Optimal auctions with ambiguity

Transcription

1 Theoretcal Economcs 1 (2006), / Optmal auctons wth ambguty SUBIR BOSE Department of Economcs, Unversty of Illnos at Urbana-Champagn EMRE OZDENOREN Department of Economcs, Unversty of Mchgan ANDREAS PAPE Department of Economcs, Unversty of Mchgan A crucal assumpton n the optmal aucton lterature s that each bdder s valuaton s known to be drawn from a unque dstrbuton. In ths paper, we study the optmal aucton problem allowng for ambguty about the dstrbuton of valuatons. Agents may be ambguty averse (modeled usng the maxmn expected utlty model of Glboa and Schmedler 1989). When the bdders face more ambguty than the seller we show that () gven any aucton, the seller can always (weakly) ncrease revenue by swtchng to an aucton provdng full nsurance to all types of bdders, () f the seller s ambguty neutral and any pror that s close enough to the seller s pror s ncluded n the bdders set of prors then the optmal aucton s a full nsurance aucton, and () n general nether the frst nor the second prce aucton s optmal (even wth sutably chosen reserve prces). When the seller s ambguty averse and the bdders are ambguty neutral an aucton that fully nsures the seller s n the set of optmal mechansms. KEYWORDS. Auctons, mechansm desgn, ambguty, uncertanty. JEL CLASSIFICATION. D44, D INTRODUCTION Optmal auctons for an ndvsble object wth rsk neutral bdders and ndependently dstrbuted valuatons have been studed by Vckrey (1961), Myerson (1981), Harrs and Ravv (1981), Rley and Samuelson (1981), and others. These papers show that the set of optmal mechansms or auctons s qute large and contans both the frst and second prce auctons wth reserve prces. One of the assumptons n ths lterature s that each Subr Bose: bose@uuc.edu Emre Ozdenoren: emreo@umch.edu Andreas Pape: apape@umch.edu We would lke to thank Bart Lpman and two referees for suggestons that led to substantal mprovements. We also thank Jeff Ely, Peter Klbanoff, Sujoy Mukerj, and audences at the 2003 RUD Conference, 2004 North Amercan Meetngs of the Econometrc Socety, Robust Mechansm Desgn Conference at the Cowles Foundaton, 2004 Decentralzaton Conference, 2005 Summer Workshop n Economc Theory at the Unversty of Brtsh Columba, Bonn Unversty, Sabancı Unversty, and Koç Unversty for helpful comments. Copyrght c 2006 Subr Bose, Emre Ozdenoren, and Andreas Pape. Lcensed under the Creatve Commons Attrbuton-NonCommercal Lcense 2.5. Avalable at

2 412 Bose, Ozdenoren, and Pape Theoretcal Economcs 1 (2006) bdder s valuaton s known to be drawn from a unque dstrbuton. In ths paper we relax ths assumpton and study how the desgn of the optmal aucton s affected by the presence of ambguty about the dstrbuton from whch the bdders valuatons are drawn. The assumpton of a unque pror s based on the subjectve expected utlty model, whch has been crtczed by Ellsberg (1961), among others. Ellsberg shows that lack of knowledge about the dstrbuton over states, often referred to as ambguty, can affect choces n a fundamental way that cannot be captured wthn the subjectve expected utlty framework. 1 Ellsberg and several subsequent researchers have demonstrated that n many stuatons decson makers exhbt ambguty averse behavor. 2 Followng Glboa and Schmedler (1989), we model ambguty averson usng the maxmn expected utlty (MMEU) model. The MMEU model s a generalzaton of the subjectve expected utlty model, and provdes a natural and tractable framework to study ambguty averson. In MMEU agents have a set of prors (nstead of a sngle pror) on the underlyng state space, and ther payoff s the mnmum expected utlty over the set of prors. Specfcally, when an MMEU bdder s confronted wth an aucton, he evaluates each bd on the bass of the mnmum expected utlty over the set of prors, and then chooses the best bd. An MMEU seller evaluates each aucton on the bass of ts mnmum expected revenue over the set of prors and chooses the best aucton. In order to better contrast our results wth the case of rsk, we assume that the bdders and the seller are rsk neutral (.e. have lnear utlty functons). Our man result, Proposton 1, s that when the bdders face more ambguty than the seller, an aucton that provdes full nsurance to the bdders 3 s always n the set of optmal mechansms. Moreover, gven any ncentve compatble and ndvdually ratonal mechansm, the seller can strctly ncrease hs revenue by swtchng to a full nsurance mechansm f the mnmum expected utlty of a bdder over the seller s set of prors s strctly larger than hs expected utlty over the bdders set of prors for a postve measure of types. To obtan some ntuton for the man result, consder the specal case where the seller s ambguty neutral,.e., hs set of prors s a sngleton. In ths case the man result says that f an ncentve compatble and ndvdually ratonal mechansm s optmal for the seller then the mnmzng set of dstrbutons for all types of the bdders must nclude the seller s pror. Suppose ths s not true for a postve measure of types and consder some such type θ. In ths case, the seller and type θ of the bdder are wllng to bet aganst each other. The seller recognzes that they have dfferent belefs about the 1 In one verson of Ellsberg s experment, a decson maker s offered two urns, one that has 50 black and 50 red balls, and one that has 100 black and red balls n unknown proportons. Faced wth these two urns, the decson maker s offered a bet on black but can decde from whch urn to draw the ball. Most decson makers prefer the frst urn. The same s true when the decson maker s offered the same bet on red. Ths behavor s nconsstent wth the expected utlty model. Intutvely, decson makers do not lke bettng on the second urn because they do not have enough nformaton or, put dfferently, there s too much ambguty. Beng averse to ambguty, they prefer to bet on the frst urn. 2 For a survey see Camerer and Weber (1992). 3 A full nsurance aucton keeps the bdders payoffs constant for all reports of the other bdders and consequently keeps them ndfferent between wnnng or losng the object.

