existence of optimal auctions in general environments

Ž .Journal of Mathematical Economics 29 1998 389–418

Existence of optimal auctions in generalenvironments

Frank H. Page Jr. )

Department of Finance, UniÕersity of Alabama, Tuscaloosa, AL 35487, USA

Received 1 August 1995; accepted 21 February 1997

Abstract

We provide a unified approach to the problem of existence of optimal auctions for awide variety of auction environments. We accomplish this by first establishing a generalexistence result for a particular Stackelberg revelation game. By systematically specializingour revelation game to cover various types of auctions, we are able to deduce the existenceof optimal Bayesian auction mechanisms for single and multiple unit auctions, as well asfor contract auctions with moral hazard and adverse selection. In all cases, we allow for riskaversion and multidimensional, stochastically dependent types. q 1998 Elsevier ScienceS.A.

Keywords: Optimal auctions; Bayesian incentive compatibility; Relative K-compactness

1. Introduction

1.1. OÕerÕiew

Much of the literature on auctions is devoted to the study of properties ofŽvarious types of auctions e.g., Milgrom and Weber, 1982 and Maskin and Riley,

.1984 . Existence questions which arise naturally in general auction environments

) Corresponding author.

0304-4068r98r$19.00 q 1998 Elsevier Science S.A. All rights reserved.Ž .PII S0304-4068 97 00015-3

( )F.H. Page Jr.rJournal of Mathematical Economics 29 1998 389–418390

have only begun to be addressed. 1 In this paper, we provide a unified approach tothe problem of existence of optimal auctions for a rich variety of auctionenvironments. Our approach involves three steps. First, building on the work of

Ž . Ž . Ž .Vickrey 1961 , Harsanyi 1967–1968 and Myerson 1979, 1982 , we construct ageneral model of a Stackelberg revelation game. 2 Second, within the context ofour model, we establish the existence of an optimal, feasible, interim individuallyrational and Bayesian incentive compatible revelation mechanism. Third, wededuce the existauctience of optimal Bayesian auction mechanisms for varioustypes of auctions by showing that our Stackelberg revelation game can besystematically specialized to cover various types of auctions. Specifically, we

Ž .show that our model covers i single unit auctions with risk neutral seller andŽbuyers e.g., Myerson, 1981; Harris and Raviv, 1981; Riley and Samuelson, 1981;

. Ž .Milgrom and Weber, 1982 and Cremer and McLean, 1988 ; ii multiple andŽsingle unit auctions with risk-averse seller and buyers e.g., Matthews, 1979, 1983;

. Ž .Maskin and Riley, 1984 and Branco, 1996 ; and iii contract auctions with moralŽ ..hazard and adverse selection e.g., Laffont and Tirole, 1987 and Page, 1994 . In

Ž .all cases, we allow private information in the form of buyer types to bemultidimensional and stochastically dependent. Thus, we show that the existenceof optimal Bayesian auction mechanisms in a wide variety of auction environ-ments follows from the existence of an optimal Bayesian revelation mechanism fora particular Stackelberg revelation game. This is the main contribution of thepaper.

1.2. Stackelberg reÕelation games

In a Stackelberg revelation game, players’ utilities depend on the privateinformation of followers and on the choice made by the leader. Following

Ž . Ž .Harsanyi 1967–1968 see also Vickrey, 1961 and Myerson, 1979, 1982 , werepresent each follower’s private information by a parameter called the follower’s

Ž .type. In the game, the leader selects a mechanism f P :T™K which specifies theŽ .leader’s choice as a function of the h-tuple of types i.e., the private information

Ž Ž .reported by followers here f P is the mechanism, T is the set of type profilesrepresenting the private information of followers, and K is the set of choices

. Ž .available to the leader . If tgT is the type h-tuple or type profile reported byŽ .followers, then the leader is committed to making choice f t gK.

1 Ž .Recent papers on existence include Menezes and Monteiro 1995 which establishes existence in aŽ .discriminatory price auction and Lebrun 1996 which addresses the problem of existence in first price

auctions assuming risk neutrality and independent types.2 Our model shares many of the characteristics of the principal-agent model with incomplete

Ž .information e.g., Myerson, 1982; Page, 1991a, 1992 and Balder, 1996 .

( )F.H. Page Jr.rJournal of Mathematical Economics 29 1998 389–418 391

ŽIn essence, the mechanism selected by the leader determines a revelation or.reporting game to be played by the followers. The leader’s problem, then, is to

choose a feasible mechanism that maximizes the his own expected utility subjectto the constraints that the revelation game determined by the mechanism inducesvoluntary participation and truthful reporting by followers. Here, we assume thateach follower, in deciding whether or not to participate in the revelation game andin choosing a reporting strategy, seeks to maximize his own conditional expectedutility, guided by his preferences and conditional probability beliefs concerning thetypes of others—assuming other followers will participate and report honestly.Thus, in order to ensure that the revelation game determined by the mechanisminduces voluntary participation and truthful reporting, the leader must choose a

Ž .mechanism that is interim individually rational IIR and Bayesian incentiveŽ .compatible BIC .

Ž .Due to the unusual nature of the Bayesian incentive compatibility BICconstraints, the resolution of the existence question which arises in connection

Žwith the leader’s maximization problem is difficult see Page, 1989, 1991a,b,. 31992, 1994 and Balder, 1996 for additional discussion . Novel existence argu-

ments are required. Here, we show that the leader’s problem has a solutionŽ .Theorem 1 . Key ingredients in our proof are a new notion of compactness for

Ž .mechanisms, called K-compactness Definition 4.2 , an abstract Komlos TheoremŽ . Ž .due to Balder 1990, 1996 also, see Komlos, 1967 , and a new result showing

that the feasible set of interim individually rational and Bayesian incentiveŽ . 4compatible revelation mechanisms is K-compact Theorem 2 . The latter result

Žextends an earlier result on K-compactness due to the author Page, 1992,.Theorem 4.1 by allowing for mechanisms that map into quite general choice sets.

Ž .In particular, the K-compactness result given here considers mechanisms f P :T™K where the choice set K is given by a convex, compact, metrizable subset of

Ž .a Hausdorff locally convex topological space. In Page 1992 , the K-compactnessresult considers only those mechanisms where K is given by the set of probability

Ž .measures P X defined over a compact metric space X of choices. This latter setof mechanisms represents an important example of the set mechanisms covered by

Ž .the K-compactness result given here Theorem 2 . The added generality affordedby Theorem 2 makes the corresponding existence result applicable in a wide

3 Ž . Ž .Page 1989, 1991a,b, 1994 and Balder 1996 treat existence questions which arise in connectionwith the leader’s maximization problem over mechanisms which are feasible, interim individually

Ž .rational, and dominant strategy incentiÕe compatible. Page 1992 treats the more difficult case ofmaximization over mechanism which are feasible, interim individually rational, and Bayesian incentiÕe

Ž .compatible. Here, we extend the existence result of Page 1992 to cover more general environmentsand use this result to give a unified treatment of existence of optimal auctions.

4 Ž .The notion of K-compactness was introduced by Page 1989 , see Definition 2.2.2. However, theŽ .term K-compactness was not used by Page 1989 . The term K-compactness was introduced by Page

Ž .1992 , see Definition 2.3.2.


variety of environments—and thus, makes possible a unified treatment of exis-tence. The K-compactness result given here also extends the earlier result by PageŽ .1992 by taking as the feasible set of mechanisms the set of all measurableselections from a feasible choice correspondence. This refinement is useful. Insome auction applications, the feasible choice correspondence specializes to an expost budget constraint for the seller and buyers. This is the case, for example, inthe multiple unit auction model developed below. In other applications, thefeasible choice correspondence serves to define ex post individual rationality—as

Ž .is the case in the contract auction model developed below. In Page 1992 , thefeasible set of mechanisms is taken to be the set of all measurable mappings frombuyer type profiles into the set of probability measures over a compact metricspace of choices. Together, Theorems 1 and 2 provide a unified approach toexistence problems involving Bayesian incentive compatibility and extend earlierresults by the author on existence in Stackelberg games with incomplete informa-

Ž .tion e.g., Page, 1989, 1992 . This is an additional contribution of the paper.

1.3. Multiple and single unit auctions

For our first application, we specialize our revelation game to a single unitŽ .auction with risk neutral seller and buyers see Section 3.1 . The resulting model

Ž . Ž .covers the classical auction models of Myerson 1981 , Harris and Raviv 1981 ,Ž . Ž .Riley and Samuelson 1981 and Cremer and McLean 1988 . Moreover, because

we allow buyer types to be multidimensional and stochastically dependent, ourmodel also covers the auction environments analyzed in, for example, Milgrom

Ž . Žand Weber 1982 where buyer types are allowed to be affiliated also, see Cremerand McLean, 1985; McAfee and Reny, 1992; Funk, 1993; Jehiel et al., 1994; Baye

.et al., 1996 .Single unit auctions in which the seller and buyers are risk neutral and buyers’

types are stochastically independent have been intensely studied in the literatureŽsee, for example, Myerson, 1981; Riley and Samuelson, 1981 and Harris and

.Raviv, 1981 . Two main conclusions emerge from this work. First, the four mostŽ . 5common forms of auctions Dutch, first-price, second-price, and English gener-

ate the same expected revenue for the seller. Second, for many common distribu-

5 The Dutch auction is conducted by an auctioneer who initially calls for a high price and thencontinuously lowers the price until some buyer stops the auction and claims the object at that price. Inan English auction, the auctioneer begins by soliciting bids at a low price and then gradually raises theprice until only one willing buyer remains. A first-price auction is a sealed-bid auction in which thebuyer making the highest bid wins and pays the amount of his bid for the object. A second-priceauction is also a sealed-bid auction in which the buyer making the highest bid wins and pays theamount of the second highest bid.


