Manipulative auction design

Transcription

1 Theoretial Eonomis 6 (2011), / Manipulative aution design Philippe Jehiel Department of Eonomis, Paris Shool of Eonomis and Department of Eonomis, University College London This paper onsiders an aution design framework in whih bidders get partial feedbak about the distribution of bids submitted in earlier autions: either bidders are asymmetri but past bids are dislosed in an anonymous way or several aution formats are being used and the distribution of bids, but not the assoiated formats, is dislosed. I employ the analogy-based expetation equilibrium (Jehiel 2005) to model suh situations. First-prie autions in whih past bids are dislosed in an anonymous way generate more revenues than seond-prie autions while ahieving an effiient outome in the asymmetri private values two-bidder ase with independent distributions. Besides, by using several aution formats with oarse feedbak, a designer an always extrat more revenues than in Myerson s optimal aution, and yet less revenues than in the full information ase whenever bidders enjoy ex post quitting rights and the assignment and payment rules are monotoni in bids. These results suggest an important role of feedbak dislosure as a novel instrument in mehanism design. Keywords. Aution design, analogy-based expetation equilibrium, manipulation. JEL lassifiation. C72, D82, D Introdution Standard equilibrium approahes of games with inomplete information (à la Harsanyi) assume that players know the distributions of signals held by other players as well as those players strategies as a funtion of their signals (see Harsanyi 1995). Yet, this requires a lot of knowledge that need not be easily aessible to players. Modern approahes to equilibrium rely on learning to justify this knowledge (see Fudenberg and Levine 1998 for an overview of the literature on learning in games). But, it is in general questionable that enough information feedbak is available to the players at the learning stage for onvergene to equilibrium to be reasonably expeted. Philippe Jehiel: jehiel@enp.fr I would like to thank B. Lipman (the editor) and two anonymous referees for helpful omments. I am also grateful to I. Esponda, L. Lamy, M. Pagnozzi, R. Spiegler, B. Wilson, J. Wooders, A. Wolinsky, and seminar partiipants at Roy seminar (Paris), Northwestern University, the Conferene in Honor of Ehud Kalai (Jerusalem, 2007), the CEMMAP Conferene on Autions (UCL, 2008), the Workshop on Biased Beliefs (UCL, 2008), SWET (Madrid, 2008), and the Third World Congress of Game Theory for useful Comments. I thank the European Researh Counil for support. Copyright 2011 Philippe Jehiel. Liensed under the Creative Commons Attribution-NonCommerial Liense 3.0. Available at DOI: /TE687

2 186 Philippe Jehiel Theoretial Eonomis 6 (2011) For the sake of illustration, onsider a series of first-prie autions of similar objets (say, iphones) involving eah time new bidders of observable harateristis i = 1 2 n (say, bidders of different age). Assume that every bidder knows his own valuation for the objet but not that of other bidders, and assume that the distribution of valuation depends on the observable harateristi i. In many ontexts, it seems unlikely that bidders would a priori know these distributions and how they depend on i. In suh ases, bidders would look at the bids submitted in earlier similar autions so as to form a judgement as to what the distribution of others bids is likely to be in the urrent aution of interest. 1 In many pratial aution designs suh as those used on ebay or treasury autions, bidders have aess only to the aggregate distribution of bids in past autions without being informed of the harateristis of the bidders of the orresponding bids. It is then dubious that bidders would be able to play a best response to the atual distribution of bids of the other bidders beause there is no way a bidder an assess the distribution of bids onditional on the observable harateristi based on the feedbak he reeives. Instead, bidders are more likely to play a best response to the onjeture that all bidders no matter what their harateristi is bid aording to the aggregate distribution of bids that mix the distribution of bids of all bidders. In the long run, assuming onvergene of the overall proess, bidders are not playing a Nash equilibrium, but an analogy-based expetation equilibrium with bidder-anonymous analogy partition (see Jehiel 2005 for the first exposition of this onept and Setion 2 for an appliation to the private value aution setup onsidered here). 2 As an alternative example, onsider promotions in an organization. Eah promotion takes the form of a ontest between several employees who eah make a proposal for the job task speifiation in ase of promotion (i.e., what they will effetively do for the organization if promoted). Eah employee has some private interest for the promotion (depending on how he values power, soial status, responsibilities...), whih is known to him only. Besides, while the riterion for the urrent promotion (based on the proposals) is typially known to the ontestants, the riterion may typially hange from one promotion to the next. 3 Nash equilibrium would require that ontestants know the distribution of others proposals given the riterion appliable to the urrent promotion. Assuming that ontestants form their expetations by looking at past promotions, this would require that employees have aess to the joint distribution of proposals and orresponding riterion. Yet, if employees have aess only to the distribution of proposals (and not of the orresponding riterion), ontestants are not able to play a Nash equilibrium. Instead, I propose that ontestants play a best response (given the urrent riterion) to the onjeture that the distribution of proposals is the same irrespetive of the 1 Typially, on ebay, one has aess to the history of bids made in autions of similar objets run in the last month. 2 In the language of eonometris, the model is not identifiable. Most applied eonometriians would partiularize the model by making an extra symmetry assumption (see Athey and Haile 2006). In a sense, this paper is assuming that bidders follow the same line as the eonometriians, even though this does not boil down to assuming that the true underlying problem is symmetri, as we shall see. 3 In the formal model to be desribed in Setion 2, the private interest should be identified with the valuation, and the proposal (even though generally multidimensional) should be identified with the bid (maybe as in soring autions).