3 Theoretcal Economcs 1 (2006) Optmal auctons wth ambguty 413 underlyng state space and wll offer sde bets usng transfers. The crucal ssue s that the modfed mechansm has to mantan overall ncentve compatblty. In our proof we address ths ssue by explctly constructng the addtonal transfers that contnue to satsfy ncentve compatblty constrants whle makng the seller better off. Essentally, we show that these addtonal transfers (to the seller) can be chosen so that n the new mechansm, under truth tellng type θ gets the mnmum expected utlty that he gets n the orgnal mechansm n every state, and thus s fully nsured aganst the ambguty. Obvously then under truth tellng, type θ s ndfferent between the orgnal mechansm and the new mechansm snce he gets the same mnmum expected utlty under both. More nterestngly, no other type wants to mtate type θ n the new mechansm. Ths s because the addtonal transfers n the new mechansm are constructed so as to have zero expected value under the mnmzng set of dstrbutons for type θ n the orgnal mechansm, but to have strctly postve expected value under any other dstrbuton. Therefore, f type θ mtates type θ n the new mechansm, he gets at best what he would get by mtatng type θ n the orgnal mechansm. Hence, snce the orgnal mechansm s ncentve compatble, the new mechansm s also ncentve compatble. Moreover, snce by assumpton the seller s dstrbuton s not n the mnmzng set for type θ n the orgnal mechansm, the addtonal transfers (to the seller) have strctly postve expected value under the seller s dstrbuton. Snce the orgnal mechansm can be modfed n ths way for a postve measure of types, the seller strctly ncreases hs revenue. In fact, for any ncentve compatble and ndvdually ratonal mechansm, by modfyng the mechansm for all types as descrbed above we can obtan a full nsurance aucton that s weakly preferred by the seller. Therefore, a full nsurance aucton s always n the set of optmal mechansms. There may be optmal sellng mechansms n addton to the full nsurance mechansm, but n some cases the full nsurance mechansm s the unque optmal mechansm. In Proposton 3 we show that f the seller s ambguty neutral and any pror that s close enough to the seller s pror s ncluded n the bdders set of prors then the optmal aucton must be a full nsurance aucton. In addton, we show n Proposton 5 that, n general, the frst and the second prce auctons are not optmal. Some real lfe auctons resemble a full nsurance aucton. 4 In general, though, full nsurance auctons are rarely observed n practce. Thus, the smplest ncorporaton of ambguty averson n auctons yelds a surprsng and somewhat negatve result. The msmatch between theory and practce could be due to several reasons ncludng the use of maxmn preferences n the theory or the relevance of ambguty n real lfe market settngs. As such the result presents a puzzle to the lterature. To hghlght some of the economc mplcatons of the above analyss, n Secton 4 we explctly derve the optmal mechansm when the seller s ambguty neutral and the bdders set of prors s the ɛ-contamnaton of the seller s pror. We show that the seller s revenue and effcency both ncrease as ambguty ncreases. We also descrbe an aucton that mplements the optmal mechansm. 4 For example, n the Amsterdam Aucton studed by Goeree and Offerman (2004), the losng bdder s offered a premum.

4 414 Bose, Ozdenoren, and Pape Theoretcal Economcs 1 (2006) When the seller s ambguty averse and the bdders are ambguty neutral we show that for every ncentve compatble and ndvdually ratonal sellng mechansm there exsts an ncentve compatble and ndvdually ratonal mechansm that provdes determnstcally the same payoff to the seller. From ths t follows that when an optmal mechansm exsts, an aucton that fully nsures the seller must be n the set of optmal mechansms. A smlar result was frst shown by Eső and Futó (1999) for auctons (n ndependent prvate value envronments) wth a rsk averse seller and rsk (and ambguty) neutral bdders. Hence, as long as bdders are rsk and ambguty neutral, ambguty averson on the part of the seller plays a smlar role to that of rsk averson. There s a small but growng lterature on aucton theory wth non-expected utlty startng wth a seres of papers by Karn and Safra (1986, 1989a,b) and Karn (1988). Salo and Weber (1995), Lo (1998), Volj (2002), and Ozdenoren (2001) study auctons wth ambguty averse bdders, and thus are closer to ths paper. These papers look at specfc aucton mechansms, such as the frst and second prce auctons, and not the optmal aucton. Matthews (1983) and Maskn and Rley (1984) study auctons wth rsk averse bdders. A more detaled comparson of our paper wth Maskn and Rley (1984) s gven n Secton 7. Fnally, a strand of lterature studes robust mechansm desgn. (See for example Bergemann and Morrs 2005, Ely and Chung forthcomng, Jehel et al. 2006, and Hefetz and Neeman 2006.) Even though there s some smlarty between that lterature and our work (as n that lterature, we relax certan assumptons of the standard mechansm desgn framework), t s mportant to pont out that our work dffers sgnfcantly from ths lterature. Standard mechansm desgn n partcular Bayesan mplementaton reles crucally on the underlyng model beng common knowledge. The focus of those papers s to study mechansm desgn whle relaxng (some of) the common knowledge assumptons. In contrast, we mantan throughout the standard methodologcal assumpton of consderng the underlyng model whch ncludes the modelng of the ambguty to be common knowledge and we relax assumptons on the preferences of the agents, n partcular allowng the agents to exhbt ambguty averson. 2. THE OPTIMAL AUCTION PROBLEM In ths secton we generalze the standard optmal aucton problem by allowng the bdders and the seller to have MMEU preferences (Glboa and Schmedler 1989). There are two bdders (denoted as bdder 1 and bdder 2) and a seller. Bdders have one of a contnuum of valuatons θ = [0,1]. Let Σ be the Borel algebra on. Each bdder knows hs true valuaton but not that of the other bdder. The set m B s a non-empty set of probablty measures on (,Σ) wth a correspondng set B of dstrbuton functons. Ths set represents each bdder s belef about the other bdder s valuaton. Bdders beleve that valuatons are generated ndependently, but they may not be confdent about the probablstc process that generates the valuatons. Ths possble vagueness n the bdders nformaton s captured by allowng for a set of prors rather than a sngle pror. We assume that both the bdders and the seller have lnear utlty functons.

5 Theoretcal Economcs 1 (2006) Optmal auctons wth ambguty 415 The seller s also allowed to be ambguty averse. The set m S s a nonempty set of probablty measures on (,Σ) wth a correspondng set S of dstrbuton functons. Ths set represents the seller s belef about the bdders valuatons. That s, the seller beleves that the bdders valuatons are generated ndependently from some dstrbuton n S. 5 Each bdder s reservaton utlty s 0. We consder a drect mechansm where bdders smultaneously report ther types. 6 The mechansm stpulates a probablty for assgnng the tem and a transfer rule as a functon of reported types. Let x ( θ,θ ) be the tem assgnment probablty functon and t ( θ,θ ) the transfer rule for bdder {1,2}. 7 The frst entry s s report and the second entry s the other bdder s report. We assume that type θ of bdder chooses a report θ to maxmze nf G B (x ( θ,θ )θ t ( θ,θ ))dg (θ ). 8 The seller s problem s to fnd a mechansm {(x 1,t 1 ),(x 2,t 2 )} that solves sup {(x 1,t 1 ),(x 2,t 2 )} nf F S (t 1 (θ,θ ) + t 2 (θ,θ ))d F (θ )d F (θ ) (1) subject to (IC) nf (x (θ,θ )θ t (θ,θ ))dg (θ ) nf (x ( θ,θ )θ t ( θ,θ ))dg (θ ) for all θ, θ and {1,2} (2) and (IR) nf (x (θ,θ )θ t (θ,θ ))dg (θ ) 0 for all θ and {1,2} (3) where x 1 (θ,θ ) + x 2 (θ,θ ) 1 for all θ,θ. As s standard, we assume that all of the above s common knowledge. The frst nequalty gves the ncentve compatblty (IC) constrants and the second nequalty gves the ndvdual ratonalty (IR) or partcpaton constrant. These are the usual constrants except that the bdders compute ther 5 Formally the seller s belef s the set of product measures µ µ on the product space (,Σ Σ) where µ m S, even though for notatonal smplcty, throughout the paper we refer to the seller s belef smply as µ m S. It s mportant, however, to keep ths n mnd, especally for a later result (Proposton 2) whch talks about the seller s belef beng n the nteror of the set of the bdders set of belefs. 6 The revelaton prncple holds n our settng. 7 Note that we do not restrct attenton to symmetrc mechansms. In the standard aucton lterature wth ambguty neutral (but not necessarly rsk neutral) bdders, the seller can restrct attenton to symmetrc auctons wthout loss of generalty. To see ths suppose an asymmetrc aucton s optmal. Snce bdders are ex ante symmetrc, by exchangng the roles of the two bdders, the seller can obtan another optmal aucton. But then by randomzng between these two asymmetrc auctons wth equal probablty, the seller can obtan a symmetrc and ncentve compatble mechansm. Ths argument does not hold when the bdders are ambguty averse. The reason s that ambguty averse bdders may strctly prefer randomzaton. Therefore, even though they prefer truth-tellng n the asymmetrc mechansm, they may prefer to msrepresent ther types n the randomzed mechansm. 8 Snce bdders preferences are lnear n transfers, we consder only determnstc transfers.