Žtions of buyers’ types including the normal, exponential, and uniform distribu-.tions , the four standard auction forms with suitably chosen reserve prices or entry

fees are optimal from the perspective of the seller. These conclusions, however,are not robust with respect to changes in the assumptions of risk neutrality or

Ž .stochastic independence. For example, Maskin and Riley 1984 show that ifbuyers’ types are independently distributed but buyers are risk averse, then fromthe seller’s viewpoint first-price and English auctions are not revenue equivalent

Ž .—nor are they optimal. Alternatively, it follows from Milgrom and Weber 1982that if buyers and the seller are risk neutral but types are dependent—and inparticular affiliated—then English auctions generate the highest expected revenueto the seller, followed by the second-price auction, and finally the Dutch andfirst-price auctions. Even more striking are the results of Cremer and McLeanŽ . Ž .1985, 1988 and McAfee and Reny 1992 . Assuming risk neutrality, they showthat if buyers’ types are dependent then by using an optimal auction the seller canobtain the same expected payoff as that obtainable under conditions of completeinformation concerning buyers’ types.

These studies indicate the importance of analyzing auction environments inwhich both risk aversion and stochastically dependent buyers’ types are present.Thus, for our second application, we specialize our revelation game to a multiple

Žunit auction with risk averse players and stochastically dependent types see. 6Section 3.2 . The resulting model automatically covers the multiple unit auction

Ž .model of Branco 1996 with risk neutral players and stochastically independenttypes. Moreover, by further specializing to a single unit auction, our model covers

Ž . Ž .the auction models of Matthews 1979, 1983 and Maskin and Riley 1984 withrisk-averse buyers and stochastically independent types.

1.4. Contract auctions

For our final application, we consider the problem faced by a risk averseprincipal seeking to choose, via a contract auction, an agent or contractor from

Žamong several competing, risk averse agents to carry out a particular project see. Ž .Section 3.3 . Laffont and Tirole 1987, 1993 characterize optimal Bayesian

contract auctions assuming risk neutrality and independent agent types, while PageŽ . Ž .1994 and Balder 1996 demonstrate the existence of an optimal, dominant-strategy incentiÕe compatible contract auction mechanism assuming risk aversionand multidimensional, dependent agent types. Here, we establish the existence ofan optimal, interim individually rational and Bayesian incentive compatible con-

6 With further minor modifications, our model becomes a model of a simultaneous pooled auctionŽ .see Menezes and Monteiro, 1996 .


Ž .tract auction mechanism—thus, extending the existence results of Page 1994 andŽ . 7Balder 1996 to the more difficult Bayesian case.

A distinguishing feature of contract auctions is the presence of both adverseselection and moral hazard. Thus, in addition to being concerned about the adverse

Ž .selection problem as reflected in the incentive compatibility constraints , theprincipal must also be concerned about the actions taken by the agent afterentering into the contract with the principal. These actions often affect contractperformance and the economic well-being of the principal as well as the winningagent. Because these actions are only indirectly controllable via contract incen-tives, the principal must choose an auction mechanism that not only selects thebest agent but also selects a contract that provides the winning agent with correctincentives.

In specializing our revelation game to a contract auction model, we show thatthe auction problem with adverse selection and moral hazard reduces to anequivalent problem with adverse selection only. While we continue to allow riskaversion and multidimensional, dependent types, in the contract auction model

Ždeveloped here like the models of Laffont and Tirole, 1987, 1993; Page, 1994 and.Balder, 1996 , we do not allow informational externalities. This assumption is

made for simplicity. In particular, in order to avoid the ‘informed principalproblem’ which would arise if information externalities were allowed, we simplyassume that each agent’s utility depends only upon his own type. We save forfuture research the treatment of informational externalities and the informedprincipal problem in contract auctions.

1.5. Outline

We shall proceed as follows: In Section 2, we present a general model of aŽ .Stackelberg revelation game and state our main existence result Theorem 1 . In

Section 3, we consider applications. In particular, in Section 3.1, we specialize ourrevelation game to a single unit auction model with risk neutral players and we

Ž .deduce the existence of an optimal Bayesian auction mechanism Corollary 1from our main existence result. In Section 3.2, we specialize our revelation gameto a multiple unit auction model with risk averse players and deduce the existence

Ž .of an optimal Bayesian auction mechanism Corollary 2 . In Section 3.3, wespecialize our revelation game to a contract auction model and show that thecontract auction problem with adverse selection and moral hazard reduces to anequivalent auction problem with adverse selection only. We also establish the

Ž .existence of an optimal Bayesian contract auction mechanism Corollary 3 . In

7 Ž .The existence result given by Balder 1996 for dominant strategy incentive compatible auctionŽ .mechanisms refines the earlier result from Page 1994 by relaxing some of the topological and

Ž .measure-theoretic assumptions made by Page 1994 .


Section 4, within the context of our general Stackelberg revelation game, we showthat the set of all feasible, interim individually rational and Bayesian incentive

Ž .compatible revelation mechanisms is K-compact Theorem 2 . Finally, in SectionŽ .4, we prove our main existence result Theorem 1 .

2. The Stackelberg revelation game

2.1. Basic elements

We will assume that the basic elements of the game are known to all players.

2.1.1. Players and types� 4To begin, let Is 0,1,2, . . . ,h denote the set of players. Elements of I will be

denoted by i or j, with 0 denoting the leader and is1,2, . . . ,h, denotingfollowers.

Now let: T s the set of ith follower types with elements denoted by t . Equipi i

T with the s-field S ; TsT =T = . . . =T with elements denoted by tsi i 1 2 hŽ .t ,t , . . . ,t . Equip T with the product s-field SsS =S = . . . =S ; T s1 2 h 1 2 h yi

T = . . . = T = T = . . . = T with elements denoted by t s1 iy1 iq1 h y iŽ .t , . . . ,t ,t , . . . ,t . Equip T with the product s-field S sS = . . . =1 iy1 iq1 h yi yi 1

Ž . Ž .S =S = . . . =S ; and t ,t s t , . . . ,t ,t ,t , . . . ,t s t.iy1 iq1 h i yi 1 iy1 i iq1 h

2.1.2. Probability beliefsŽ < .For each follower is1, . . . ,h, let q P P denote the stochastic kernel repre-i

senting the ith follower’s conditional probability beliefs concerning other follow-Ž < .ers’ types. Thus, for each t gT , q P t is a probability measure defined on Si i i i i

specifying for each type t , the ith follower’s conditional probability beliefsiŽ < .concerning other followers’ types, and for each EgS , q E P is a real-valued,yi i

S -measurable function defined on T specifying for each of the ith follower’si i

types, the probability weight the ith follower assigns to subset E.Ž .For the leader, let p be a probability measure defined on T , S , the measure0

space of type profiles, representing the leader’s probability beliefs concerningfollowers’ type profiles.

Assumption 1:

We will assume that there is a product measure msm = . . . =m defined on1 hŽ . Ž .T , S with each m s-finite on S , such that: 1 the leader’s probability measurei i

Ž . Ž .p over T is absolutely continuous with respect to m denoted p <m ; and 20 0Ž < .for each i and t gT , the ith follower’s conditional probability measure q P ti i i

Ž Ž < . .over T is absolutely continuous with respect to m denoted q P t <m ,yi yi i i yi


where m is the product measure m = . . . =m =m = . . . =m . We willyi i iy1 iq1 h

refer to m as the dominating measure.Assumption 1 is satisfied automatically in any revelation game in which the

leader’s probability beliefs concerning type profiles are specified via a probabilitya density function and followers’ conditional probability beliefs are specified via a

Žconditional probability density function i.e., Lebesgue measure then serves as the.dominating measure .

w .Example: Suppose there are two followers, is1,2, with types T s O,` for eachiw . w .i. Thus, the set of type profiles is given by Ts 0,` = 0,` . Let S be the Borel

w . w . Ž .product s-field, B 0,` =B 0,` , in T and equip T , S with the Lebesgueproduct measure msm =m . Suppose now that the leader’s probability beliefs1 2

Ž .p are given via a joint density function h P , defined on T , so that for EgS,0 0

p E s h t m d t .Ž . Ž . Ž .H0 0E

Thus, p <m. Suppose also that each follower’s conditional probability beliefs0Ž < . Ž < .q P t are given via a conditional density r P t corresponding to a jointi i i i

Ž .probability density h P so that for SgS ,i yi

< <q S t s r t t m d t .Ž .Ž . Ž .Hi i i yi i yi yiS

Ž < .Thus, q P t <m for all t gT .i i yi i i

2.1.3. Choices and reÕelation mechanismsLet K be a subset of a Hausdorff locally convex topological space E. Elements

of K , denoted by f , will be called choices and measurable functions from typeprofiles T into K will be called revelation mechanisms. If the leader selects

Ž . Ž .revelation mechanism f P :T™K , then the leader is committed to choice f t gKwhenever followers report type profile tgT. Here, measurability is defined with

Ž .respect to the product s-field S on T and the Borel s-field B K on K. InŽ . Ž . y1Ž .particular, f P :T™K is said to be measurable if for each EgB K , f E :s

� Ž . 4 Ž .tgT : f t gE gS. We will denote by M T , K the set of all revelationmechanisms.

