0. First, suppose � = 1 so there is no coordination. Thus, we can just workwith payoff beliefs and associate a strategy � with its posterior distribution µ = K�

p0as in Sections

B.1 and B.2. Consider a sequence of decreasing search costs cn ! 0 and associated equilibria µn

such that the buyer value functions Vcn (µn, ·) converges to the limit function V ⇤. Since the set ofsimple strategies are compact, we can assume µn ! µ for some simple strategy µ. Let E be thepartition induced by µ and µE be the simple strategy corresponding to revealing E . We will showthat V ⇤

(p0) = W (µE , p0).Let tn be the largest integer less than c

� 12

n so the buyer’s value function is at least his payofffrom visiting tn sellers, that is

Vcn (µn, q) � �tncn +W tm(µn, q) � �c

12n +W tn

(µn, q)

Since tn ! 1 as cn ! 0, we can assume tn > T for some T without loss of generality. Thus,

Vcn (µn, q) � �c12n +W T

(µn, q)

41

as tn rounds of information is more informative than T rounds of information. Since µn ! µ andW T is continuous, by taking limits, we have V ⇤

(q) � W T(µ, q). Since this is true for all T , we can

take T ! 1 so W T(µ, q) ! W (µE , q) by Lemma 7. Thus, V ⇤

(q) � W (µE , q). To see why V ⇤(p0)

is weakly less than W (µE , p0), note that since µn is an equilibrium, by Lemma 5,

Vcn (µn, p0) = �cn +

Z

�Sq · u (q)µn (dq) = �cn +W (µn, p0)

Taking limits, we have V ⇤(p0) = W (µ, p0) W (µE , p0) where the inequality follows from the fact

that µE is more informative than µ. This proves that V ⇤(p0) = W (µE , p0) so all limit equilibria

are partitional.Now, suppose � < 1. Consider a sequence of equilibria �n =

�

�gn,�i

n

�

and let µn be thecorresponding posteriors distributions on �S for �i

n. As above, let µ be the limit of µn and E beits induced partition. We will now show that V ⇤

(p0) � W (µE , p0). Let tn be the largest integerless than c

� 12

n and for any s tn, let P tn() denote the probability that the buyer receives at most

draws from the independent policy �in out of a total of tn total draws. Hence, the buyer’s value

function is at least his independent payoffs from visiting tn sellers, that is

Vcn (�n,q) � �tncn + P tn() · 0 +

�

1� P tn()�

W (µn, q)

� �c12n +

�

1� P tn()�

W (µn, q)

Recall that � is the mean of this binomial distribution and suppose �tn. By Hoeffding’sinequality,

P tn() e�2 (tn��)2

tn

andVcn (�n,q) � �c

12n +

✓

1� e�2 (tn��)2

tn

◆

W (µn, q)

If we let n = �tn for � < �, then

Vcn (�n,q) � �c12n +

⇣

1� e�2(���)2tn⌘

W �tn(µn, q)

Since tn ! 1 as cn ! 0, we can assume tn > T for some T without loss of generality. Thus,

Vcn (�n,q) � �c12n +

⇣

1� e�2(���)2tn⌘

W T(µn, q)

as tn rounds of information is more informative than T rounds of information. Since µn ! µ andW T is continuous, taking limits we have V ⇤

(q) � W T(µ, q). Since this is true for all T , we can

take T ! 1 so W T(µ, q) ! W (µE , q) by Lemma 7. Thus, V ⇤

(q) � W (µE , q) and in particularV ⇤

(p0) � W (µE , p0) as desired.

42

B.4 Proof of Theorem 4

Suppose ↵ > 0 and � > 0. Let E be the partition induced by the limit µ of the independent policiesµn as in the proof of Lemma 4 in Section B.3. We will prove that all events in E are trivial bycontradiction. Suppose otherwise and let F ⇢ E denote the set of non-trivial events in the partition.By the premise, for every E 2 F , pE is fully improvable, i.e. ⇡ (pE) < ¯

⇧ (pE) where ¯

⇧ is the full-information profit. Define the strategy ⌫ that is essentially the same as the partitional strategy µE

but provides another round of information according to full-information at pE for every E 2 F . Asa result,

Z

�S⇡ (q)µE (dq) =

X

E2E⇡ (pE) p (E) <

Z

�S⇡ (q) ⌫ (dq) (7)

Now, since ⌫ only has support on either trivial or degenerate beliefs, ⌫ (Qcn (�n)) = 1. Given that�n is an equilibrium, µn must be optimal given �n so

Z

�S⇡ (q)µ (dq) � lim sup

n

Z

�S⇡ (q)µn (dq) �

Z

�S⇡ (q) ⌫ (dq) (8)

We now show that the monopoly profits from µ and µE are the same. First, note that by Lemma5, we have that µn (Qcn (�n)) = 1 for all n. Since Vcn (�n, ·) ! V ⇤ and µn ! µ, we can use Theorem17.13 of Aliprantis and Border (2006) as in the proof of Lemma 15 to obtain that

µ {q 2 �S | q · u (q) � V ⇤(q)} = 1

Moreover, since V ⇤(q) � W (µE , q) from the proof of Lemma 4 above, this implies that

µ {q 2 �S | q · u (q) � W (µE , q)} = 1

so q · u (q) = W (µE , q) µ-a.s.. Thus, if we let JEq , denote the posterior distribution corresponding

to revealing E to a buyer with belief q, then we have µ-a.s.

q · u (q) =Z

�Sr · u (r) JE

q (dr) =X

i

Z

Hi

r · uiJEq (dr) =

X

i

JEq (Hi) qi · ui

where qi =1

J(q,Hi)

R

HirJE

q (dr) for JEq (Hi) > 0. If we let q 2 Hj , then this implies that JE

q (Hj) = 1.Thus,

Z

�S⇡ (q)µE (dq) =

Z

�S

✓

Z

�S⇡ (r) JE

q (dr)

◆

µ (dq) =

Z

�S⇡ (q)µ (dq)

However, this contradicts inequalities (7) and (8) above. Hence, all events in E are trivial so byLemma 4, all limit equilibria must be full-information as desired.

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