On the Generic (Im)possibility of Full Surplus Extraction in Mechanism Design

Transcription

1 On the Generc (Im)possblty of Full Surplus Extracton n Mechansm Desgn Avad Hefetz vka Neeman Prelmnary Verson: February 27, 2004 Abstract A number of studes, most notably Crémer and McLean (985, 988), have shown that n Harsany type spaces of a fxed fnte sze, t s genercally possble to desgn mechansms that extract all the surplus from players, and as a consequence, mplement any outcome as f the players prvate nformaton were commonly known. In contrast, we show that wthn the set of common prors on the unversal type space, the subset of prors that permt the extracton of the players full surplus s shy. Shynesssanoton of smallness for convex subsets of nfnte-dmensonal topologcal vector spaces (n our case, the set of common prors), whch generalzes the usual noton of zero Lebesgue measure n fnte-dmensonal spaces. Journal of Economc Lterature Classfcaton numbers: D44, D82, H4. Keywords: Surplus extracton, nformaton rents, mechansm desgn, prvate nformaton, unversal type space, genercty, prevalence. Department of Economcs and Management, The Open Unversty of Israel, avadhe@openu.ac.l Department of Economcs, Boston Unversty, and Center for Ratonalty and Department of Economcs, Hebrew Unversty of Jerusalem, zvka@bu.edu

2 Introducton Does the holdng of relevant prvate nformaton necessarly confers a postve economc rent? Surprsngly, the answer gven by the lterature to ths queston s negatve. A number of studes, ncludng, most notably, Crémer and McLean (985, 988), have shown that under standard assumptons such as the exstence of a common pror, a fxed fnte number of types, rsk neutralty, and no lmted lablty, t s genercally possble to mplement any outcome as f the players prvate nformaton were commonly known. In partcular, a seller, for example, should genercally be able to extract the full surplus of any number of bdders n an aucton. As these full-surplus-extracton results mply that the players prvate nformaton s (genercally) rrelevant, they have been sad to cast doubt on the value of the current mechansm desgn paradgm as a model of nsttutonal desgn (McAfee and Reny, 992, p. 400). Snce full-surplus-extracton results make heavy use of the fxed fnte type space assumpton, t s natural to ask whether or not the possblty of full-surplus-extracton extends to the most general consstent prvate nformaton type space magnable, or to the collecton of consstent (n the sense of Harsany, ) subspaces of the unversal type space (Mertens and amr, 985), n whch t s common knowledge that each player knows her own sgnal, and n whch every other consstent prvate nformaton type space can be embedded. Supposng that any common pror on ths unversal type space could just as well serve as a plausble model of a stuaton nvolvng asymmetrc nformaton, s t typcally the case that full-surplus-extracton s possble? Ths s the queston addressed n ths paper. The assumpton of rsk neutralty and no lmted lablty whch s the other mportant assumpton necessary for full surplus extracton s mantaned throughout ths paper. If belefs are endowed wth the mnmal topology that allows for the formulaton of each player s belefs about the state of nature and the belefs of other players (the topology of weak convergence), then the set of common prors wth fnte support s dense n the space of all common prors (Mertens and amr, 985). Combnng ths observaton wth the results of Crémer and McLean (985, 988) for fnte type spaces mples that the set of prors that permt full-surplus-extracton s dense n the space of all possble common prors. Recently, Neeman (200) showed that full-surplus-extracton s possble only f the type space has the property that every possble belef of every player about other players types s assocated (wth probablty one) wth a unque valuaton or prvate sgnal of the player (Neeman called ths property belefs determne preferences ). Neeman (200) also showed that arbtrarly close to any consstent fnte type space, there s another consstent fnte type The reason we confne our attenton to consstent subspaces of the unversal type space s twofold. Frst, confnng attenton to Harsany-consstent subspaces of the unversal type space and ther assocated common prors s standard practce n nformaton economcs. Indeed, ths practce, whch has been called the Harsany doctrne, s consdered by some to be a hallmark of Bayesan ratonalty (see the dscusson n Aumann 998, Gul 998 and Morrs 995). Second, the unversal space has a product structure t ncludes all the possble combnatons of players belefs and prvate sgnals. Hence, as should become clear below when the terms are defned, belefs do not determne preferences n the unversal type space, and consequently full-surplus-extracton s genercally mpossble there. Thus the queston s settled n ths case. 2