3 Theoretial Eonomis 6 (2011) Manipulative aution design 187 riterion (and that it orresponds to the aggregate distribution of proposals they have aess to). This is again an analogy-based expetation equilibrium with appropriately hosen analogy partitions. In this paper, I generalize the above two examples. I onsider one-objet private values aution environments in whih the valuations are independently distributed aross bidders, bidders reeive oarse feedbak about the distribution of previous bids, and every single bidder partiipates in just one aution. Speifially, I onsider situations in whih, as in the first-prie aution example, only the aggregate distribution of bids with no referene to the harateristis of bidders is dislosed, and also situations in whih, as in the promotion example, different aution formats are being used (this is the analog of different riteria being used) and only the aggregate distribution of bids aross the various aution formats is being dislosed. Any ombination of suh oarse dislosures is also allowed. The equilibrium onept that is used to desribe the interation of bidders with suh oarse feedbak dislosure the analogy-based expetation equilibrium requires that bidders play a best response to the aggregate distribution of bids, as given by the feedbak they reeive. I explore whether and when it is the ase that the designer is better off when bidders play an analogy-based expetation equilibrium rather than a Nash equilibrium and how the answer is affeted by the speifi forms of feedbak dislosure and aution rules. Addressing suh questions opens new avenues in mehanism design. As it turns out, providing oarse feedbak about past behaviors as onsidered in this paper an enhane the designer s objetive, whih suggests the relevane of an instrument not previously onsidered in mehanism design. 4 To highlight the potential role of feedbak dislosure in mehanism design, I assume that (in addition to the format(s)) the designer is free to hoose whih kind of feedbak (within the lass speified above) to dislose to bidders. In the promotion example, hoosing whih feedbak to dislose sounds natural for the designer given that the feedbak is, in priniple, under the ontrol of the organization. In the aution example, this is an idealization that would fit well if a single seller had many similar objets to aution off over time. My framework in the aution ase an more usefully be interpreted as providing insights as to what kind of feedbak poliy an aution house suh as ebay should adopt so as to inrease the revenues of sellers (whih in turn determine the revenues of the aution house through the fees). I first assume that the main objetive of the designer is welfare maximization whereas the auxiliary goal is revenue (as is the ase in many government autions). The main question I address with this objetive in mind is, Can the designer do better than using a seond-prie aution (or equivalently an asending-prie aution) by using a oarse feedbak devie? In the lassi setup (relying on Nash equilibrium), the so-alled revenue equivalene theorem implies that the designer an do no better. This is so beause the seond-prie aution indues an effiient outome and any effiient mehanism that respets the 4 For a related investigation in the ontext of symmetri first-prie autions with affiliated signals, see the independent work of Esponda (2008a), whih is disussed in Setion 5.1.

4 188 Philippe Jehiel Theoretial Eonomis 6 (2011) partiipation onstraints of bidders must ahieve a revenue no greater than that of the seond-prie aution (see, for example, Milgrom 2004 for an exposition of the revenue equivalene theorem). In my setup, I show that the designer an sometimes do better, thereby illustrating a failure of the revenue equivalene theorem when the solution onept is the analogybased expetation equilibrium. Speifially, in the ase of two bidders with asymmetri distributions of valuations, I show that the first-prie aution in whih the designer provides as the aggregate distribution of bids feedbak, with no referene to the harateristi of the bidders (as onsidered above), always indues an effiient outome and always generates an expeted revenue that is stritly greater than that of the seond-prie aution no matter what the distributions of valuations are. I also provide onditions ensuring that suh a result holds true when there are more than two bidders. 5 I next onsider the ase in whih the designer is only interested in revenues and I assume that bidders an always veto the transation ex post (thereby limiting the sope for manipulation). The main question I am interested in is, Can the designer generate more revenues than in the lassi optimal aution haraterized by Myerson (1981)and Riley and Samuelson (1981)? I show that this is always so and that the best revenue so obtained lies stritly below the full information maximal revenue. In other words, a lever use of oarse feedbak oupled with a judiious hoie of format(s) may redue the informational rent left to bidders even though some rent must be left to bidders. 6 The rest of the paper is organized as follows. In Setion 2, the model is introdued together with the analogy-based expetation equilibrium and the mehanism design problem. In Setion 3, the first-prie aution with bidder-anonymous feedbak partition is analyzed. In Setion 4, it is shown how one an generate more revenues than in Myerson s optimal aution. Setion 5 offers a disussion, inluding how the paper relates to the literature and what would happen if the transation need not be approved ex post or if the designer ould use shill bidders. Setion 6 onludes. All proofs are given in the Appendix. 2. Basi definitions An objet is to be autioned off and there are n bidders i = 1 n. Eah bidder i knows his own valuation v i for the objet, but not that of the other bidders j i. The distribution of valuations is independent aross bidders. The valuation v i is drawn from a distribution with support [ d] and (ontinuous) density f i ( ),wheref i (v) > 0 for all v [ d] and d> 0. Bidders have quasilinear preferenes and they are risk neutral. That is, if a bidder with valuation v expets to win the objet with probability p and expets to make 5 Suh lear-ut revenue omparisons (at least in the two-bidder ase) should be ontrasted with the ambiguous revenue ranking between the first-prie aution and the seond-prie aution obtained with Nash equilibrium when bidders are asymmetri (see Maskin and Riley 2000). 6 The aution design and feedbak poliy used to prove that one an go stritly beyond Myerson s optimal aution revenue require the use of several (in fat two) aution formats and that bidders be informed only of the aggregate distribution of bids over the various formats. Suh aution designs do not learly resemble existing ones, and more work is required to map this theoretial result to pratial aution design. Suh a onstrution may possibly shed light on why transpareny need not be optimal in promotion ontexts.