Assumption 2:

We shall assume that the set of choices K is convex, compact, and metrizable forthe relative topology inherited from E.

Ž .For each h-tuple tgT of reported types, let K t be the set of feasible choicesin K given reported types t.


Assumption 3:

Ž . KWe will assume that the feasible choice correspondence K P :T™2 is such thatŽ .for each tgT , K t is a closed convex subset of K. A revelation mechanism

Ž . Ž .f P gM T , K is said to be feasible if:

w xf t gK t a.e. m .Ž . Ž .Ž .Note that the qualifier ‘a.e.’ almost everywhere is with respect to the dominating

Ž . Ž .measure, m see Assumption 1 . Thus, for each feasible mechanism, f P , theleader and the followers will agree upon the set of type profiles where themechanism may fail to produce a feasible choice and all players will assign thisset probability measure zero.

We shall denote by G the feasible set of revelation mechanisms. Thus,

w xGs f P gM T , K : f t gK t a.e. m . 1� 4Ž . Ž . Ž . Ž . Ž .Ž .In some applications, the mapping K P specializes to an ex post budget constraint

for the leader and followers. This is the case in the multiple unit auction modelŽ .developed in Section 3.2. In other applications, K P serves to define ex post

individual rationality. This is the case in the contract auction model developed inSection 3.3.

2.1.4. Followers’ utility functionsŽ .For is1,2, . . . ,h, let Õ P,P :T=K™R be the ith follower’s utility function.i

Ž .Given type profile tgT and choice fgK , Õ t, f is the corresponding utilityi

level for the ith follower.

Assumption 4:

Ž . Ž .For each is1,2, . . . ,h, we will assume that: 1 Õ t,P is affine and continuousiŽ . Ž . Ž Ž .. Ž . Ž .on K t for each tgT ; 2 Õ P, f P is S-measurable for each f P gM T , K ;iŽ . Ž < . Ž .and 3 for each t gT there exists a q P t -integrable function j P :T ™Ri i i i t yii

< Ž Ž .. < Ž . w Ž < .x Ž . Ž .such that Õ t ,t , f t ,t Fj t a.e. q P t on T for f P gM T , K .i i yi i yi t y i i i yii

2.1.5. Bayesian incentiÕe compatibility and interim indiÕidual rationalityŽ .Let B denote the subset of mechanisms in M T , K such that corresponding to

Ž . Ž .each f P gB, there are h sets C , . . . ,C , . . . ,C , with C gS and m C s0,1 i h i i i i

such that for each follower is1,2, . . . ,h:

< X <Õ t ,t , f t ,t q d t t G Õ t ,t , f t t q d t t 2Ž . Ž . Ž .Ž . Ž .Ž . Ž .H Hi i yi i yi i yi i i i yi i yi i yi iT Tyi y i

for all t gT _C and all tX gT . Note that the measure m is the ith component ofi i i i i iŽ .the dominating product measure msm = . . . =m see Assumption 1 . Under1 h


any revelation mechanism in B, given each follower’s conditional probabilitybeliefs concerning other followers’ types, and given almost any follower type,each follower will do at least as well by reporting his true type t to thei

mechanism as by reporting tX given that all other followers report truthfully. WeiŽ .will refer to B as the set of Bayesian incentiÕe compatible BIC mechanisms.

Thus, for any mechanism in B, truthful reporting is a Nash equilibrium of therevelation game induced by the mechanism.

Ž .Let L denote the subset of mechanisms in M T , K such that corresponding toŽ . Ž .each f P gL, there are h sets Q , . . . ,Q , . . . ,Q with Q gS and m Q s0,1 i h i i i i

such that for each follower is1,2, . . . ,h:

<Õ t ,t , f t ,t q d t t Gr t 3Ž . Ž . Ž .Ž .Ž .H i i yi i yi i yi i i iTyi

Ž .for all t gT _Q . Here r P :T ™R is the ith follower’s interim reservationi i i i i

utility function. Under any revelation mechanism in L, given each follower’sconditional probability beliefs concerning other followers’ types, and given almostany follower type, each follower will do at least as well by participating in themechanism as by abstaining from participation, given that all other followersparticipate and report truthfully. We will refer to L as the set of interim

Ž .indiÕidually rational IIR mechanisms.

Assumption 5:

Ž . Ž .We shall assume the following: 1 GlBlL is nonempty; and 2 The set ofchoices K contains an f such that for each follower is1,2, . . . ,h:

<Õ t ,t , f q d t t Fr t for all t gT .Ž .Ž .Ž .H i i yi i yi i i i i iTyi

Ž .We shall refer to any choice fgK satisfying Assumption 5 2 as a penaltychoice.

Ž .Assumption 5 1 is nontriviality assumption: without it, the revelation game isŽ .uninteresting. Assumption 5 1 will be satisfied if, for example, the set K of

Y Y Ž .choices contains an f such that for each tgT , f gK t and for each followeris1,2, . . . ,h:

Y <Õ t ,t , f q d t t Gr t for all t gT .Ž . Ž .Ž .H i i yi i yi i i i i iTyi

2.1.6. The leader’s utilityŽ .Now, let u P,P :T=K™R be the leader’s utility function. Given type profile

Ž .tgT and choice fgK , u t, f is the corresponding level of utility for the leader.


Assumption 6:

Ž . Ž . Ž .We will assume that: 1 u t,P is concave and upper-semicontinuous on K t forŽ . Ž Ž .. Ž . Ž .each tgT ; 2 u P, f P is S-measurable for each f P gGlBlL; and 3

Ž . Ž Ž .. Ž . w xthere exists a p -integrable function z P such that u t, f t Fz t a.e. p on T0 0Ž .for f P gGlBlL.

2.2. The leader’s problem and the existence result

Ž .The leader’s problem is to choose a revelation mechanism f P from the set offeasible, interim, individually rational and Bayesian incentive compatible mecha-nisms GlBlL that maximizes the leader’s expected utility:

u t , f t p d t .Ž . Ž .Ž .H 0T

Stated compactly, the leader’s problem is given by:

max u t , f t p d t . 4Ž . Ž . Ž .Ž .Hf ŽP.g G l B l L 0T

( )Theorem 1 Existence of an Optimal Revelation Mechanism :

Under Assumptions 1–6, there exists an optimal reÕelation mechanism solÕing the( ( ))leader’s problem Eq. 4 .

3. Auctions

By carefully specifying the choice set K , the feasible choice correspondence,Ž . KK P :T™2 , and the leader’s and followers’ utility functions, many auction

Ž Ž ..problems can be formally treated as special cases of the leader’s problem Eq. 4 .Thus, in many auction environments, the existence of an optimal auction mecha-

Ž .nism for the seller acting as the leader can be deduced from Theorem 1. In orderto illustrate this, we begin by considering single unit auctions of an indivisiblegood.

3.1. Bayesian single unit auctions with risk neutral players

For our first application, we specialize our revelation game to a single unitauction with risk neutral seller and buyers. In particular, we consider a sellerŽ . Ž .leader who faces h risk neutral potential buyers followers and is interested in

Ž .adopting an auction procedure a revelation mechanism that maximizes theexpected revenue from the sale of a single object. The resulting model covers the

Ž . Ž .classical auction models of Myerson 1981 , Harris and Raviv 1981 , Riley and


Ž . Ž .Samuelson 1981 and Cremer and McLean 1988 . Moreover, because we allowbuyer types to be multidimensional and stochastically dependent, our model alsocovers the auction environments analyzed in, for example, Milgrom and WeberŽ . Ž1982 where buyer types are allowed to be affiliated also, see Cremer andMcLean, 1985; McAfee and Reny, 1992; Funk, 1993; Jehiel et al., 1994 and Baye

.et al., 1996 .We begin by assuming that each players’ probability beliefs concerning type

profiles satisfy Assumption 1. A standard assumption in the auction literature isŽthat buyers’ types are identically and independently distributed e.g., see Myerson,

.1981 and Riley and Samuelson, 1981 . Here, we depart from this assumption,allowing for stochastically dependent types.

Ž . Ž .Following Myerson 1981 see also Page, 1991b , a single unit auctionŽ . Ž .mechanism is a pair of S-measurable functions, p P :T™D, x P :T™C, such

Ž . Ž .that if ts t , . . . ,t is the h-tuple of reported buyer types then p t is the1 h iŽ .probability that the ith buyer wins the object and x t is the amount of money thei

ith buyer must pay to the seller. Here, C is a closed bounded convex subset of Rh

containing the origin and:

hhDs p , . . . , p gR : p G0 and p F1 . 5Ž . Ž .Ý1 h i i½ 5

is1

Ž h h.Thus, the choice set K is given by KsD=C EsR =R , and the set ofŽ .revelation mechanisms M T ,D=C consists of functions of the form:

f P s p P , x P s p P , . . . , p P , x P , . . . , x P . 6Ž . Ž . Ž . Ž . Ž . Ž . Ž . Ž .Ž . Ž .1 h 1 h

Ž . D=CWe assume that the feasible choice correspondence K P :T™2 is given byŽ .K t sD=C for all t. Thus, Assumptions 2 and 3 are satisfied.