6 416 Bose, Ozdenoren, and Pape Theoretcal Economcs 1 (2006) utltes n the mechansm usng the MMEU rule. For example, the IC constrant requres that the nfmum expected utlty a bdder of type θ gets reportng hs type truthfully s at least as much as the nfmum expected utlty that he gets reportng any other type θ. One nterpretaton of the set of prors n the above formulaton s subjectve : the players preferences are common knowledge and the sets are subjectve representatons of the uncertanty (as well as the averson to ths uncertanty) the players face about the stochastc process that generates the valuatons. Another nterpretaton s objectve : the players learn everythng they can about the stochastc process that generates the types, but there are hard-to-descrbe factors that prevent them from learnng the process completely. The objectve nterpretaton s more restrctve than the subjectve one for two reasons. Frst, when the set of prors s objectvely fxed, bdders atttudes to ambguty are represented by the mnmum functonal only, whch may be vewed as extreme. Second, the objectve nterpretaton makes sense when both the seller and the buyers have the same set of prors (whch s covered n our framework), snce the set of prors s assumed to be common knowledge. Note that our formulaton dffers slghtly from that of Glboa and Schmedler snce we use the nfmum (supremum) nstead of the mnmum (maxmum). However, we contnue to refer to these preferences as maxmn snce ths termnology s standard. At the end of the next secton we provde condtons on preferences and mechansms that guarantee that the mnmum over the sets of prors and an optmal aucton exst. 3. FULL INSURANCE AUCTION In ths secton we show that when m S m B,9 a full nsurance aucton s always n the set of optmal auctons and dscuss when the seller can make strct gans by swtchng to a full nsurance aucton. In what follows, for a gven mechansm {(x 1,t 1 ),(x 2,t 2 )}, t s convenent to defne q (θ,θ ) x (θ,θ )θ t (θ,θ ) for all θ,θ. So q (θ,θ ) s the ex post payoff to type θ of bdder from truth tellng n the mechansm {(x 1,t 1 ),(x 2,t 2 )} when the other bdder reports θ. 10 We say that an event has postve measure f nf µ m µ( ) > 0 and zero measure otherwse. Next, we formally defne a full nsurance S aucton. DEFINITION 1. A full nsurance mechansm s one where the (ex post) payoff of a gven type of a bdder does not vary wth the report of the competng bdder. That s, {(x 1,t 1 ), (x 2,t 2 )} s a full nsurance mechansm f, for almost all θ, q (θ,θ ) s constant as a functon of θ. Next, we gve the formal statement of the man proposton. All proofs are n the Appendx. 9 In partcular, ths covers two nterestng cases. If m S s a sngleton set, then the seller s ambguty neutral and the bdders are (weakly) ambguty averse. On the other hand, f m S = m B, then both the seller and the bdders are (weakly) ambguty averse wth a common set of prors. 10 Formally q should be ndexed also by {(x 1,t 1 ),(x 2,t 2 )}. Snce the mechansm to whch we are referrng s always clear from the context, we drop ths ndex to smplfy the notaton.

7 Theoretcal Economcs 1 (2006) Optmal auctons wth ambguty 417 PROPOSITION 1. Suppose that the seller s set of prors s S and the bdders set of prors s B wth S B. Let {(x 1,t 1 ),(x 2,t 2 )} be an arbtrary ncentve compatble and ndvdually ratonal mechansm. There s always a full nsurance mechansm, also satsfyng ncentve compatblty and ndvdual ratonalty, that generates at least as much mnmum expected revenue over the set of prors S for the seller. Moreover f there exsts a bdder {1,2} and some postve measure event such that, for all θ, nf q (θ,θ )dg (θ ) > nf H B q (θ,θ )d H(θ ), then {(x 1,t 1 ),(x 2,t 2 )} s not optmal. In fact, the seller can strctly ncrease hs mnmum expected revenue over the set of prors S by usng a full nsurance mechansm. To understand ths result consder the case where the seller s ambguty neutral,.e., S = {F }. Let (θ ) = arg mn H B q (θ,θ )d H(θ ) where for ease of exposton we assume that the mnmum exsts so that we wrte mn nstead of nf. 11 In ths case Proposton 1 mples that f a mechansm {(x 1,t 1 ),(x 2,t 2 )} s optmal then F must be n (θ ) for almost all θ and {1,2}. Suppose to the contrary that for some {1,2} there exsts a postve measure of types for whch ths s not true. Consder some such type θ for whch F / ( θ ). The seller can always adjust the transfers of type θ so that, under truth tellng, type θ gets the same mnmum expected utlty as he gets n the orgnal mechansm n every state, and thus s fully nsured aganst ambguty n the new mechansm. Furthermore, by constructon, the dfference between the transfers n the new mechansm and the orgnal mechansm has weakly postve expected value 12 for any dstrbuton n B. Ths s true because ths dfference has zero expected value under ( θ ), the mnmzng set of dstrbutons n the orgnal mechansm, and strctly postve expected value under any other dstrbuton,.e., for dstrbutons n B ( θ ). Obvously, under truth tellng, type θ s ndfferent between the orgnal mechansm and the new mechansm snce he gets the same mnmum expected utlty under both. More nterestngly, no other type wants to mtate type θ n the new mechansm. Ths s true snce the orgnal mechansm s ncentve compatble and mtaton n the new mechansm s even worse gven that the dfference n transfers has weakly postve expected value under any dstrbuton n B. Moreover, by assumpton, the seller s dstrbuton s not n the mnmzng set for the orgnal mechansm, whch means the addtonal transfers (to the seller) must have strctly postve expected value under the seller s dstrbuton. Thus the seller s strctly better off n the new mechansm, contradctng the optmalty of the orgnal mechansm. When the nfma and suprema n equatons (1), (2), and (3) are replaced wth mnma and maxma, we can prove a stronger verson of Proposton 1. The next proposton provdes suffcent condtons for ths. 11 See Proposton 2 below for condtons that guarantee that ths assumpton holds. 12 Recall that these are transfers to the seller.