5 Theoretial Eonomis 6 (2011) Manipulative aution design 189 (an expeted) transfer t to the seller, his expeted utility is pv t. The seller s valuation is v s. The objet is autioned off through possibly different aution formats M k, k K = {1 r}, whereformatm k is used with probability λ k. Eah aution format M k (whih together with λ k is hosen by the designer) is restrited to take the following form. Bidders i = 1 nsimultaneously submit a bid b i [0 d]. Based on the profile of bids b = (b i ) i=n i=1, with probability ϕk i (b) where i ϕk i (b) 1, bidder i is offered to obtain the objet in exhange for a payment τi k (b) to the seller. The transation takes plae if after being informed of τi k (b), bidder i approves the terms of the ontrat. For every i, k, ϕ k i (b) is a nondereasing funtion of b i and a noninreasing funtion of b j, j i; τi k(b) is a nondereasing funtions of b i and of b j, j i. Moreover, if b i <v s,thenϕ k i (b) = 0. The aution formats onsidered in this paper require that the bidders approve the terms of the ontrat ex post. That is, assuming that bidder i with valuation v i is delared the winner with a tentative transation prie τi k (b), there is effetively a transation (at prie τi k(b))onlyifv i >τi k (b), and otherwise there is no transation and the seller keeps the objet. 7 The monotoniity assumptions made on ϕ k i (b) and τk i (b) are in line with the aution interpretation: As a bidder inreases his bid, his probability of winning (weakly) inreases and so does the prie this bidder must pay. As a ompeting bidder inreases his bid, the probability of winning (weakly) dereases and the prie paid in ase of transation (weakly) inreases. Moreover, the requirement that if b i <v s,thenϕ k i (b) = 0 is meant to represent the idea that the valuation v s of the seller is the minimum starting point of the aution (as the seller would refuse to sell at a prie lower than v s ). It should be mentioned that the first-prie aution and the seond-prie aution with reserve prie no smaller than v s both belong to the above lass of autions, and that it is always possible to pik an aution in this lass that ahieves Myerson s optimal revenue no matter what the densities f i ( ) are. Moreover, the requirement that bidders should approve the terms of the ontrat ex post ensures that a bidder, no matter what bidding strategy he employs and no matter what he expets the distribution of other bids to be, is better off partiipating in the aution rather than staying outside. I have in mind situations in whih bidders have no prior idea about what the densities of valuations f i ( ), i I, are and in whih every individual bidder partiipates in just one aution. To form their expetations about the distribution of other bids, bidders rely on the feedbak about past play that is made available by the designer. Speifially, when aution format M k prevails, bidder i is assumed to be informed of the funtions ϕ k i (b) and τk i (b) that apply to him in this format (but he need not be informed of other harateristis of M k ; see more on this below). Thus, if bidder i with 7 I also assume that there is a tiny (rather than exatly zero) ost to anelling out the transation. This allows me to rule out razy behaviors from bidders with valuations v<v s.

6 190 Philippe Jehiel Theoretial Eonomis 6 (2011) valuation v i bids b i and expets the bid profile b i = (b j ) j i to be distributed aording to the random variable b i, his pereived expeted utility in M k is u k i (v i b i ; b i ) = Ẽ b i [ ϕ k i (b i b i ) max(v i τ k i (b i b i ) 0) ] Astrategyofbidderi is a family of bid funtions β i = (β k i ) k, one for eah aution format M k,whereβ k i (v i) denotes bidder i s bid in format M k when i s valuation is v i. 8 Nash equilibrium would require that for eah k and v i, bidder i plays a best response to the atual distribution of bids of bidders j i in M k.thatis, β k i (v i) arg max u k i (v i b i ; β k i ) b i where (with some slight abuse of notation) β k i stands for the random variable of bids (β k j (v j)) j i as generated by the densities (f j ( )) j i. As already highlighted, bidders are not assumed to know (or have aess to) β k j for every j and k. Instead, eah bidder i reeives partial feedbak about the distribution of bids observed in past autions. They play a best response to this feedbak (in a sense to be defined next) and a steady state is assumed to have been reahed. Speifially, I onsider the following lass of partial feedbak. Endow eah bidder i with a partition P i of the set {(j k): j I and k K} referred to as the analogy partition of bidder i. A typial element of P i is denoted by α i and referred to as an analogy lass of bidder i. The element of P i that ontains (j k) is denoted by α i (j k). The interpretation of P i is that bidder i gets informed only of the empirial distribution of bids of bidders engaged in past autions where all bids b j submitted in M k with (j k) α i are treated alike (i.e., they are not distinguished). I further assume that a steady state (in whih new sets of bidders with newly drawn valuations arrive eah time) has been reahed so that the empirial distributions of previous bids orrespond to the atual distributions. When making his hoie of strategy in aution format M k, bidder i is thus assumed to know only (in addition to ϕ k i (b) and τi k(b)) the aggregate distribution of bids in every α i. He is further assumed to play a best response to the onjeture that bidder j in format M k bids aording to the aggregate distribution of bids in α i (j k), the analogy lass to whih (j k) belongs. Formally, let A = (M k λ k P i ) i I k K denote an aution design. The solution onept is defined as follows. Definition 1. An analogy-based expetation equilibrium of A = (M k λ k P i ) i I k K is a strategy profile β = (β i ) i I suh that for every k and v i, β k i (v i) arg max u k i (v i b i ; β k i ) b i 8 Stritly speaking, allowing for mixed strategies β k i (v i) should be a distribution over bids. Yet, for my purpose, onsidering pure strategies is enough.