Ž . ŽAs in Myerson 1981 , each buyer is risk neutral with utility function Õ P,P,Pi. Ž .:T= D=C ™R given by:

Õ t , p , x sp Py t yx . 7Ž . Ž . Ž .Ž .i i i i

Ž .We also assume that the h-tuple of buyer valuation functions y P sŽ Ž . Ž ..y P , . . . ,y P is such that the range of valuations:1 h

V :s y t , . . . ,y t gRh :tgT , 8Ž . Ž . Ž .� 4Ž .1 h

is contained in C and that each buyer’s valuation function is S-measurable. TheŽ .seller—also risk neutral—has valuation function y P :T™R, bounded and S-0

Ž . Ž .measurable, and utility function u P,P,P :T= D=C ™R given by:

h h

u t , p , x s 1y p Py t q x . 9Ž . Ž . Ž .Ž . Ý Ýi 0 iž / ž /is1 is1

Thus, Assumptions 4 and 6 are satisfied.


To complete our description of the single unit auction model, we assume thatŽ .each buyer’s interim reservation utility function r P :T ™R is identically zero.i i

Ž Ž . Ž .. Ž .A single unit auction mechanism p P , x P gM T ,D=C is Bayesian incen-Ž .tive compatible i.e., is contained in the set B if there are h sets C , . . . ,C , . . . ,C1 i h

Ž .with C gS and m C s0, such that for each buyer is1,2, . . . ,h:i i i i

<p t ,t Py t ,t yx t ,t q d t tŽ . Ž . Ž . Ž .H i i yi i i yi i i yi i yi iTyi

X X <G p t ,t Py t ,t yx t ,t q d t t 10Ž . Ž . Ž . Ž .Ž .H i i yi i i yi i i yi i yi iTyi

for all t gT _C and all tX gT .i i i i iŽ Ž . Ž .. Ž .A single unit auction mechanism p P , x P gM T ,D=C is interim individ-

Ž .ually rational i.e., is contained in the set L if there are h sets Q , . . . ,Q , . . . ,Q1 i hŽ .with Q gS and m Q s0, such that for each buyer is1,2, . . . ,h:i i i i

<p t ,t Py t ,t yx t ,t q d t t G0 11Ž . Ž . Ž . Ž .Ž .H i i yi i i yi i i yi i yi iTyi

Ž .for all t gT _Q , where Q gS and m Q s0.i i i i i i iŽ Ž . Ž .. Ž Ž . Ž .. Ž .Note that the auction mechanism p P , x P given by p t , x t s 0,0 for

all t is feasible, interim individually rational and Bayesian incentive compatibleŽ . Ž Ž . Ž ..i.e., is contained in GlBlL . Note also that under the mechanism p P , x P ,we have for each buyer:

<p t ,t Py t ,t yx t ,t q d t t s0 for all t gT .Ž . Ž . Ž . Ž .H i i yi i i yi i i yi i yi i i iTyi

12Ž .

Thus, Assumption 5 is satisfied.The seller’s single unit auction problem can be stated compactly as:

h h

max 1y p t Py t q x t p d t .Ž . Ž . Ž . Ž .Ý ÝHŽ pŽP. , xŽP..g G l B l L i 0 i 0ž / ž /T is1 is1

13Ž .Ž Ž ..It is easy to see that the seller’s single unit auction problem Eq. 13 is a version

Ž Ž ..of the leader’s problem Eq. 4 . Moreover, because our single unit auction modelsatisfies Assumptions 1–6, it follows directly from Theorem 1 that the seller’sproblem has a solution.

( )Corollary 1 Existence of an Optimal Single Unit Auction Mechanism :

GiÕen the single unit auction model aboÕe, there exists an auction mechanism( ) ( ) ) ( )) ( ( ))p P ,x P gGlBlL solÕing the seller’s problem Eq. 13 .


The result aboÕe extends to the Bayesian case the single unit existence resultfor dominant strategy incentiÕe compatible auction mechanisms giÕen by Page( )1991b .

3.2. Bayesian multiple unit auctions with risk aÕerse players

Now, consider a risk averse seller who faces h risk averse buyers and isinterested in adopting an auction procedure that maximizes his expected utilityfrom the sale of seÕeral indivisible units of a homogeneous good.

Again, we assume that players’ probability beliefs concerning type profilessatisfy Assumption 1. As indicated by our example in Section 2, this assumption iseasily satisfied.

Let m denote the total number of units the seller wishes to put up for auction,and let:

� 4Qs 0,1,2, . . . ,m

Ž .Each player, is0,1, . . . ,h seller as well as buyers , in the auction can win from 0to m units of the good. Now, let:

Ž .with typical element zs z , z , . . . , z . Each hq1-tuple zgZ gives a profile of0 1 h

winnings by the players in the auction. Note that if z )0, then the seller keeps0Ž .some of the good i.e., some units are not sold during the auction .� 4Now, let Cs cgR:0FcFw for some large positive number w, and for

Ž .each buyer i and buyer type t gT , let w t gC and:i i i i

C t s cgR :0FcFw t .� 4Ž . Ž Ž .i i i i

Ž .Here w t represents the maximum amount a type t ith buyer can pay at auction.i i iŽ . CThe mapping C P :T ™2 is the ith buyer’s budget correspondence. Now, let:i i

Ž .The set D equipped with the usual Euclidean metric here, denoted by d is aD

compact metric space. Finally, for each type profile t, let:

D t sC t = . . . =C t .Ž . Ž . Ž .1 1 h h

Ž . Ž .Each h-tuple cs c , . . . ,c gD t represents a profile of payments from buyers1 hŽ . Dto the seller given reported types t and D P :T™2 is the ex post budget

correspondence for buyers.


Ž . Ž . Ž .A tuple z,c s z , z , . . . , z ,c , . . . ,c gZ=D t is a feasible outcome of0 1 h 1 h

the auction if the sum of the units allocated via the auction S h z equals m andis0 i

if the sum of the payments from buyers to seller S h c equals or exceeds someis1 iŽ .minimum revenue requirement r t , dependent on the types reported by buyers.0

Formally, the feasible set of auction outcomes corresponding to each reported typeprofile t is given by:

F t s z ,c s z , z , . . . , z ,c , . . . ,cŽ . Ž . Ž .0 1 h 1 h½h h

gZ=D t : z sm and c Gr t . 14Ž . Ž . Ž .Ý Ýi i 0 5is0 is1

Ž . 8Note that for each t, F t is a closed subset of the compact metric space Z=D.We are now in a position to define the choice set K and the feasible choice

Ž . Kcorrespondence, K P :T™2 corresponding to the multiple unit auction. LetŽ .P Z=D denote the set of all probability measures defined on the product s-fieldŽ . Ž .I Z =B D of the compact metric space Z=D of auction outcomes. Here,Ž . Ž .B D denotes the Borel s-field generated by the Euclidean metric d and I ZD

denotes the s-field consisting of all subsets of the finite set Z. Also, let the set ofŽ .choices K be given by P Z=D . Here, the space E is taken to be the set of all

Žbounded signed measures on Z=D equipped with the narrow topology i.e., thetopology of weak convergence measures—see Dellacherie and Meyer, 1975,

. Ž .III.54 . The choice set KsP Z=D is clearly convex. By III.60 in DellacherieŽ . Ž .and Meyer 1975 , KsP Z=D satisfies Assumption 2. Denote by w a typical

Ž . Ž . Ž Ž . < .element of the choice set P Z=D and denote by w P or w d z,c P a typicalŽ . Ž Ž ..element of the choice set of auction revelation mechanisms M T , P Z=D .

Ž . KLet the feasible choice correspondence, K P :T™2 be given by:

K t :s wgP Z=D :w F t s1 . 15� 4Ž . Ž . Ž . Ž .Ž .Ž .Given type profile t, any probability measure wgK t assigns probability 1 to the

Ž . Ž . Ž .set of auction outcomes F t . For each t, K t is convex. Thus, since each F t isŽ .compact, it follows from III.58 and III.60 in Dellacherie and Meyer 1975 that the

Ž . K Ž .feasible choice correspondence K P :T™2 defined via Eq. 6 satisfies Assump-tion 3. The feasible set of auction mechanisms is thus given by:

w xGs w P gM T , P Z=D :w t gK t a.e. m . 16� 4Ž . Ž . Ž . Ž . Ž .Ž .Ž .For each potential buyer is1, . . . ,h let y P,P,P :T=Z=D™R be the ithi

buyer’s utility function over type profiles and auction outcomes. We will assume

8 The results of this section remain unchanged if in the definition of the set of feasible outcomes weh Ž . Ž Ž ..require S c s r t i.e., we require the sum of buyer payments to the seller to equal r t . Withis1 i 0 0

Ž .this modification, the feasible set of auction outcomes F t remains a closed subset of the compactmetric space Z= D.