8 418 Bose, Ozdenoren, and Pape Theoretcal Economcs 1 (2006) PROPOSITION 2. Suppose the seller can use only mechansms such that transfers are unformly bounded and suppose that m B m S s weakly compact and convex and ts elements are countably addtve probablty measures. Then the sets of mnmzng prors n equatons (1), (2), and (3) and the set of optmal mechansms are nonempty. The followng corollary strengthens Proposton 1 when the hypothess of Proposton 2 holds. COROLLARY 1. Suppose that the hypothess of Proposton 2 holds. For a gven mechansm {(x 1,t 1 ),(x 2,t 2 )}, let mn S = arg mn t1 (θ,θ ) + t 2 (θ,θ ) dg (θ )dg (θ ). If there exsts a bdder {1,2} and some postve measure event such that for all θ and for all G mn S, q (θ,θ )dg (θ ) > mn H B q (θ,θ )d H(θ ), (4) then the seller can strctly ncrease hs mnmum expected revenue over the set of prors S usng a full nsurance mechansm. Moreover, there s a full nsurance mechansm that s optmal for the seller. Corollary 1 s stronger than Proposton 1 n two ways. Frst, the nequalty n (4) s checked only for the mnmzng dstrbutons for the seller (not all the dstrbutons n S ). Second, snce an optmal mechansm exsts, a full nsurance mechansm s always optmal for the seller. 13 In general there may be optmal sellng mechansms that dffer from the full nsurance mechansm. But f the seller s belef set s a sngleton F wth strctly postve densty and f any pror that s close enough to F s n B, then the set of dstrbutons that gve the mnmum expected utlty does not nclude F unless the ex post payoffs are constant. In ths case the unque optmal aucton s a full nsurance aucton. The next proposton states ths observaton. PROPOSITION 3. Suppose that the seller s ambguty neutral wth S = {F } where F has strctly postve densty. If there exsts ɛ > 0 such that for any dstrbuton H on, (1 ɛ)f +ɛh B, then the unque optmal aucton s a full nsurance aucton. The usual (ndependent and prvate values) aucton model, where the bdders and the seller are both rsk and ambguty neutral, corresponds n our model to the case where S and B are both sngleton sets. In the usual model, every bdder takes the expectaton of hs payoffs (for dfferent reports of hs type) over hs opponent s types and these nterm expected payoffs are enough to characterze ncentve compatblty 13 In contrast, Proposton 1 says that the seller can get arbtrarly close to the supremum n equaton (1) usng a full nsurance mechansm.

9 Theoretcal Economcs 1 (2006) Optmal auctons wth ambguty 419 and ndvdual ratonalty and hence optmal auctons. Thus, n the usual model the dependence of bdders (ex post) payoffs on the types of hs opponents s not pnned down. In our model, when the buyers and the seller are possbly ambguty averse, ths dependence on the opponent s type s pnned down (unquely under the condtons of Proposton 3). The requrements mposed by ths dependence elmnate many auctons that are optmal n the usual model such as frst and second prce auctons (under the condtons of Proposton 5). In the next two sectons we provde some applcatons of the results n ths secton. 4. FULL INSURANCE UNDER ɛ-contamination In ths secton we explctly derve the optmal mechansm n the case of ɛ-contamnaton when the seller s ambguty neutral wth S = {F }. We assume that the seller s dstrbuton F s a focal pont, and bdders allow for an ɛ-order amount of nose around ths focal dstrbuton. We make the common assumptons that F has a strctly postve densty f and L(θ ) = θ 1 F (θ ) f (θ ) s strctly ncreasng n θ. We construct B as follows: B = {G : G = (1 ɛ)f + ɛh for any dstrbuton H on } where ɛ (0,1]. By Proposton 3 we know that the unque optmal mechansm for the ɛ-contamnaton case s a full nsurance mechansm. Ths mples that we can restrct ourselves to full nsurance mechansms n our search for the optmal mechansm. Let {(x 1,t 1 ),(x 2,t 2 )} be a full nsurance mechansm. I.e., for a gven θ, q (θ,θ ) does not vary wth θ. Let u (θ ) = q (θ,θ ) for {1,2}. Next we defne some useful notaton. Let X (θ ) = X mn (θ ) = nf X max (θ ) = sup Usng the IC constrant we obtan u (θ ) = nf nf u ( θ ) + nf x (θ,θ )d F (θ ) x (θ,θ )dg (θ ) x (θ,θ )dg (θ ). (x (θ,θ )θ t (θ,θ ))dg (θ ) (x ( θ,θ )θ t ( θ,θ ))dg (θ ) (θ θ )x ( θ,θ )dg (θ ). (5)

10 420 Bose, Ozdenoren, and Pape Theoretcal Economcs 1 (2006) If θ > θ then Exchangng the roles of θ and θ n (5) we obtan Agan f θ > θ then u (θ ) u ( θ ) + (θ θ )X mn ( θ ). (6) u ( θ ) u (θ ) + nf ( θ θ )x (θ,θ )dg (θ ). u ( θ ) u (θ ) + ( θ θ )X max (θ ). (7) Now observe that u s non-decreasng, snce for θ > θ by the IC constrant we have u (θ ) u ( θ ) + (θ θ )X mn ( θ ) u ( θ ). The next lemma s useful n characterzng the optmal aucton. LEMMA 1. The functon u s Lpschtz. Snce u s Lpschtz, t s absolutely contnuous and therefore s dfferentable almost everywhere. For θ > θ we use (6) and (7) to obtan X max (θ ) u (θ ) u ( θ ) θ θ X mn ( θ ). We take the lmt as θ goes to θ to obtan for almost all θ that X max (θ ) u θ X mn (θ ). Snce an absolutely contnuous functon s the defnte ntegral of ts dervatve, θ 0 X max (y )d y u (θ ) u (0) Inequalty (8) suggests that the auctoneer may set and u (θ ) = θ t (θ,θ ) = x (θ,θ )θ 0 θ 0 X mn (y )d y. (8) X mn (y )d y (9) θ 0 X mn (y )d y, (10) snce for a gven allocaton rule x, transfers as n (10) are the hghest transfers the auctoneer can set wthout volatng (8). Of course, (8) s only a necessary condton and for gven allocaton rules x 1 and x 2, the resultng mechansm {(x 1,t 1 ),(x 2,t 2 )} may not be ncentve compatble. Fortunately, ths dffculty does not arse f the allocaton rules x 1 and x 2 are chosen optmally for transfers gven as n (10). In other words, our strategy