7 Theoretial Eonomis 6 (2011) Manipulative aution design 191 where β k i = (βk j ) j i and β k j is the aggregate distribution of bids in α i(j k). Thatis,β k j is the distribution of bids that assigns weight λ k / (j k ) α i (j k) λ k to the distribution β k j (v j ) as generated by the density f j ( ) for every (j k ) α i (j k), and the distributions, j i, are pereived by bidder i to be independent of eah other. β k j Remarks. (i) It should be mentioned that the feedbak reeived by bidders is about the distribution of individual bids and not about the distribution of bid profiles. 9 (ii) The analogy-based expetation equilibrium was first introdued in Jehiel (2005) for extensive form games and in Jehiel and Koessler (2008) for stati games of inomplete information. It is further interpreted in the ontext of private value autions in Setion 2.2, and it is related to other approahes in the literature, in partiular, the self-onfirming equilibrium in Setion 5.1. Various objetives for the designer are onsidered. The first objetive is a lexiographi riterion with welfare ranked first and revenues ranked seond. The seond objetive is the seller s expeted utility as measured by the revenue she gets when there is a transation and her valuation v s when there is no transation. In all ases, the designer is assumed to be risk neutral. She assesses an aution design A = (M k λ k P i ) i I k K aording to the expeted value of her objetive, assuming that bidders behave aording to an analogy-based expetation equilibrium of A Examples of analogy partitions and aution designs The lass of analogy partitions onsidered in Definition 1 is quite large, as it allows for both the bundling of bidders and the bundling of formats in any possible way. For the possibility results that show that the designer an do stritly better than using the finest analogy partition, I either onsider the bundling of bidders (Propositions 1 and 2) or the bundling of formats (Proposition 3), but not a ombination of the two. The more general formulation allowed by Definition 1 is useful for the impossibility result (Proposition 4), expressing that the full information benhmark is a strit upper bound on what the designer an hope to ahieve whatever the manipulation. Speifially, the following lasses of aution designs with publi feedbak (all P i are thesamearossi) play a entral role in the analysis of Propositions 1, 2,and3. Class 1: Bidder-anonymous analogy partition. Inthisase,thereisonlyoneaution format, and the feedbak is about the aggregate distribution of bids aross all bidders. That is, K ={1} and for all i I, P i ={ j I {(j 1)}}. For example, the 9 Aordingly, every bidder i treats every bidder j s distribution of bids, j i, as being independent of eah other. This fits in well when bidders do not have aess (or pay attention to) whether the past bids they observe were submitted at the same or different times. 10 As in Myerson (1981), one also impliitly assumes that the designer an hoose the analogy-based expetation equilibrium she likes best. Yet, whih analogy-based expetation equilibrium is played turns out to be inessential for the main results provided weakly dominated strategies are never played.

8 192 Philippe Jehiel Theoretial Eonomis 6 (2011) objet ould be sold through a first-prie aution and bidders would reeive feedbak about the aggregate distribution of bids with no referene to the harateristis of the bidders who generated the various bids. This is the situation studied in Setion 3. Class 2: Format-anonymous analogy partition. In this ase, bidders know the aggregate distribution of bids aross the different aution formats M k, k K, butthey differentiate the distribution of bids for the various bidders i I. That is, for all i I, P i ={ k K {(j k)}} j I. This situation orresponds to the ontest appliation mentioned in the Introdution anditisonsideredinsetion Interpretation The interpretation of an analogy-based expetation equilibrium is that it stands for the limiting outome of a learning proess in whih (1) at every stage there is a new aution and new bidders of observable harateristis i I, 11 (2) aution format M k is used with frequeny λ k, and (3) a bidder with harateristi i reeives as feedbak the aggregate empirial distribution of past bids in every α i. 12 If behaviors stabilize in suh a learning proess, it must be to an analogy-based expetation equilibrium, provided bidders onsider the simplest theory that is onsistent with the feedbak they reeive. The mehanism design perspetive onsidered in this paper orresponds to the idealization that a single designer an optimize on the aution formats M k, their frequenies λ k, and the analogy partitions P i provided to bidders, and that behaviors have stabilized to a orresponding analogy-based expetation equilibrium of A. As mentioned in the Introdution, suh a view is appropriate in situations in whih a single seller repeatedly sells similar objets or in situations in whih a single organization repeatedly organizes ontests for promotion. It may also be useful to understand the inentives of an aution house suh as ebay, whih is obviously interested in inreasing the sellers revenues (through the fees they generate), and whih an ontrol both the formats and the feedbak about past autions that is dislosed to bidders. The approah developed in this paper has a non-bayesian element in the sense that upon learning the oarse feedbak the designer reports to them, bidders do not update their belief about the distribution of others bids based on some (possibly subjetive) prior. Instead, bidders are assumed to onsider the simplest theory onsistent with the feedbak they reeive: They play a best response to the onjeture that the distribution of bids is uniformly the same over the various (j k) that are bundled in the same analogy lass. 13 I believe this is a natural assumption in many pratial situations of interest in whih (1) subjets would have no other data than the feedbak they reeive to form their 11 The profile of bidders harateristis is assumed to stay the same throughout the proess (see Setion 5 for some elaboration on the ase in whih the number of bidders may vary stohastially). 12 That is, a bidder with harateristi i is informed of the aggregate empirial distribution of past bids {b k j (j k) α i} with no referene to whih (j k) generated the bid. 13 Tehnially speaking, the analogy-based expetation equilibrium an be viewed as a refinement of some variant of the self-onfirming equilibrium in whih bidders adopt the simplest onjeture as just desribed; see elaborations in Setion 5.1.