Ž . Ž . Ž . Ž Ž ..that: a y t,P,P is continuous on Z=D for each tgT ; b y P, z,c isi iŽ . Ž .S-measurable on T for each feasible auction outcome z,c gZ=D; and c for

Ž < . Ž .each t gT there exists a q P t -integrable function c P :T ™R such thati i i i t yii

< Ž Ž .. < Ž . Ž Ž ..y t ,t , z,c Fc t for all t , z,c gT =Z=D.i i yi t i yi yiyi

Ž . Ž .Given choice wgP Z=D , the ith buyer’s expected utility is given by:

Õ t ,w sH y t , z ,c w d z ,c . 17Ž . Ž . Ž . Ž .Ž . Ž .i Z=D i

Ž . Ž .For each tgT , w™Õ t,w is continuous on P Z=D with respect to theiŽ . Žnarrow topology Dellacherie and Meyer, 1975, III.54 , and for each wgP Z=

. Ž .D , t™Õ t,w is S-measurable on T. Thus, each buyer’s expected utilityiŽ . Ž . Ž Ž .. Žfunction Õ P,P :T=P Z=D ™R is S=B P Z=D -measurable Castaingi

. Ž Ž ..and Valadier, 1977, III. 14 . Here, B P Z=D denotes the Borel s-field inŽ .P Z=D generated by the narrow topology. Given these observations and the fact

Ž . Ž . Ž .that each buyer’s utility function, y P,P,P , satisfies assumptions a – c above, itiŽ . Ž .is easy to verify that each buyer’s expected utility function Õ P,P :T=P Z=Di

Ž .™R given in Eq. 17 satisfies Assumption 4.Ž . Ž Ž ..Under auction mechanism w P gM T , P Z=D , the ith buyer’s interim

expected utility given true type t gT and reported type tX gT is:i i i i

X <Õ t ,t ,w t t q d t t . 18Ž . Ž .Ž .Ž .H i i yi i yi i yi iTyi

Ž . Ž Ž ..A multiple unit auction mechanism w P gM T , P Z=D is Bayesian incen-Ž .tive compatible i.e., is contained in the set B if there are h sets C , . . . ,C , . . . ,C ,1 i h


< <H H y t ,t , z ,c w d z ,c t ,t q d t tŽ . Ž . Ž .Ž . Ž .T Z=D i i yi i yi i yi iyi

< X <GH H y t ,t , z ,c w d z ,c t ,t q d t t , 19Ž . Ž . Ž .Ž .Ž . Ž .T Z=D i i yi i yi i yi iyi

for all t eT _C and all tX gT .i i i i iŽ . Ž Ž ..A multiple unit auction mechanism w P gM T , P Z=D is interim individ-

Ž .ually rational i.e., is contained in the set L if there are h sets Q , . . . ,Q , . . . ,Q ,1 i hŽ .with Q gS and m Q s0, such that for each buyer is1,2, . . . ,h:i i i i

< <H H y t ,t , z ,c w d z ,c t ,t q d t t Gr t , 20Ž . Ž . Ž . Ž .Ž .Ž . Ž .T Z=D i i yi i yi i yi i i iyi

for all t gT _Q .i i iŽ .We shall assume that for auction outcome m,0, . . . ,0,c , . . . ,c in which the1 h

seller keeps all units of the good and the ith buyer pays a positive amount c ,iŽ . Ž .c , . . . ,c gD t for all t and:1 h

<H y t ,t , m ,0, . . . ,0,c , . . . ,c q d t t Fr t for all t gT .Ž .Ž .Ž .Ž .T i i yi 1 h i yi i i i i iyi


Ž . Ž .Thus, F t is nonempty for all t and Assumption 5 2 is satisfied. We shall alsoY Ž . Y Ž .assume that there is a probability measure w gP Z=D such that w gK t

w xa.e. m and for each buyer is1,2, . . . ,h:

Y < w xy t ,t , z ,c w d z ,c q d t t Gr t a.e. m .Ž . Ž . Ž .Ž . Ž .Ž .H H i i yi i yi i i i iT Z=Dy1

Ž .Thus, Assumption 5 1 holds.Ž .The buyer’s underlying utility functions y P,P,P :T=Z=D™R can takei

several different forms. For example, we might suppose that:

y t , z ,c sr t , z yc ,Ž . Ž .Ž .i i i i

Ž .where r t, z is the value to the ith buyer of z units of the good giveni i i

information t and c is the total amount the ith buyer must pay to the seller toi

obtain z units. Specializing further, we might assume that:i

y t , z ,c sß t Pz yc ,Ž . Ž .Ž .i i i i

Ž .where ß t is the per unit value of the good to the ith buyer given information t.iŽ .To complete our description of the multiple unit auction, let p P,P,P :T=Z=

D™R be the seller’s utility function over type profiles and auction outcomes. WeŽ . Ž .will assume that: a p t,P,P is upper semicontinuous on Z=D for each tgT ;

Ž . Ž . Ž . Ž . Ž .b p P,P,P is S=I Z =B D -measurable; and c there exists a p -integrable0Ž . Ž Ž .. Ž . Ž Ž ..function c P :T™R such that p t, z,c Fc t for all t, z,c gT=Z=D.0 0

Ž . Ž .Given choice wgP Z=D , the seller’s expected utility is given by:

u t ,w s p t , z ,c w d z ,c . 21Ž . Ž . Ž . Ž .Ž . Ž .HZ=D

Ž . Ž .For each tgT , w™u t,w is upper semicontinuous on P Z=D with respect toŽ . . Ž . Ž .the narrow topology Nowak, 1984 , Lemma 1.5 . Moreover, t,w ™u t,w is

Ž Ž .. Ž .S=B P Z=D -measurable Nowak, 1984, Lemma 1.6 . Given these observa-Ž .tions and the fact that the seller’s utility function, p P,P,P , satisfies assumptions

Ž . Ž .a – c above, it is easy to verify that the seller’s expected utility functionŽ . Ž . Ž .u P,P :T=P Z=D ™R given in Eq. 21 satisfies Assumption 6.

Ž .Under auction mechanism w P gGlBlL, the seller’s expected utility is:

u t ,w t p d t . 22Ž . Ž . Ž .Ž .H 0T

Ž .The seller’s underlying utility function p P,P :T=Z=D™R can also takeseveral different forms. For example, we might suppose that:

h

p t , z ,c sr t , z q c yh t , z ,Ž . Ž . Ž .Ž . Ž .Ý0 o i i iis1

Ž . Žwhere r t, z is the value to seller of z units of the good not sold at0 0 0. Ž Ž ..auction—but kept by the seller given information t, c yh t, z is the profiti i i


earned by the seller from the sale of z units of the good to the ith buyer at a priceih Ž Ž ..of c given information t, and S c yh t, z is the total profit earned by thei is1 i i i

h Ž .seller from the sale of S z units to the buyers. Here the function h t, z canis1 i i i

be interpreted as the cost to the seller of supplying z units of the good to the ithi

buyer given information t.It is easy to see that the seller’s multiple unit auction problem is a version of

Ž Ž ..the leader’s problem Eq. 4 . Moreover, because our multiple unit auction modelsatisfies Assumptions 1–6, it follows directly from Theorem 1 that the seller’sproblem has a solution.

( )Corollary 2 Existence of an Optimal Multiple Unit Auction Mechanism :

Given the multiple unit auction model above, there exists an auction mecha-) Ž .nism w P gGlBlL solving the seller’s problem:

<max p t , z ,c w d z ,c t p d t . 23Ž . Ž . Ž . Ž .Ž . Ž .HHwŽP.g G l B l L 0T Z=D

3.2.1. Single unit auctions with risk aÕerse players: a special caseThe multiple unit auction model above can be easily specialized to the single

unit case. Keeping all elements in the model above the same and letting ms1Ž .recall m is the total number of units the seller puts up for auction , we obtain asingle unit auction model with risk averse players satisfying Assumptions 1–6.Existence for the single unit case with risk aversion then follows immediately

� 4from Corollary 2. Note that with ms1 the set Q becomes Qs 0,1 and a typicalfeasible auction outcome has the form:

z ,c s 0, . . . ,0,1,0, . . . ,0,c ,c , . . . ,c ,Ž . Ž .1 2 h

Ž .where the 1 in the ith component indicates that player i for is0,1, . . . ,h haswon the auction and therefore receives the single unit being auctioned off.

3.3. Bayesian contract auctions

Now, we consider the problem faced by a principal seeking to choose, via acontract auction, an agent or contractor from among h competing agents to carryout a particular project.

We begin by assuming that each player’s probability beliefs concerning typeprofiles satisfies Assumption 1.

Working backward, we first analyze the winning agent’s action choice problemŽ .i.e., the moral hazard stage . Let A be a compact metric space of actionsavailable to agents and let F be a compact metric space of contracts available to

Ž .the principal. For each agent is1,2, . . . ,h, let y P,P,P :T =F=A™R be thei i

ith agent’s utility defined over types T , contracts F , and actions A. Note that herei


the ith agent’s utility depends only on his own type and not on the types of others.Ž . AFinally for is1,2, . . . ,h let A P :T ™2 be a set-valued mapping specifying fori i

agent i the set of actions available to the agent depending on his type. We shallŽ . Ž . Ž . Ž .assume that: a y t ,P,P is continuous on F=A for each t gT ; b y P,s,a isi i i i i

Ž . Ž . Ž .S -measurable on T for each contract–action pair s,a gF=A; and c A P isi i i

S -measurable.i

Suppose now that the ith agent wins the auction and is awarded contractsgF . If the ith agent is type t gT , he will choose an action from the feasiblei i

Ž .set A t so as to solve the problem:i i

max y t ,s,a . 24Ž . Ž .ag A Ž t . i ii i

Let:

y ) t ,s :smax y t ,s,a . 25Ž . Ž . Ž .i i ag A Ž t . i ii i

) Ž .The quantity y t ,s is a type t ith agent’s optimal expected utility underi i i) Ž . Ž . Žcontract sgF . The function y P,P is S =B F -measurable on T =F seei i i

. ) Ž . Ž .Schal, 1974 , and y t ,P is continuous on F for t gT see Berge, 1963 . Herei i i iŽ .B F is the Borel s-field in the compact metric space of contracts F .