11 Theoretcal Economcs 1 (2006) Optmal auctons wth ambguty 421 s to fnd the optmal allocaton rules x 1 and x 2 assumng that the transfers are gven by (10), and then show that the resultng mechansm {(x 1,t 1 ),(x 2,t 2 )} s ncentve compatble. For a transfer functon gven by (10), we can rewrte the seller s revenue as R = 2 = θ x (θ,θ ) Usng ntegraton by parts we obtan Defne R = 2 =1 1 0 θ 0 θ X (θ )f (θ )d θ X mn (y )d y 1 0 (1 F (θ ))X mn L ɛ (θ ) = θ (1 ɛ) 1 F (θ ) f (θ ) d F (θ )d F (θ ). (θ )d θ. (11) and let r (0,1) be such that L ɛ (r ) = 0. The followng proposton characterzes the optmal allocaton when the transfer functon s gven by (10). PROPOSITION 4. The unque optmal mechansm s symmetrc and for any θ and θ the allocaton rule x 1 = x 2 = x s gven by 1 f θ > θ and θ r x (θ,θ 1 ) = f θ = θ 2 and θ r 0 otherwse and the correspondng transfer functon t = t 1 = t 2 s gven by equaton (10). Note that the correspondence that maps ɛ to the set of optmal mechansms s upper semcontnuous but not lower semcontnuous n ɛ at ɛ = 0. To see ths note that when ɛ = 0, we are n the standard (ndependent and prvate value) settng wth rsk and ambguty neutral bdders and seller. In ths case, the set of optmal mechansms conssts of all mechansms {(x 1,t 1 ),(x 2,t 2 )} where x 1 = x 2 = x satsfes (12), r s gven by r (1 F (r ))/f (r ) = 0, and the transfer functons satsfy 1 0 t (θ,θ )d F (θ ) = X (θ )θ θ 0 X (y )d y. Note that transfers are pnned down only at the nterm level. Notce that settng ɛ = 0 and ntegratng over (10), we mmedately see that the mechansm n Proposton 4 s optmal for the seller at ɛ = 0. Thus the correspondence that maps ɛ to the set of optmal mechansms s upper semcontnuous at ɛ = 0. Yet, as s well known there are many other mechansms (such as the second prce aucton wth reserve prce r ) that are n the (12)

12 422 Bose, Ozdenoren, and Pape Theoretcal Economcs 1 (2006) set of optmal mechansms at ɛ = 0. Obvously these mechansms are not attaned n the lmt, so the correspondence s not lower semcontnuous. It s nterestng to note economc mplcatons of the above analyss for revenue and effcency. Frst, the seller s revenue ncreases as ambguty ncreases. To see ths note that under the above allocaton rule X mn (θ ) = (1 ɛ)x (θ ) for all θ < 1. Pluggng ths nto the revenue expresson (11) we see that the revenue ncreases as ɛ ncreases. In fact, when ambguty becomes extreme,.e., when ɛ equals one, the seller extracts all the surplus. Second, an ncrease n ambguty helps effcency. To see ths note that L ɛ shfts up as ɛ ncreases and snce L ɛ (θ ) s an ncreasng functon of θ, the cutoff type r decreases as ɛ ncreases. Agan n the case of extreme ambguty the seller does not exclude any types, and full effcency s acheved. Fnally, a natural queston to ask at ths stage s how to mplement the optmal mechansm descrbed above. Several auctons mplement the mechansm, and we descrbe one such aucton here. Consder an aucton where bdders submt bds for the object and the allocaton rule s the usual one, namely, the hghest bdder who bds above the reservaton value r obtans the object. The payment scheme s as follows: the wnnng bdder pays to the auctoneer an amount equal to hs bd, and all bdders (regardless of havng won or lost) who have bd above the reservaton prce receves a gft from the seller. For a bdder who bds, say, b (where b s greater than r ), the amount of the gft s S(b) = (1 ɛ) b F (y )d y. In ths aucton, the equlbrum strategy of a bdder wth valuaton θ s to bd hs valuaton. To see ths note that the allocaton rule s the same as the r one n Proposton 4. Moreover, a bdder who bds θ pays θ (1 ɛ) θ F (y )d y f he wns r the aucton and (1 ɛ) θ F (y )d y f he loses the aucton, and these transfers are also r the ones n Proposton 4. Snce reportng one s true value s ncentve compatble n the optmal mechansm, t s also optmal to bd one s true value n ths aucton as well. 5. THE FIRST AND SECOND PRICE AUCTIONS Lo (1998) shows that the revenue equvalence result does not hold when bdders are ambguty averse. In partcular, the frst prce aucton may generate more revenue than the second prce aucton. In ths secton we show that the frst prce aucton s n general not optmal ether. 14 In fact under rather general condtons, the frst and second prce auctons, as well as many other standard auctons are not optmal n ths settng. The followng proposton gves a weak condton on B that s suffcent for the nonoptmalty of a large class of auctons ncludng the frst and second prce auctons. PROPOSITION 5. Suppose that S and B are weakly compact and convex wth elements that are countably addtve probablty measures. Suppose that for any G S there exsts some dstrbuton H B such that H frst-order stochastcally domnates G. Now, f under some mechansm {(x 1,t 1 ),(x 2,t 2 )} wth unformly bounded transfers there exsts a 14 When the type space s dscrete, nether the frst nor the second prce aucton s the optmal aucton for reasons completely unrelated to the ssues beng studed n ths paper.

13 Theoretcal Economcs 1 (2006) Optmal auctons wth ambguty 423 bdder and a postve measure subset such that for all θ, q ( θ,θ ) s weakly decreasng n θ and q ( θ,θ ) < q ( θ,θ ) for some θ,θ, then {(x 1,t 1 ),(x 2,t 2 )} s not optmal. To apply the above proposton to the frst and second prce auctons, we need to show that n the drect mechansms that correspond to these aucton forms q ( θ,θ ) s weakly decreasng n θ and q ( θ,θ ) < q ( θ,θ ) for some θ,θ for a postve measure subset. Frst note that the payoff q ( θ,θ ) of all types of a bdder s weakly decreasng n the report of the other bdder. Next consder a type θ that s greater than the reserve prce and less than one, whch s the hghest possble valuaton. The payoff of θ s strctly larger f the other bdder reports a type θ that s less than θ as opposed to a type θ greater than θ. Ths s because n both of these auctons f the other bdder reports more than θ the payoff of θ s zero, but f the other bdder reports less than θ the payoff of θ s strctly postve. Ths shows that q ( θ,θ ) < q ( θ,θ ). Therefore under the hypothess of Proposton 5 the frst and second prce auctons are not optmal. 6. AMBIGUITY AVERSE SELLER Next, we consder the case where the seller s ambguty averse and the bdders are ambguty neutral. We restrct attenton to symmetrc mechansms. 15 Our next result complements Proposton 1. PROPOSITION 6. Suppose that the seller s ambguty averse, wth a set of prors S, and the bdders are ambguty neutral, wth a pror F S. For every ncentve compatble and ndvdually ratonal sellng mechansm (x, t ) there exsts an ncentve compatble and ndvdually ratonal mechansm (x, t ) that provdes determnstcally the same revenue to the seller,.e. t (θ,θ ) + t (θ,θ ) s constant for all θ,θ. Moreover, f nf t (θ,θ ) + t (θ,θ ) dg (θ )dg (θ ) < t (θ,θ ) + t (θ,θ ) d F (θ )d F (θ ) then (x, t ) strctly ncreases the mnmum expected revenue of the seller over the set of prors S. When an optmal mechansm exsts, Proposton 6 mples that an aucton that fully nsures the seller must be n the set of optmal mechansms. Eső and Futó (1999) prove a smlar result for auctons wth a rsk averse seller n ndependent prvate values envronments wth rsk (and ambguty) neutral bdders. 15 Snce bdders are ambguty neutral, we can now restrct attenton to symmetrc mechansms wthout loss of generalty. The argument s very smlar to the standard one. If an asymmetrc aucton s optmal than by exchangng the roles of the bdders the seller obtans another optmal aucton. Randomzng between these two asymmetrc mechansms generates a symmetrc mechansm. Snce bdders are ambguty neutral, ths symmetrc mechansm s ndvdually ratonal and ncentve compatble. Moreover, t s easy to see that an ambguty averse seller weakly prefers the symmetrc mechansm.