9 Theoretial Eonomis 6 (2011) Manipulative aution design 193 prior, 14 and (2) the funtions ϕ k j and τk j that govern bidder j s inentive in format M k are either the same aross all (j k) that belong to the same analogy lass or they are not known to subjets with harateristi i, i j. 15 In suh situations, it would seem rather hard (in fat impossible) for subjets to understand how the distribution of bids varies aross different (j k) that belong to the same analogy lass simply based on the aggregate empirial distribution they are informed about: assuming that the distribution is the same aross these various (j k) seems foal and I propose it gives a good aount for bidders mode of thinking in suh situations. 2.3 Preliminaries A few preliminary observations follow. First, by piking a single aution format M and by adopting the finest analogy partition, the designer an always repliate the revenue generated in M when behaviors are assumed to be governed by Nash equilibrium. Thus, if the designer seeks to maximize revenues, she an always ahieve a revenue at least as large as Myerson s (1981) optimal revenue. The question is whether she an ahieve larger revenues. Seond, onsider an aution format M in whih bidder i has a dominant strategy. Then in any aution design inluding format M, an analogy-based expetation equilibrium requires that bidder i plays his dominant strategy in M (remember that bidders are informed of the alloation rule and the payment rule that applies to them in the format they are in). This is a straightforward observation, sine bidder i finds his strategy best no matter what his expetation about the distribution of others bids is and thus no matter how the aution design is further speified. Third, one of the aution designs that is studied falls in the following lass. There is one aution format M, whih respets the anonymity of bidders. That is, onsider two bid profiles b and b obtained by permuting the bids of players i and j. Then ϕ i (b) = ϕ j (b ) and τ i (b) = τ j (b ),andforallm i j, ϕ m (b) = ϕ m (b ), τ m (b) = τ m (b ). Consider the bidder-anonymous analogy partition defined above, and all A the orresponding aution design. One an relate the analogy-based expetation equilibria of A to the Nash Bayes equilibria of the game Ɣ ba (A) defined by the aution format M in whih, for eah i, the distribution of bidder i s valuation has the average density f(v i ) = i j f j(v i )/n instead of f i (v i ). Claim 1. A symmetri strategy profile is an analogy-based expetation equilibrium of A if and only if it is a Bayes Nash equilibrium of Ɣ ba (A). Fourth, another lass of aution designs A onsidered below is suh that the various aution formats M k in A satisfy ϕ k i (b i b i ) = ϕ i (b i b i ) for all k K (for example, in all formats, the objet is alloated to the player who submitted the highest bid). When the 14 This in partiular requires that bidders have no prior knowledge about the densities f i, not even about the frequenies λ k. 15 In the first-prie aution with bidder-anonymous analogy partition, the former property applies (φ k j and τj k are the same aross all bidders). It may be argued that in the ontest for promotion example mentioned in the Introdution, the latter property is often met given that the riteria used for different promotions are often not very transparent to outsiders.

10 194 Philippe Jehiel Theoretial Eonomis 6 (2011) format-anonymous analogy partition prevails, one an relate the analogy-based expetation equilibria of suh aution designs A to the Nash Bayes equilibria of the following game referred to as Ɣ fa (A). Game Ɣ fa (A). Eah bidder i (simultaneously) submits a bid b i. The objet is assigned to bidder i with probability ϕ i (b i b i ). Prior to bidding, bidder i is privately informed of his valuation v i drawn from f i ( ) and of his method of payment k defined by τ k i (b i b i ). The methods of payment k are identially and independently drawn aross bidders and every bidder i is subjet to the method of payment k with probability λ k. 16 Claim 2. Suppose that the format-anonymous analogy partitions prevail and that in all aution formats M k of A, ϕ k i (b i b i ) = ϕ i (b i b i ) for all k K and i I. Thenastrategy profile β is an analogy-based expetation equilibrium of A if and only if it is a Bayes Nash equilibrium of Ɣ fa (A). 3. Effiieny and revenues Assume the designer s valuation v s is 0 and that the designer is interested in both effiieny and revenues, and suppose that the primary objetive of the designer is effiieny while revenue is only the seondary objetive. In the standard Myerson s paradigm, the so-alled revenue equivalene result holds. That is, if two mehanisms result in the same alloation rule and the expeted payment made by any bidder i with minimal valuation v i = is 0, then both mehanisms must yield the same revenues. Sine an effiient outome an be ahieved by a seond-prie aution SPA, the standard approah (i.e., relying on Nash equilibrium) onludes that the designer an do no better than using a SPA. I now observe that, within the framework introdued in Setion 2, the designer an sometimes ahieve stritly larger revenues (than that obtained through the SPA) while still preserving effiieny, thereby illustrating a failure of the alloation equivalene in a manipulative aution design setup. Besides, this gain in revenues is ahieved by using a fairly standard aution format (with the bidder-anonymous analogy partition). Proposition 1. Consider a two-bidder i = 1 2 aution setup with asymmetri distributions (F 1 ( ) F 2 ( ) on a set of stritly positive measure). There is a unique analogy-based expetation equilibrium of the first-prie aution with bidder-anonymous analogy partition. Moreover, this analogy-based expetation equilibrium indues an effiient outome and it generates a stritly higher revenue than the seond-prie aution. The revenue gain is 1 4 (F 1(v) F 2 (v)) 2 dv dβ(v) dv (F 1(v) F 2 (v)) 2 dv > 0 where β(v) = v xf(x)dx/f(v), f(x)= (f 1(x) + f 2 (x))/2 and F(v)= v f(x)dx Compared to the true aution design, the differene is that the methods of payment are independently distributed aross bidders in Ɣ fa whereas they are (perfetly) orrelated in Ɣ. 17 The funtion β( ) is the equilibrium bid funtion in a symmetri two-bidder first-prie aution with density of valuations f.assuh,β( ) is an inreasing funtion.