The ith agent’s reaction correspondence given by:

A) t ,s :s agA t :y t ,s,a Gy ) t ,s . 26� 4Ž . Ž . Ž . Ž . Ž .i i i i i i i i

Ž . ) Ž . Ž .By Lemma 1.10 of Nowak 1984 , A P,P is S =B F -measurable on T =Fi i iŽ . ) Ž .and by Theorem 1 of Robinson and Day 1974 , A t ,P is upper semicontinuousi i

on F for t gT .i i

Suppose now that type profile t is reported by agents and that agent i wins theauction, is type t , and is awarded contract s. The principal’s next problem then isi

) Ž .to request that the winning agent take an action from the set A t ,s that is in thei i

principal’s best interest. Thus, the principal’s problem is to solve:

max ) p t ,s,a . 27Ž . Ž .ag A Ž t , s.i i

) Ž .Note that any action request by the principal from the set A t ,s is incentivei i

compatible provided the winning agent is type t and contract s is awarded.iŽ .We shall assume that the principal’s utility function p P,P,P :T=F=A™R

Ž . Ž . Ž .is such that: a p t,P,P is upper semicontinuous on F=A for each tgT ; bŽ . Ž . Ž . Ž .p P,P,P is S=B F =B A -measurable; and c there exists a p -integrable0

Ž . Ž . Ž . Ž .function c P :T™R such that p t,s,a Fc t , for all t,s,a gT=F=A.0 0

Now, let:

p ) t , i ,s smax ) p t ,s,a . 28Ž . Ž . Ž .Ž . ag A Ž t , s.i i

) Ž Ž ..Thus p t, i,s is the principal’s optimal utility if agent i wins the auction, is) Ž .awarded contract s, and takes the action in A t ,s requested by the principal.i i


ŽIn order to take into account the possibility that no contracting occurs i.e., that.no agent wins the auction , we must extend the definition of the principal’s utility

function. To begin, let:

g ) t , j,s sp ) t , j,s 1yh j qg t h j , 29Ž . Ž . Ž . Ž . Ž . Ž .Ž . Ž . Ž .0 0 0 0

where:

1 if js0h j sŽ .0 ½0 if j/0.

Ž .Given truthfully reported types t, if agent j wins the auction j/0 and is) Ž Ž .. ) Ž Ž ..awarded contract s, the principal’s utility is given by g t, j,s sp t, j,s ,0

Ž . ) Ž Ž ..and if no contracting occurs js0 , the principal’s utility is given by g t, j,s0Ž . Ž .sg t . We shall assume that g P :T™R is S-measurable and bounded from0 0

above by some p -integrable function. Thus, the principal’s utility function0) Ž Ž .. Ž . ) Ž .g P, P,P :T= I=F™R is such that: a g t,P,P is upper semicontinuous on0 0

Ž . ) Ž . Ž . Ž . 9 Ž .I=F for each tgT ; b g P,P,P is S=I I =B F -measurable ; and c0) Ž .g P,P,P is bounded from above on T= I=F by some p -integrable function.0 0

In order to take into account the possibility that the ith agent looses the auction,we must also extend the definition of agents’ utility functions. For agent is1, . . . ,h, let:

g ) , t , j,s sy ) t ,s h j qg t 1yh j , 30Ž . Ž . Ž . Ž . Ž . Ž .Ž . Ž .i i i i i i i i

where:

1 if js ih j sŽ .i ½0 if j/ i .

Ž .Given that agent i is type t , if agent i wins the auction js i and is awardedi) Ž Ž .. ) Ž .contract s, agent i’s utility is given by g t , j,s sy t ,s , and if agent ii i i i

Ž . ) Ž Ž .. Ž .looses the auction j/ i , agent i’s utility is given by g t , j,s sg t . Notei i i i) Ž Ž ..that for is1, . . . ,h, the utility function g P, P,P :T = I=F™R is such thati i

) Ž . Ž � 4.g t ,P,P is continuous on I=F for each t gT recall that is 0,1, . . . ,h .i i i i

For each agent is1, . . . ,h, we shall assume that:

g t Fr t for all t gT ,Ž . Ž .i i i i i i

Ž .where r P :T ™R is the ith agent’s interim reservation utility function. Thus ifi i

agent i looses the auction, his utility level is at most his interim utility level.For each type profile tgT , let:

F t s j,s g I=F : js1, . . . ,h and y ) t ,s Ge tŽ . Ž .� 4Ž . Ž .j j j j

j 0,s : sgF . 31� 4Ž . Ž .Ž .Here, e P :T ™R is the jth agent’s ex post reservation utility function, andj j

9 Ž .Here I I denotes the s-field consisting of all subsets of the finite set I.


) Ž .y t ,s is the jth agent’s optimal level of utility under contract s given type tj j jŽ Ž .. X Xsee Eq. 25 . If agent j wins the auction and is awarded contract s and if:

jX ,sX g j,s g I=F : js1, . . . ,h and y ) t ,s Ge t ,Ž . Ž .� 4Ž . Ž .j j j j

then agent jX will be able to achieve his reservation utility level under the awardedX Ž .contract s via an optimal action choice from A t . Thus, from the perspective ofi i

X Ž X X .the winning agent j , auction outcome j ,s is ex post rational—that is, undercontract sX, it is rational for agent jX to enter into the contract with the principal.

Ž Y . YAlternatively, if the auction outcome is 0,s for some s gF , then no agent isŽ .selected and hence no contracting occurs. The set F t is the feasible set of

Ž .auction outcomes given truthfully reported types t. Moreover, for each t, F t is aclosed subset of the compact metric space I=F of playerrcontract pairs. This

Ž .follows from the definition of F t and the fact that for each tgT , the functionŽ . ) Ž . � 4 Ž .j,s ™y t ,s is continuous on 1,2, . . . ,h =F and the function j™e t isj j j j

� 4continuous on 1,2, . . . ,h .We are now in a position to define the choice set K and the feasible choice

Ž . Kcorrespondence, K P :T™2 corresponding to the contract auction. To begin, letŽ .P I=F denote the set of all probability measures defined on the product s-fieldŽ . Ž . ŽI I =B F of the compact metric space I=F of auction outcomes i.e.,

. Ž .playerrcontract pairs . Here, B F denotes the Borel s-field generated by theŽ .metric on the contract space F and I I denotes the s-field consisting of all

Ž .subsets of the finite set I. Also, let the set of choices K be given by P I=F .Here, the space E is taken to be the set of all bounded signed measures on I=F

Ž .equipped with the narrow topology. The set of choices KsP I=F is clearlyŽ . Ž .convex. By III.60 in Dellacherie and Meyer 1975 , KsP I=F satisfies

Ž .Assumption 2. Denote by w a typical element of the choice set P I=F , and byŽ . Ž Ž . < . Ž .w P or w d j,s P a typical element of the set of auction revelation mecha-

Ž Ž ..nisms M T , P I=F .Ž . KNow, let the feasible choice correspondence, K P :T™2 be given by:

K t :s wgP I=F :w F t s1 . 32� 4Ž . Ž . Ž . Ž .Ž .Ž .Given type profile t, probability measure wgK t assigns probability 1 to the set

Ž . Ž . Ž .of playerrcontract pairs F t . For each t, K t is convex. Since each F t isŽ .compact, it follows from III.58 and III.60 in Dellacherie and Meyer 1975 that the

Ž . K Ž .feasible choice correspondence K P :T™2 defined via Eq. 32 above satisfiesŽAssumption 3. The feasible set of auction mechanisms i.e., the set of ex post

.individually rational auction mechanisms is then given by:

w xGs w P gM T , P I=F :w t gK t a.e. m . 33� 4Ž . Ž . Ž . Ž . Ž .Ž .Ž . Ž .Given choice wgP I=F , the ith agent’s expected utility is given by:

Õ t ,w sH g ) t , j,s w d j,s . 34Ž . Ž . Ž . Ž .Ž .Ž .i i I=F i i


Ž . ŽRepeating arguments similar to those given after Eq. 17 above defining buyers’.expected utilities in the multiple unit auction case , we conclude that each agent’s

Ž . Ž . Ž .expected utility function Õ P,P :T =P I=F ™R given in Eq. 34 satisfiesi i

Assumption 4.Ž . Ž Ž ..Under auction mechanism w P gM T , P I=F , the ith agent’s interim

expected utility given true type t gT and reported type tX gT is:i i i i

X <Õ t ,w t t q d t t . 35Ž . Ž .Ž .Ž .H i i i yi i yi iTi

Ž . Ž Ž ..A contract auction mechanism w P gM T , P I=F is Bayesian incentiveŽ .compatible i.e., is contained in the set B if there are h sets C , . . . ,C , . . . ,C1 i h


) <g t , j,s w d j,s t ,t q d t tŽ . Ž . Ž .Ž . Ž .H H i i i yi i yi iT I=Fyi

) < X <G g t , j,s w d j,s t ,t q d t t , 36Ž . Ž . Ž .Ž .Ž . Ž .H H i i i yi i yi iT I=Fyi

for all t gT _C and all tX gT .i i i i iŽ . Ž Ž ..A contract auction mechanism w P gM T , P I=F is interim individually

Ž .rational i.e., is contained in the set L if there are h sets Q , . . . ,Q , . . . ,Q , with1 i hŽ .Q gS and m Q s0, such that for each buyer is1,2, . . . ,h:i i i i

) < <g t , j,s w d j,s t ,t q d t t Gr t , 37Ž . Ž . Ž . Ž .Ž .Ž . Ž .H H i i i yi i yi i i iT I=Fyi