14 424 Bose, Ozdenoren, and Pape Theoretcal Economcs 1 (2006) The basc dea of the proof s smple. For any ndvdually ratonal and ncentve compatble mechansm (x,t ), one can defne a new mechansm ( x, t ) where the allocaton rule x s the same as x, but wth the followng transfers: t (θ,θ ) = T (θ ) T (θ ) + T ( )d F ( ) where T (θ ) = t (θ,θ )d F (θ ). Note that n the new mechansm t (θ,θ )+ t (θ,θ ) s always 2 T ( )d F ( ), whch s constant. It s straghtforward to check that ths mechansm s ncentve compatble and ndvdually ratonal as well. The reason ths mechansm works n both rsk and ambguty settngs s that snce the bdders are rsk and ambguty neutral ( x, t ) s ncentve compatble n ether settng (rsk or ambguty) and provdes full nsurance to the seller aganst both rsk and ambguty. 7. COMPARISON OF OPTIMAL AUCTIONS WITH RISK AVERSE VS. AMBIGUITY AVERSE BIDDERS Matthews (1983) and Maskn and Rley (1984), henceforth, MR, relax the assumpton that bdders are rsk neutral and replace t wth rsk averson. Even though there s some smlarty between rsk averson and ambguty averson, the two are dstnct concepts. In partcular, an envronment wth rsk-averse bdders gves rse to optmal auctons that dffer from the optmal auctons when bdders are ambguty averse. In ths secton we contrast our results wth those n MR, to hghlght ths dstncton. To facltate comparson, we assume, lke MR, that the seller s rsk and ambguty neutral. Bdders, on the other hand, are rsk averse and ambguty neutral n MR and rsk neutral and ambguty averse n ths paper. MR defne u ( t,θ ) as the utlty of a bdder of type θ when he wns and pays t, and w ( t ) as the utlty when the bdder loses the aucton (and pays t ). Assumng u and w to be concave functons, they note that f the aucton mechansm s such that the margnal utlty u 1 s dfferent from w 1, then keepng other thngs constant, a seller can gan by rearrangng the payments n such a way that the bdder s expected utlty remans the same whle the expected value of the revenue ncreases. They note, however, that provdng ths nsurance can change the ncentves of the bdders; n partcular when the margnal utlty u 1 vares wth θ, the seller can explot ths to earn hgher revenue by exposng all but the hghest type to some rsk, thus, n effect, screenng types better. MR defne a mechansm called a perfect nsurance aucton where the margnal utlty u 1 s equal to margnal utlty w 1 for all types. Ther results show that n general the optmal aucton s not perfect nsurance, the excepton beng the stuaton when bdders preferences satsfy the condton u 12 = 0,.e. when the margnal utlty u 1 does not vary wth θ. (See ther dscusson followng ther Theorem 11.) To contrast ther result wth ours, notce frst that n our model (usng ther notaton), u ( t,θ ) = θ t and w ( t ) = t, so that u 12 = 0, and more mportantly, the

15 Theoretcal Economcs 1 (2006) Optmal auctons wth ambguty 425 margnal utltes when a bdder wns and when he loses are equal to each other n all stuatons. (Ths s just restatng the fact that we assume rsk-neutral bdders n our model.) Wth ambguty averse bdders, our results show that a full nsurance aucton s always wthn the set of optmal auctons and n some stuatons t s the unquely optmal one. Wth rsk-averse bdders, MR show that for the specal case when u ( t,θ ) = θ v (t ) and w ( t ) = v (t ), so that u 12 = 0, the optmal aucton s a perfect nsurance aucton (gven that v s a convex functon). 16 Notce however, that our full nsurance aucton s dfferent from ther perfect nsurance aucton, snce n a full nsurance aucton x (θ,θ )θ t (θ,θ ) s a functon of θ only (.e., does not vary wth θ ), whch means that the realzed payoff when the bdder wns, θ t (θ,θ ), s the same as the realzed payoff when he loses, t (θ,θ ). Hence, the optmal auctons under the two stuatons are dfferent mechansms even when preferences n ther model satsfy the restrcton u 12 = Fnally, note that n ther framework, perfect nsurance auctons do become full nsurance auctons when preferences satsfy what they call Case 1. Ths s when u ( t,θ ) = U(θ t ) and w ( t ) = U( t ) (wth U a concave functon), so that equatng margnal utltes mples equatng utltes. However, n ths stuaton, the perfect nsurance aucton (and hence the full nsurance aucton) s revenue equvalent to the second prce aucton (MR, Theorem 6). When U s strctly concave, both full nsurance and second prce auctons generate expected revenue that s strctly less than the expected revenue from the hgh bd aucton (MR, Theorem 4 and Theorem 6; see n partcular, the dscusson at the bottom of page 1491). Hence the full nsurance aucton, whch s the optmal (and n some cases, as mentoned above, the unquely optmal) mechansm under ambguty averson s not the optmal mechansm under rsk averson. 8. CONCLUSION We have analyzed the auctons that maxmze a seller s proft when agents may not know the dstrbuton from whch the bdders valuatons are drawn. We have shown that when the bdders face more ambguty than the seller, an aucton that provdes full nsurance to the bdders s optmal and sometmes unquely optmal. We have shown also that standard auctons such as the frst and the second prce auctons wth reserve prces are not optmal n ths settng. In addton, we have shown that when the bdders are ambguty neutral but the seller s ambguty averse, t s the seller who s perfectly nsured. The methods developed here maybe be used n other mechansm desgn problems wth ncomplete nformaton n whch agents are ambguty averse. We beleve that the results n ths paper wll naturally extend to these stuatons, especally n envronments where the payoffs are quaslnear. For example n a barganng problem (see Myerson 1979) we conjecture that the most effcent (from the mechansm desgner s pont of 16 Put dfferently, lettng a and b be the payments when a bdder wns and loses the aucton respectvely, MR show that convexty of v mples that a = b under the optmal mechansm. 17 A further dfference s that n our settng optmal mechansm may be asymmetrc. See the dscusson n footnote 6.