11 Theoretial Eonomis 6 (2011) Manipulative aution design 195 What is the intuition for the above result? First, observe that the use of the bidderanonymous analogy partition leads the bidders (whatever their harateristi) to best respond to the same distribution of bids, whih given the anonymous harater of the first-prie aution, ensures effiieny. Seond, the use of the bidder-anonymous analogy partition leads the bidders to feel that they are in ompetition with a fititious bidder who has a distribution of valuations that is the average distribution between the distributions of the various bidders (this essentially follows from Claim 1 above). In the two-bidder ase, the prie level in the seond-prie aution is determined by the lowest valuation, hene by the weak bidder. Roughly, the manipulation generated by the bidder-anonymous analogy partition enhanes revenues beause it makes the strong bidder feel that the weak bidder is less weak than he really is. When there are more than two bidders, the first-prie aution with bidderanonymous analogy partition remains effiient, but the revenue omparison with the seond-prie aution an go either way, depending on the form of the asymmetry of the distributions. 18 The following result in whih R SPA denotes the expeted revenue generated in the seond-prie aution with densities of valuations f i ( ), i I, andr denotes the expeted revenue generated in the fititious seond-prie aution with symmetri densities of valuations f i (v) f(v)= (f 1 (v) + f 2 (v) + +f n (v))/n provides a generalization of Proposition 1 to the n-bidder ase. Proposition 2. Consider an n-bidder setup and assume that R R SPA. There is a unique analogy-based expetation equilibrium of the first-prie aution with bidderanonymous analogy partition. Moreover, this analogy-based expetation equilibrium indues an effiient outome and it generates a stritly higher revenue than the seond-prie aution. Proposition 2 is proven by noting that (1) the revenues in the seond-prie aution or in the first-prie aution with symmetri densities oinide (this is the standard revenue equivalene result) and that (2) the expeted revenue in the first-prie aution with bidder-anonymous analogy partition is stritly above the expeted revenue in the firstprie aution with symmetri densities of valuations f(v)= (f 1 (v) + f 2 (v) + +f n (v))/ n where the latter result follows beause (i) bidders employ the same strategy in both ases (by Claim 1) and (ii) the distribution of highest valuation in the asymmetri ase first-order stohastially dominates the distribution of highest valuation in the 18 To see that revenues an go either way, onsider first a situation with two bidders whose distribution of valuations is onentrated around d and a third bidder whose distribution of valuation is onentrated around. It is readily verified that the first-prie aution with bidder-anonymous feedbak partition generates less revenues than the seond-prie aution (whih ahieves a revenue approximately equal to d). Consider next a situation with one bidder whose distribution of valuations is onentrated around d while other bidders have a distribution of valuations onentrated around. The first-prie aution with bidderanonymous feedbak partition generates more revenues than the seond-prie aution (whih generates a revenue very lose to ). Thus the revenue omparison an go either way.

12 196 Philippe Jehiel Theoretial Eonomis 6 (2011) symmetri ase. While R R SPA always holds in the two-bidder ase (thereby explaining why Proposition 1 holds with no restrition on f 1 ( ) and f 2 ( )), this need not be so in the more than two-bidder ase The above insights (Propositions 1 and 2) are distint even though related to Myerson s insight about how to inrease revenues in asymmetri autions. An important impliation of Myerson s analysis is that in the asymmetri ase, ompetition between bidders should be biased in favor of weak bidders so as to inrease revenues. The net effet of suh biased autions is that some ineffiienies are indued, letting the weak bidder sometimes win the objet. As Proposition 1 (and to some extent Proposition 2) illustrates, the use of the bidder-anonymous feedbak partitions allows one, in some ases, to symmetrize (a bit) the ompetition without sarifiing on effiieny. Of ourse, this is ahieved by moving away from the Nash equilibrium paradigm, whih the use of partial feedbak permits. 21 Comment. In some appliations, the distribution of winning bids as opposed to the aggregate distribution of all bids is available to bidders. From this feedbak, bidders an ompute a best response strategy based on the assumption that all bidders bid aording to the same distribution (this might be argued to be the simplest onjeture in this ase). Applied to the asymmetri first-prie aution format, the analog of Claim 1 would reveal that in this ase bidders would bid as if the distribution of valuations of eah bidder were F(v) = ( n i=1 F i (v)) 1/n. 22 It is interesting to note that in this ase, no matter how many bidders are around, revenues would be higher in the first-prie aution (in whih only the winning bids are observed) than in the seond-prie aution Intuitively, one would expet to have R R SPA when there is one stronger bidder who is faing symmetrially weak bidders, but the required notion of strong bidder does not boil down to the first-order stohasti dominane relation in the distributions of valuations. 20 It should also be noted that when the distributions of valuations are nearly the same aross bidders (say the umulative funtions differ up to ε), then the revenues of the two aution designs differ aording to a smaller magnitude (of order ε 2 ). When the distributions are very asymmetri, the differene of revenues an be quite substantial. For example, in the two-bidder ase onsidered in Proposition 1, assume that the distribution of valuations of one bidder is onentrated around d, whereas the distribution of valuations of the other bidder is onentrated around. In this ase, the first-prie aution with bidder-anonymous feedbak partition provides a revenue of (3 + d)/4, whih should be ompared with the revenue of the seond-prie aution. Clearly, as d gets large relative to, the revenue gain, (d )/4, an be quite substantial in suh asymmetri setups. 21 There is some experimental evidene that in asymmetri first-prie autions, the observed bidding strategy is less asymmetri than Nash equilibrium requires (see Güth et al. 2005). Suh an experimental finding is onsistent with the view that bidders when onsidering whih bid to submit may look at previous bids without paying attention to the strength of the bidder who submitted the bid (whih was modeled here through the apparatus of the bidder-anonymous analogy partition). 22 Indeed, symmetry implies that all bidders bid aording to the same inreasing bid funtion. Thus, only the bid of the highest valuation bidder would be observed, thereby providing the desired result. 23 The analog of Step 2 in the proof of Proposition 1 is Theorem 1 in Cantillon (2008), and Step 3 holds with an equality beause the distribution of the highest valuation is the same with (F 1 F n ) and (F F).