Ž .for all t gT _Q . As before, r P :T ™R is the ith agent’s interim reservationi i i i i

utility function.Ž . ŽNow, consider the auction mechanism w P that selects the principal and

.hence selects no contracting with probability 1 for all reported types. Thus, underŽ . Ž�Ž . 4 < .mechanism w P , w 0,s :sgF t s1 for all t. Given the definition of the ith

Ž Ž ..agent’s utility function see Eq. 30 , we have:

) < <g t , j,s w d j,s t ,t q d t t sg t for all t gT .Ž . Ž . Ž .Ž .Ž . Ž .H H i i i yi i yi i i i i iT I=Fyi

Ž . Ž .By assumption g t Fr t for all t gT . Thus, we have:i i i i i i

) < <g t , j,s w d j,s t ,t q d t t Fr t for all t gT .Ž . Ž . Ž .Ž .Ž . Ž .H H i i i yi i yi i i i i iT I=Fyi

Ž .and thus, our contract auction model satisfies Assumption 5 2 . We shall alsoY Ž . Y Ž .assume that there is a probability measure w gP I=F such that w gK t

w xa.e. m and for each agent is1,2, . . . ,h:

)Y < w xg t , j,s w d j,s q d t t Gr t a.e. m .Ž . Ž . Ž .Ž . Ž .Ž .H H i i i yi i i i i

T I=Fyi

Ž .Thus, Assumption 5 1 holds.


To complete our description of the contract auction model, observe that givenŽ . Ž .choice wgP I=F , the principal’s expected utility is given by:

u t ,w s g ) t , j,s w d j,s . 38Ž . Ž . Ž . Ž .Ž . Ž .H 0I=F

Ž . ŽRepeating arguments similar to those given after Eq. 21 above defining the.seller’s expected utility in the multiple unit auction case , we conclude that the

Ž . Ž . Ž .principal’s expected utility function u P,P :T=P I=F ™R given in Eq. 38satisfies Assumption 6.

Ž .Under contract auction mechanism w P gGlBlL, the principal’s expectedutility is:

u t ,w t p d t . 39Ž . Ž . Ž .Ž .H 0T

As is true in the case of the multiple unit auction, it is easy to see that theŽ Ž ..principal’s contract auction problem is a version of the leader’s problem Eq. 4 .

Moreover, because our contract auction model satisfies Assumptions 1–6, itfollows directly from Theorem 1 that the principal’s problem has a solution.

( )Corollary 3 Existence of an Optimal Contract Auction Mechanism :

Given the contract auction model above, there exists an auction mechanism) Ž .w P gGlBlL solving the principal’s problem:

) <max g t , j,s w d j,s t p d t . 40Ž . Ž . Ž . Ž .Ž . Ž .HHwŽP.g G l B l L 0 0T I=F

4. Proof of the main existence result

Given the nature of the Bayesian incentive compatibility constraints, the proofof Theorem 1 is nonstandard. The proof given here rests on a new result showingthat the feasible set of interim individually rational and Bayesian incentive

Ž .compatible revelation mechanisms is K-compact Theorem 2, below . As dis-cussed in the introduction, this result refines and extends Theorem 4.1 of PageŽ .1992 . As in the proof of Theorem 4.1, the proof of Theorem 2 given here utilizesthe notions of K-convergence and K-compactness, as well as the abstract Komlos

Ž . Ž .Theorem due to Balder 1996 , Theorem 4.1 i , also see Theorem 2.1 of BalderŽ .1990 . The proof of Theorem 2 also avoids a minor error in the proof of Theorem

Ž . 104.1 by Page 1992 . We begin with the definitions K-convergence and relative

10 Ž .The basic steps in the proofs of Theorem 2 and Theorem 4.1 by Page 1992 are similar. However,because a minor error in the proof of Theorem 4.1 is avoided here in the proof of Theorem 2, the stepsin the proof of Theorem 2 do not precisely match the steps in the proof of Theorem 4.1.


Ž .K-compactness and then proceed to a statement of Theorem 4.1 i by BalderŽ . Ž1996 with the notation and numbering of the assumptions modified to fit the

.notation and numbering of assumptions given here .Ž .Let T , S be a type space equipped with a s-finite measure m.

( )Definition 4.1 K-convergence :

{ ( )} ( ) [ ]A sequence of mechanisms f P ;M T,K is said to K-conÕerge m to an n(̂ ) ( ) { ( )}K-limit f P gM T,K if for each subsequence f P , there is a m-null setnk k

( ( ) )NgS i.e., m N s0 such that:m1

ˆlim f t s f t for all tgT_N.Ž . Ž .Ý nkmm™` ks1

( )Definition 4.2 Relative K-compactness :

( ) [ ]A subset C;M T,K is said to be relatiÕely K-compact m if eÕery sequence in C

ˆ[ ] ( ) ( )contains a subsequence K-conÕerging m to some mechanism f P gM T,K .A( ) [ ]subset C;M T,K is said to be K-compact m if eÕery sequence in C contains a

ˆ[ ] ( )subsequence K-conÕerging m to some mechanism f P gC .

( )BalderrrrrrKomlos Theorem K-compactness of G :

� Ž .4Let f P be a sequence of revelation mechanisms in G . Under Assumptions 2n nˆ 11� Ž .4 Ž .and 3, there exists a subsequence f P and a mechanism f P gG such that:nk k

m1ˆ w xlim f t s f t a.e. m . 41Ž . Ž . Ž .Ý nkmm™` ks1

Ž .In Theorem 2, we show that given type space T , S,m with m s-finite, underŽ .Assumptions 1–4 and Assumption 5 2 , the set of feasible, Bayesian incentive

compatible, and interim individually rational mechanisms GlBlL is K-com-w xpact m

( )Theorem 2 K-compactness of GlBlL :

{ ( )}Let f P be a sequence of reÕelation mechanisms in GlBlL. Undern n( ) { ( )}Assumptions 1–4 and Assumption 5 2 , there exists a subsequence f P and ank k

) ( )mechanism f P GlBlL such that:m1

) w xlim f t s f t a.e. m .Ž . Ž .Ý nkmm™` ks1

11 Ž̂ .This pointwise convergence of averages continues to hold with the same f P but with varying� Ž .4m-null sets for every further subsequence of f P .n k k


Proof of Theorem 2First note that GlBlL is convex. The convexity of G follows from

Assumptions 2 and 3. The convexity of BlL follows from Assumption 4 and inŽ . Ž .particular from the affinity of Õ t,P on K t for each i and t.i

� Ž .4Let f P be a sequence of mechanisms in GlBlL. By the BalderrKomlosn n

Theorem, we can assume without loss of generality that for some m-null setŽ Ž . .NgS i.e., m N s0 :

n ˆlim f t s f t for all tgT_N ,Ž . Ž .n™`

Ž̂ .where f P gG and where for all n:

f P q PPP qf PŽ . Ž .1 nnf P s . 42Ž . Ž .n

) ) ˆŽ . Ž . Ž . w xWe will show that there exists a mechanism f P with f t s f t a.e. m) Ž .such that f P gGlBlL. To begin, let:

N t s t gT : t ,t gN . 43� 4Ž . Ž . Ž .i yi yi i yi

Ž .For each i, we have see Ash, 1972, Section 2.6 :

m N s m N t m d t s0, 44Ž . Ž . Ž . Ž .Ž .H yi i i iTi

Ž .so that for some N gS with m N s0:i i i i

m N t s0 for all t gT _N . 45Ž . Ž .Ž .yi i i i i

Now, define:

1 if tgj N =T ,i i yih t s 46Ž . Ž .½0 if tgT_ j N =T ,Ž .i i yi

and consider the auction mechanism:

) ˆf t s f t P 1yh t q fPh t , 47Ž . Ž . Ž . Ž . Ž .Ž .Ž .where fgK is the ‘penalty’ choice given in Assumption 5 2 .

) ) ˆŽ . Ž . Ž . Ž . Ž .Since h P is S-measurable, f P gM T , K . Moreover, since f t s f th ) ˆŽ . Ž Ž .. Ž . Ž .for all tgT_ j N =T , where m j N =T s0, f t s f t a.e.i i yi is1 i yi

w x Ž . Ž .m . Recall that the N are the m -null sets given in Eqs. 44 and 45 . Thus,i i) Ž . ) Ž . Žf P is contained in G and f P is a K-limit with respect to the dominating

. � Ž .4measure m of the sequence f P .n n) Ž . Ž Ž . Ž ..From the definition of f P see Eqs. 46 and 47 , it follows that if tgT is

Ž . Ž . Ž .such that for some follower i, ts t ,t gN =T then h t sh t ,t s1i yi i yi i yi


) Ž . Ž .and f t s f i.e., the mechanism chooses the penalty f . Note also that becauseŽ Ž .. Ž Ž . Ž .. nŽ .t gN implies that m N t )0 see Eqs. 44 and 45 , f t ,t may fail toi i yi i i yi

ˆ ˆŽ . � Ž .4 Ž .converge to f t ,t . Thus, f P may fail to K-converge to f P for tgj Ni yi n n i i

=T .yi

Now, fix t gT and let:i i

t gT _ j N =T jN t ,Ž .Ž .yi yi j , j/ i j yi , j i

where:

N =T sT = PPP =T =T = PPP =T =N =T = PPP =T .j yi , j 1 iy1 iq1 jy1 j jq1 h

wŽ . Ž Ž ..xFrom the perspective of follower i, T _ j N =T j N t is the setyi j , j/ i j yi , j i

of other follower type profiles t such that no other follower has type t gNyi j jŽ .j/ i triggering the penalty choice and such that no type profile t causes ayi

Ž � Ž .4failure of K-convergence recall that K-convergence of the sequence f P mayn nŽ . Ž . Ž ..fail to hold at t ,t if t gN t , see Eq. 43 .i yi yi i

For is1, . . . ,h, let C jQ gS be the m -null set of ith follower typesin in i i

where for types t gC jQ interim individual rationality andror Bayesiani in in

incentive compatibility may fail to hold for the ith follower under the mechanismŽ . Ž Ž . Ž ..f P see Eqs. 2 and 3 . Now, let:n

`F s j C jQ jN , 48Ž . Ž .i` ns1 in in i

Ž . Ž .where again the sets N are the m -null sets given in Eqs. 44 and 45 . Note thati iŽ . Ž Ž ..m F s0 and for t gT _F , m N t s0. Observe also that if:i i` i i i` yi i

t gT _Fi i i`

and if:

t gT _ j N =T jN t ,Ž .Ž .yi yi j , j/ i j yi , j i

then:

t ,t gT_ j h N =T .Ž . Ž .Ž .i yi is1 i yi

We can now conclude the following: for is1, . . . ,h, t gT _F , and t gT _i i i` yi yi) ˆ nwŽ . Ž .x Ž . Ž . Ž . Ž . Ž .j N = T j N t : a f t ,t s f t ,t ; b lim f t ,t sj , j/ i j yi , j i i yi i yi i yi

n™`

Ž̂ . Ž . ŽŽ . Ž . < . Ž .f t ,t ; and c q j N =T jN t t s0. Conclusion c is an im-i yi i j , j/ i j yi , j i iŽ .mediate consequence of Assumption 1 2 and the fact that for t gT _F ,i i i`

ŽŽ . Ž ..m j N =T jN t s0.yi j , j/ i j yi , j iŽ . Ž .It now follows from observations b and c , Assumption 4 concerning

Žfollowers’ utility functions, and the Dominated Convergence Theorem see Ash,.1972 that for is1, . . . ,h and t gT _F :i i i`

n <lim Õ t ,t , f t ,t q d t tŽ . Ž .Ž .H i i yi i yi i yi in™` Tyi

ˆ <s Õ t ,t , f t ,t q d t t . 49Ž . Ž .Ž .Ž .H i i yi i yi i yi iTyi


Ž . Ž .Given observations a and c :

ˆ )< <Õ t ,t , f t ,t q d t t s Õ t ,t , f t ,t q d t tŽ . Ž .Ž . Ž .Ž .Ž .H Hi i yi i yi i yi i i i yi i yi i yi iT Tyi y i

50Ž .

Ž Ž ..Given the definition of the m -null set F see Eq. 48 , the convexity ofi i`� Ž .4GlBlL and the fact that f P ;GlBlL, we have for all n, is1, . . . ,h,n

and t gT _F :i i i`

n <Õ t ,t , f t ,t q d t t Gr t .Ž . Ž .Ž .Ž .H i i yi i yi i yi i i iTyi

Ž . Ž .By Eqs. 49 and 50 , we have for all is1, . . . ,h, and t gT _F :i i i`

) <Õ t ,t , f t ,t q d t t Gr t . 51Ž . Ž . Ž .Ž .Ž .H i i yi i yi i yi i i iTyi

) Ž .Thus, f P gL.) Ž .In order to show that f P gB, we will show that for all is1, . . . ,h, and

t gT _F :i i i`

) < )X <Õ t ,t , f t ,t q d t t G Õ t ,t , f t ,t q d t tŽ . Ž .Ž . Ž .Ž . Ž .H Hi i yi i yi i yi i i i yi i yi i yi i

T Tyi y i

52Ž .

for all tX in T .There are two cases to consider.i i

Case 1: Follower i is type t gT _F and reports his type as tX gN implying thati i i` i iX X

)Ž Ž .. Ž . Ž Ž ..m N t )0.Under case 1, f t ,t s f for all t gT see Eq. 47 .yi i i i yi yi yiŽ .Thus, on the right-hand side of Eq. 52 , we have:

X) < <Õ t ,t , f t ,t q d t t s Õ t ,t , f q d t t Fr t .Ž . Ž .Ž . Ž .Ž . Ž .H Hi i yi i yi i yi i i i yi i yi i i i

T Tyi y i

Ž .The latter inequality holds by Assumption 5 2 . Since t gT _F , it follows fromi i i`Ž . Ž Ž ..Eq. 51 above that inequality Eq. 52 holds for case 1.

Case 2: Follower i is type t gT _F and reports his type as tX gT _N implyingi i i` i i iX

)X ˆ XŽ Ž .. Ž . Ž .that m N t s0.Under case 2, f t ,t s f t ,t for all t gT _yi i i i yi i yi yi yi

wŽ . Ž X.x Ž Ž X..j N =T jN t . More importantly, since m N t s0, we havej , j/ i j yi , j i yi i iŽŽ . Ž X.. Ž . ŽŽm j N =T jN t s0. Thus by Assumption 1 2 , q j N =yi j , j/ i j yi , j i i j , j/ i j. Ž X. < .T jN t t s0 for all t gT and in particular for t gT _F . Finally,yi , j i i i i i i i`

Ž X. Ž Ž ..given the definition of the set N t again see Eq. 43 , we have for alli iX n X ˆ XwŽ . Ž .x Ž . Ž .t gT _ j N =T jN t , lim f t , t s f t , t . Given theseyi yi j , j/ i j yi , j i i yi i yi

n™`


observations and Assumption 4 concerning followers’ utility functions, it followsŽ .from the Dominated Convergence Theorem see Ash, 1972 that:

n X <lim Õ t ,t , f t ,t q d t tŽ . Ž .Ž .H i i yi i yi i yi in™` Tyi

ˆ X <s Õ t ,t , f t ,t q d t t . 53Ž . Ž .Ž .Ž .H i i yi i yi i yi iTyi

Moreover,

ˆ X)

X< <Õ t ,t , f t ,t q d t t s Õ t ,t , f t ,t q d t t .Ž . Ž .Ž . Ž .Ž .Ž .H Hi i yi i yi i yi i i i yi i yi i yi iT Tyi y i

54Ž .

� nŽ .4Since f P ;GlBlL and t gT _F , by the above we have each n:i i i`

n < n X <Õ t ,t , f t ,t q d t t G Õ t ,t , f t ,t q d t t .Ž . Ž .Ž . Ž .Ž . Ž .H Hi i yi i yi i yi i i i yi i yi i yi iT Tyi y i

55Ž .

Ž . Ž . Ž . Ž .Taking limits on both sides of Eq. 55 , it follows from Eqs. 49 , 50 , 53 andŽ .54 that:

) < )X <Õ t ,t , f t ,t q d t t G Õ t ,t , f t ,t q d t t .Ž . Ž .Ž . Ž .Ž . Ž .H Hi i yi i yi i yi i i i yi i yi i yi i

T Tyi y i

Ž Ž ..Thus, inequality Eq. 52 holds for case 2. Having established that inequalityŽ Ž .. ) Ž .Eq. 52 holds cases 1 and 2, we conclude that f P gB. Q.E.D.

RemarkBecause strategic misreporting by followers can cause a failure of K-conver-

X ˆŽ Ž Ž .. . Ž .gence on a set of positive measure i.e., m N t )0 , the K-limit f P of ayi i� Ž .4sequence f P of Bayesian incentive compatible mechanisms is not necessarilyn n

Bayesian incentive compatible. In the proof above, we use the dominating measureŽ .m to identify mathematically inconvenient lies or misreports by followers and we

use the penalty choice f to punish followers for such misreports. Following this) Ž .line of reasoning, we are able to construct a new mechanism f P contained in

Ž̂ .the m-equivalence class determined by the K-limit f P and show that thismechanism is Bayesian incentive compatible.


Proof of Theorem 1

Let:

U ) ssup u t , f t p d t ,Ž . Ž .Ž .Hf ŽP.g G l B l L 0T

� Ž .4and let f P be a sequence of revelation mechanisms in GlBlL such that:n n

u t , f t p d t ™U ) .Ž . Ž .Ž .H n 0T

� Ž .4By Theorem 2, we can assume without loss of generality that the sequence f Pn n) Ž . Ž .K-converges to K-limit f P gGlBlL. By Theorem 4. 1 ii of Balder

Ž .1996 :

lim u t , f t p d t sU ) F u t , f ) t p d t .Ž . Ž . Ž . Ž .Ž .Ž .H Hn 0 0n T T

) Ž .Since f P gGlBlL, we have:

u t , f ) t p d t smax u t , f t p d t .Ž . Ž . Ž . Ž .Ž . Ž .H H0 f ŽP.g G l B l L 0T T

Q.E.D.

Acknowledgements

The author thanks two anonymous referees for helpful comments. The authoralso thanks Erik Balder, Andrzej Nowak, and Myrna Wooders for many helpfulcomments and discussions. An earlier version of this paper, entitled ‘OptimalAuction Design with Risk Aversion and Correlated Information,’ was completedwhile the author was visiting the Center for Economic Research at TilburgUniversity and appears as Discussion Paper No. 94109, Center for EconomicResearch, Tilburg University. The author gratefully acknowledges Center’s sup-port and hospitality.

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