16 426 Bose, Ozdenoren, and Pape Theoretcal Economcs 1 (2006) vew) mechansm wll requre that some agent be fully nsured aganst the ambguty. In any case, and unlke the standard unque pror envronment, the transfer and not just the allocaton rule wll play a crucal role n the desgn of the optmal mechansm n the presence of ambguty. We hope to explore these extensons n future research. Fx a mechansm {(x 1,t 1 ),(x 2,t 2 )}. Let APPENDIX A. PROOF OF PROPOSITION 1 K (θ ) = nf q (θ,θ )dg (θ ), so that K (θ ) s the maxmn expected payoff of type θ of bdder. For any θ, defne the functon δ (θ, ) : by δ (θ,θ ) = q (θ,θ ) K (θ ) for all θ. Let t (θ,θ ) = t (θ,θ ) + δ (θ,θ ) and consder the mechansm {(x 1,t 1 ),(x 2,t 2 )}. We prove the proposton n several steps. In the frst step we show that {(x 1,t 1 ), (x 2,t 2 )} s a full nsurance mechansm. Furthermore, t leaves the bdders payoffs unchanged under truth-tellng and therefore s ndvdually ratonal. To see that {(x 1,t 1 ),(x 2,t 2 )} s a full nsurance mechansm consder an arbtrary type θ of bdder and note that x (θ,θ )θ t (θ,θ ) = x (θ,θ )θ t (θ,θ ) δ (θ,θ ) = q (θ,θ ) q (θ,θ ) + K (θ ) = K (θ ). Thus bdders payoffs under truth tellng are unchanged snce nf x (θ,θ )θ t (θ,θ ) dg (θ ) = K (θ ). In the second step of the proof we show that {(x 1,t 1 ),(x 2,t 2 )} s ncentve compatble. The payoff for θ to devate to an arbtrary θ, θ θ, s nf x ( θ,θ )θ t ( θ,θ ) dg (θ ) = nf nf x ( θ,θ )θ t ( θ,θ ) δ ( θ,θ ) dg (θ ) x ( θ,θ )θ t ( θ,θ ) dg (θ ) nf δ ( θ,θ )dg (θ ). (13)

17 Theoretcal Economcs 1 (2006) Optmal auctons wth ambguty 427 The nequalty above follows snce the sum of the nfmum of two functons s (weakly) less than the nfmum of the sum of the functons. But note that nf δ ( θ,θ )dg (θ ) = nf q ( θ,θ ) K ( θ ) dg (θ ) = 0. Combnng ths wth (13) mples that nf x ( θ,θ )θ t ( θ,θ ) dg (θ ) nf x ( θ,θ )θ t ( θ,θ ) dg (θ ). Now the payoff for type θ of bdder from truth-tellng n {(x 1,t 1 ),(x 2,t 2 )} must be weakly larger than the last expresson, because the mechansm {(x 1,t 1 ),(x 2,t 2 )} was assumed to be ncentve compatble, and by the frst step the truth tellng payoffs are unchanged. Thus {(x 1,t 1 ),(x 2,t 2 )} s ncentve compatble. In the thrd step we show that the seller s weakly better off usng {(x 1,t 1 ),(x 2,t 2 )}. To see ths frst note nf = nf + nf t 1 (θ,θ ) + t 2 (θ,θ ) dg (θ )dg (θ ) t1 (θ,θ ) + t 2 (θ,θ ) dg (θ )dg (θ ) + nf Moreover for any G S, δ 1 (θ,θ )dg (θ )dg (θ ) + t1 (θ,θ ) + t 2 (θ,θ ) dg (θ )dg (θ ) δ (θ,θ )dg (θ )dg (θ ) = = δ 1 ( θ,θ )dg (θ )dg ( θ ) + nf q (θ,θ dg (θ ) K (θ ))dg (θ ) q (θ,θ )dg (θ ) nf G B δ 2 (θ,θ )dg (θ )dg (θ ) δ 2 ( θ,θ )dg (θ )dg ( θ ) (14) q (θ,θ )dg (θ ) dg (θ ) 0 (15) where the nequalty holds snce by assumpton S B and thus G B. Combnng equatons (14) and (15) we see that the seller s weakly better off usng {(x 1,t 1 ),(x 2,t 2 )}. Fnally we show that f there exsts a bdder and some postve measure such that for any θ nf q ( θ,θ )dg (θ ) > nf H B q ( θ,θ )d H(θ )

18 428 Bose, Ozdenoren, and Pape Theoretcal Economcs 1 (2006) then the seller strctly prefers {(x 1,t 1 ),(x 2,t 2 )} to {(x 1,t 1 ),(x 2,t 2 )}. To see ths note that nf δ ( θ,θ )dg (θ )dg ( θ ) = nf q ( θ,θ )dg (θ ) nf H B q ( θ,θ )d H(θ ) dg ( θ ) > 0. (16) The strct nequalty follows because for all G S the expresson nsde the ntegral s greater than zero for all θ and, by assumpton, the event gets strctly postve weght for all dstrbutons n S. Combnng equatons (14) and (16) we conclude that the seller strctly prefers the mechansm {(x 1,t 1 ),(x 2,t 2 )}. Ths completes the proof. B. PROOF OF PROPOSITION 2 Note that snce n ths paper we deal n envronments where the bdders valuatons are drawn ndependently, restrctng attenton to mechansms where the transfers are unformly bounded s wthout any loss of generalty as far as a search for an optmal sellng mechansm s concerned. In our proof we use the followng defntons and results. Suppose that p and q are conjugate ndces,.e. 1/p + 1/q = 1. If p = 1 then the conjugate s q =. Suppose that f n L p (,Σ, µ) for n {1,2,...}. (From now on we wll wrte L p nstead of L p (,Σ, µ) for notatonal smplcty.) We say that f n converges weakly to f L p f g f n d µ converges to g f d µ for all g L q. Denote by ca(σ) the set of countably addtve probablty measures on (, Σ). Chateauneuf et al. (2005) prove that when ca(σ) s weakly compact and convex then there s a measure µ such that all measures n are absolutely contnuous wth respect to µ. Usng ths result we fx µ to be a measure such that µ µ for all µ m B m S. For each µ m B m S there exsts a Radon Nkodym dervatve f L 1( µ). By the Radon Nkodym Theorem, there s an sometrc somorphsm between ca( µ) and L 1 ( µ) determned by the formula µ(a) = f d µ (see Dunford and Schwartz 1958, p. 306 and A Marnacc and Montruccho 2004, Corollary 11). Hence, a subset s weakly compact n ca( µ) f and only f t s n L 1 ( µ) as well. Let B and S be the set of Radon Nkodym dervatves of measures n m B and m S wth respect to µ respectvely. Fnally, let r = {g L : g r }. By Theorem 19.4 n Bllngsley (1995), r s weakly compact. Now we turn to the proof. PROOF OF PROPOSITION 2. In ths proof, to smplfy notaton, we drop the bdder subscrpt. It s clear that all the arguments go through for asymmetrc mechansms as well. Frst we show that the mnmzng set of prors n (1) s nonempty. Let g θ θ (θ ) = x( θ,θ )θ t ( θ,θ ).