13 Theoretial Eonomis 6 (2011) Manipulative aution design Optimal autions Assume now that the designer seeks to maximize her expeted payoff as measured by the revenue she gets when there is transation and her valuation v s when there is no transation. Proposition 3 omprises our first main observation. 24 Proposition 3. The largest expeted payoff that the designer an ahieve in a manipulative aution design is stritly larger than her expeted payoff in Myerson s optimal aution (denoted by R M ). The intuition for Proposition 3 is as follows. Myerson s optimal aution an always be implemented in suh a way that every bidder has a (weakly) dominant strategy and ex post quitting rights of bidders are fulfilled (think of the seond-prie aution with well hosen reserve prie in the symmetri regular ase). One an now think of an aution design in whih this aution format all it Myerson design (MD) is mixed with a little bit of first-prie aution (FPA) with v s reserve prie, and bidders get to know only the aggregate distribution of bids over the two aution formats. In format MD, the strategies are the same as in the standard ase (beause bidders have a weakly dominant strategy in MD). In format FPA, bidders play a best response to the aggregate distribution of bids over the two formats. For many hoies of MD, this onstrution need not indue an expeted payoff to the seller that is higher than R M. 25 But there are many variants of MD in whih submitted bids are first monotonially transformed before the original format is applied. For a suitable hoie of suh a variant, the onstrution leads bidders in FPA to bid very aggressively beause they are led to think that by shading their bid too muh, the hane of winning in FPA gets too small. In the limit, a bidder with valuation v may be indued to bid very lose to v whenever v>v s. Given that suh bidding strategies in FPA indue an expeted payoff to the seller that is lose to the full information optimal expeted payoff R F = E(max i (v i v s )) and given that R F >R M, the result of Proposition 3 follows. Proposition 3 establishes that the designer an do better than using Myerson s optimal aution (with fine analogy partitions), but how muh an she gain? Clearly, the best that she an hope to get in aution designs with ex post quitting rights annot exeed the maximal full information expeted payoff R F = E(max i (v i v s )). This trivially follows from the observation that a winner of the aution would always objet if he were asked to pay more than his valuation. As it turns out, the designer s best expeted payoff in our manipulative design setup lies stritly below R F. Proposition 4. The best expeted payoff that the designer an ahieve in a manipulative aution design with ex post quitting rights is stritly smaller than the full information expeted payoff R F if <v s <d. 24 By inspeting the proof of Proposition 3, one an see that all analogy-based expetation equilibria (not employing weakly dominated strategies) of the aution design onsidered there are suh that the designer obtains higher revenues than in Myerson s optimal aution. Thus, the onlusion of Proposition 3 would hold under the stronger full implementation requirement (provided one restrits oneself to equilibria not employing weakly dominated strategies). 25 It an be heked, for example, that in the ase of uniform distributions, a mix of seond-prie autions and first-prie autions would have no effet on revenues.