19 Theoretcal Economcs 1 (2006) Optmal auctons wth ambguty 429 Recall that we assume t (θ,θ ) K for some K > 0. In other words transfers are unformly bounded. Therefore by assumpton g θ θ L. Now suppose that f n B s such that g θ θ f n d µ converges to nf f B g θ θ f d µ. Snce B s weakly compact, by passng to a subsequence we can fnd f B such that f n weakly converges to f. Thus f arg mn f B g θ θ f d µ, provng that the mnmzng set of prors n the IC and IR constrants s nonempty. Now, we show that the mnmzng set of prors n the seller s objectve functon s nonempty. Suppose f n S s such that t (θ,θ )f n (θ )f n (θ )d µ(θ )d µ(θ ) approaches nf f S t (θ,θ )f (θ )f (θ )d µ(θ )d µ(θ ). Snce S s weakly compact, by passng to a subsequence we can fnd f S such that f n weakly converges to f. Thus t (θ,θ )f n (θ )d µ(θ ) converges to t (θ,θ ) f (θ )d µ(θ ). Let g n (θ ) = t (θ,θ )f n (θ )d µ(θ ) and g (θ ) = t (θ,θ ) f (θ )d µ(θ ). Consder g n (θ )f n (θ )d µ(θ ). Note that g n (θ )f n (θ )d µ(θ ) g (θ ) f (θ )d µ(θ ) g n (θ )f n (θ )d µ(θ ) g n (θ ) f (θ )d µ(θ ) + g n (θ ) f (θ )d µ(θ ) g (θ ) f (θ )d µ(θ ) K + 1 (f n (θ ) f (θ ))d µ(θ ) + (g n (θ ) g (θ )) f (θ )d µ(θ ).

20 430 Bose, Ozdenoren, and Pape Theoretcal Economcs 1 (2006) The frst term goes to zero. To see that the second term also goes to zero note (g n (θ ) g (θ )) f (θ )d µ(θ ) = t (θ,θ )f n (θ )d µ(θ ) t (θ,θ ) f (θ )d µ(θ ) f (θ )d µ(θ ) = t (θ,θ ) f (θ )d µ(θ ) f n (θ )d µ(θ ) t (θ,θ ) f (θ )d µ(θ ) f (θ )d µ(θ ). Thus f arg mn f S t (θ,θ )f (θ )f (θ )d µ(θ )d µ(θ ), provng that the mnmzng set of prors n the seller s objectve functon s nonempty. Next, we show that there exsts a mechansm (x,t ) that satsfes the IC and IR constrants and acheves the optmal revenue for the seller. Snce transfers are bounded, the seller s revenue s bounded. Suppose that the value of the seller s problem (1) s R. Ths means that there exst a sequence of mechansms {(x n,t n )} such that (x n,t n ) satsfes the IC and IR constrants for each n, and f we let R n = mn µ m S t n (θ,θ ) + t n (θ,θ ) d µ(θ )d µ(θ ), then R n R. Note that x n 1 and t n K. Therefore passng to subsequences, x n converges weakly to x and t n converges weakly to t. Clearly x(θ,θ ) + x (θ,θ ) 1 for all θ,θ. Next, we show that (x,t ) satsfes the IC and IR constrants. Note that t s suffcent to show that for any θ, θ, lm mn n µ m B x n ( θ,θ )θ t n ( θ,θ ) d µ(θ ) = mn µ m B To smplfy notaton let x( θ,θ )θ t ( θ,θ ) d µ(θ ). g ñ θ θ (θ ) = x n ( θ,θ )θ t n ( θ,θ ) and g θ θ (θ ) = x( θ,θ )θ t ( θ,θ ) for all θ, θ. Observe that g ñ and g θ θ θ θ are both bounded by K + 1 and thus are both n L. Moreover snce x n and t n converge weakly to x and t, g ñ converges weakly to θ θ g θ θ. Now note that for all ˆµ m B, lm mn n µ m B g ñ θ θ (θ )d µ(θ ) lm n g ñ θ θ (θ )d ˆµ(θ ) = where the equalty follows snce g ñ converges weakly to g θ θ θ θ (θ ). Thus lm mn n µ m B g ñ θ θ (θ )d µ(θ ) mn µ m B g θ θ (θ )d ˆµ(θ ), (17) g θ θ (θ )d µ(θ ). (18)

21 Theoretcal Economcs 1 (2006) Optmal auctons wth ambguty 431 On the other hand, for each n let f n B be such that g ñ θ θ (θ )f n (θ )d µ(θ ) = mn f B g ñ θ θ (θ )f (θ )d µ(θ ). (19) (We know such an f n exsts snce the mnmzng set of prors s nonempty.) Snce B s weakly compact agan by passng to a subsequence, f n converges weakly to f B. Note that g ñ (θ )f n (θ )d µ(θ ) g θ θ θ θ (θ ) f (θ )d µ(θ ) g ñ (θ )f n (θ )d µ(θ ) g ñ (θ ) f (θ )d µ(θ ) θ θ θ θ + g ñ (θ ) f (θ )d µ(θ ) g θ θ θ θ (θ ) f (θ )d µ(θ ) K + 1 (f n (θ ) f (θ ))d µ(θ ) + g ñ (θ ) f (θ )d µ(θ ) g θ θ θ θ (θ ) f (θ )d µ(θ ). The last nequalty follows from the fact that g ñ (θ ) K + 1. Snce f n weakly θ θ converges to f B and g ñ converges weakly to g θ θ θ θ both terms on the rght hand sde of the last nequalty approach 0. Ths mples that by takng lmts n equaton (19), x g θ θ (θ ) f (θ )d µ(θ ) = lm mn n ( θ,θ )θ t n ( θ,θ ) f (θ )d µ(θ ), n f B whch n turn mples that mn f B g θ θ (θ )f (θ )d µ(θ ) lm n mn f B x n ( θ,θ )θ t n ( θ,θ ) f (θ )d µ(θ ). The prevous nequalty together wth (18) mples (17), whch concludes the proof. C. PROOF OF COROLLARY 1 Suppose that for some mechansm {(x 1,t 1 ),(x 2,t 2 )}, there exsts a bdder and some postve measure event such that for all θ and for all G mn S, q (θ,θ )dg (θ ) > mn H B q (θ,θ )d H(θ ). (20) We need to show that there exsts a full nsurance aucton that s strctly preferred by the seller. Let {(x 1,t 1 ),(x 2,t 2 )} be defned as n the proof of Proposton 1. We know that nf t G 1 (θ,θ ) + t 2 (θ,θ ) dg (θ )dg (θ ) S nf t1 (θ,θ ) + t 2 (θ,θ ) dg (θ )dg (θ ) + nf δ 1 ( θ,θ )dg (θ )dg ( θ ) + nf δ 2 ( θ,θ )dg (θ )dg ( θ ). (21)