14 198 Philippe Jehiel Theoretial Eonomis 6 (2011) In the proof of Proposition 3, some (Myerson-optimal) mehanism MD implementable in (weakly) dominant strategy was mixed with a little bit of first-prie aution with reserve prie v s, FPA, and the seller s expeted payoff obtained in FPA was shown to be lose to R F. However, suh a onstrution requires that the frequeny with whih FPA is used is set suffiiently small. As one inreases the frequeny of FPA, the manipulation loses its fore and, of ourse, in the limit as the designer almost always piks FPA, one gets the standard seller s expeted payoff generated in the first-prie aution with reserve prie v s, whih following Myerson s analysis annot be greater than R M. What Proposition 4 establishes is that within the lass of mehanisms under study one an never reah R F whatever the manipulation. This observation is of partiular interest beause if the designer ould freely hoose the belief of bidders (with no onstraint), she ould get a payoff arbitrarily lose to R F (see the arguments surrounding Proposition 7). Thus, Proposition 4 establishes that there is some limitation to manipulation imposed by the requirement that the feedbak should be orret (even if oarse). To get an intuition for Proposition 4, think of a symmetri senario in whih the aution design A uses the format-anonymous feedbak partition. To get lose to R F,it would be required that, in all aution formats M k used in A, every bidder with valuation v>v s pays a prie lose to his valuation when he wins. This implies that every bidder should pereive making almost zero profit. 26 Yet a bidder with valuation v d an always onsider submitting a bid β k ((d + v s )/2) in format M k, and he should expet there to win and make a payment no more than (d + v s )/2, thereby making a net gain of no less than (d v s )/2 > 0 whenever other bids b j are smaller than β k ((d + v s )/2). 27 Now, it is not a priori lear what pereived probability bidder i attahes in format M k to the event that all b j, j i, are smaller than β k ((d + v s )/2). But one an always rank the formats M k by inreasing order of β k ((d + v s )/2). For those formats M k suh that β k ((d + v s )/2) is above the median of β k ((d + v s )/2) (in the distribution in whih β k ((d + v s )/2) has probability λ k ), it is lear that in the format-anonymous analogy partition, this pereived probability is no less than 1 2 Pr(max j i v j <(d+ v s )/2). A ontradition is obtained given that, in format M k, a bidder with valuation v d annot pereive his payoff from following β k (d) (this should be approximately 0) tobe stritly smaller than his payoff from following β k ((d + v s )/2) (this has been shown to be no less than 1 2 (d v s)/2pr(max j i v j <(d+ v s )/2), whih is stritly positive). Comments. (1) From the proof of Proposition 4,ifv s, it is not lear whether a similar onlusion arises, beause it is not then a priori guaranteed that for the full information payoff to arise, a bidder should neessarily pereive making negligible profit. It should be noted though that with the additional requirement that in every format M k,thepayment made by the winner should lie in between the largest and the seond largest bid, 26 The pereived payoff might a priori differ from the atual one due to the manipulative harater of the design. To establish this, I make use of v s >and of the monotoniity of τ k (b i b i ), but I suspet this holds muh more generally. 27 This is beause in format M k, the atual bid of bidder i with valuation v<(d+ v s )/2 is β k (v), and a bidder with valuation (d + v s )/2 should win and pay a prie approximately equal to his valuation whenever he meets bidders with lower valuations.

15 Theoretial Eonomis 6 (2011) Manipulative aution design 199 then the same onlusion as in Proposition 4 would arise even if v s <, beause the rules of the formats would imply that it is weakly dominated to bid above one s own valuation (and this property an easily be used to establish that bidders annot pereive making signifiant gains if R F is to be ahieved). (2) If the designer were allowed to ommit to offering positive payments to losers and if the payments from the winner were not assumed to be monotoni in bids, then the designer ould get a revenue lose to R F while still preserving the ex post partiipation onstraints of bidders. 28 The restrition on mehanisms (i.e., not allowing positive payments to losers and imposing that payments from winners be monotoni in bids) an then be thought of as resulting from the regulatory desire to protet bidders from manipulation. 5. Disussion 5.1 Related literature This paper is related to several strands of literature. First, the onept used in this paper follows Jehiel (2005), Jehiel and Koessler (2008), and Ettinger and Jehiel (2010) who define the analogy-based expetation equilibrium for extensive form games of omplete information, stati games of inomplete information, and multistage games of inomplete information, respetively. The analogy partitions onsidered in this paper are slightly less general than those onsidered in Jehiel and Koessler (2008) exeptfor the fat that bidder i s own bids an be lumped together with others bids in bidder i s analogy partition. 29 The main novelty of the approah pursued here is that the feedbak partitions are viewed as a hoie made by the designer. That is, they are not exogenously given as in Jehiel (2005)orJehiel and Koessler (2008) To see this, onsider a symmetri two-bidder senario in whih bidders valuations are identially distributed on ( d). Consider an aution design with format-anonymous analogy partition and two formats M and M used in equal proportion. In format M, the equilibrium bids lie in [0 d]; in format M,theequilibrium bids lie in {0} [d 2d]. In eah format, a bidder with negative valuation bids 0 in equilibrium. In both M and M, the bidder with highest bid wins the aution if this bid is stritly positive. In M, ifb i (0 d) for i = 1 2, the winner pays his own bid and the loser reeives no transfer. In M, ifb i {0} [d 2d] for i = 1 2, the winner i pays b i d and the loser reeives no transfer. The idea is to augment the transfers in M and M to over all bid profile onfigurations, even for bid realizations that never our in the respetive formats. So in M, a (losing) bidder submitting b i (0 d)is offered a promise of transfer h(b i ) if b j (d 2d), and in M, a (winning) bidder submitting b i (d 2d) is offered a transfer h(b i ) if b j (0 d).bysuitablehoiesofh and h, one an ensure that for v i > 0, bidding β(v i ) = v i in M and bidding β(v i ) = v i + d in M is an analogybased expetation equilibrium. [For example, in the uniform distribution ase, h(b) = b 2 /(2d) b/d and h(b) = (b d) 2 /(2d) (b d)/d. These funtions are determined so that the expeted pereived transfers orrespond to those that would be made in a SPA with 0 reserve prie.] With suh bidding strategies, the expeted revenues generated in eah format are R F, and thus the designer gets R F in expetation. 29 As already mentioned, suh an extension is partiularly adapted if eah individual bidder i partiipates in just one aution. 30 By assuming that the designer an ontrol the analogy lasses of the bidders, this paper takes the more rational interpretation of the onept, that is, assuming that players make maximal use of the information provided to them. In Jehiel (2005), it is alternatively suggested that an analogy-based expetation equilibrium might arise beause players may not pay attention to all of the details of their past experienes, whih amounts to a bounded rationality interpretation of the onept.