3 space n whch belefs do not determne preferences, and consequently full surplus cannot be extracted. It thus follows that both the set of prors that allow for full-surplus-extracton (henceforth, FSE prors), and ts complement the set of prors that do not allow for fullsurplus-extracton (henceforth, NFSE prors), are dense n the space of all fnte-support prors, and hence also n the space of all prors. In partcular, wth the topology of weak convergence, there s no topologcal sense n whch one of these sets can be sad to be larger (.e., open and dense) than the other. However, just as both the ratonals and the rratonals are dense n the set of real numbers although the set of rratonals s larger n other senses (cardnalty, Lebesgue measure), t s also concevable that one of the subsets of common prors above s larger than the other n some meanngful sense. One may thnk of two general approaches that may permt such a sharper result. Frst, t may be argued that the topology of weak convergence on prors s not the natural topology to apply n a strategc settng. 2 Intutvely, two prors can be sad to be close f and only f they nduce smlar equlbrum behavor. Indeed, prelmnary results (Kaj and Morrs, 998) suggest that such a noton of strategc proxmty may nduce a stronger topology on the set of prors. However, a proper defnton of strategc proxmty n general type spaces and ts characterzaton n terms of belefs s not yet avalable. If and when such a characterzaton s obtaned, then t may turn out that the resultng stronger topology renders one of the subsets of common prors above both open and dense, whle ts complement would not be dense. Second, t s also possble to consder non-topologcal notons of genercty, such as the measure-theoretc noton of full Lebesgue measure. Unfortunately, the noton of full Lebesgue measure cannot be appled drectly because t can meanngfully capture the dea of a set beng large only n fnte-dmensonal spaces. In contrast, the space of common prors on the unversal type space (or even the smaller space of common prors wth a fnte support) s not only nfnte but nfnte-dmensonal. It s therefore necessary to consder a measure-theoretc noton of genercty that can be appled n nfnte-dmensonal topologcal vector spaces. Such an approprate noton, called prevalence, was orgnally conceved by Chrstensen (974) and Hunt et al. (992) and further developed by Anderson and ame (200) as an extenson of the dea of full Lebesgue measure to nfnte-dmensonal spaces. The complement of a prevalent set s called shy. A collecton of shy sets n a fnte-dmensonal space s dentcal toacollectonofsetswthlebesguemeasurezero. Inannfnte-dmensonal space, shy sets retan the propertes of zero-probablty events: no open set s shy, and the collecton of shy sets s closed under subsets, under translatons, and under countable unons. We show that the subset of FSE prors s shy wthn the set of common prors on the unversal type space. It therefore follows that the complement of the set of FSE prors, or the subset of NFSE prors, s generc. The same result also obtans f attenton s restrcted to the subset of prors wth fnte-support. The proof s based on the followng lemma, whch stems from Neeman s (200) observaton that full-surplus-extracton requres that players belefs determne ther prvate sgnals 2 See Morrs (2002) for such an argument. 3

4 or preferences. Only weghted averages of FSE prors yeld FSE prors, whle a weghted average of a NFSE and any other pror yelds a NFSE pror. Ths asymmetry n favor of the NFSE prors delvers the result. What makes ths result mathematcally non-trval s the fact that the set of FSE prors s dense n the consstent unversal type space. The rest of the paper proceeds as follows. For smplcty, nstead of consderng a general mechansm desgn problem wth nterdependent players types, we begn n the next secton wth the consderaton of the classc problem of a seller of an object who desgns an aucton for n rsk neutral bdders wth prvate valuatons, wth the goal of maxmzng hs expected revenue. In Secton 3, we explan how our results can be appled to any mechansm desgn problem wth nterdependent types. Secton 4 surveys the related lterature, and contans a dscusson of the relatonshp of our results to those of Crémer and McLean. 2 Surplus Extracton n Sngle Object Auctons wth Prvate Values We consder the problem of a seller who wshes to desgn an aucton that would maxmze the expected revenue he obtans by sellng some object to one out of a set of n rsk neutral bdders wth prvate valuatons for the object. Each bdder or player may refuse to partcpate n the seller s aucton, but f she agrees to partcpate, then she s bound by the outcome of the aucton. Let N = {,..., n} denote the set of bdders or players. Each bdder has a certan nonnegatve valuaton or wllngness to pay for the object whch we denote by v V. The set of bdder s valuatons V s assumed to be a complete, separable, metrc space (n partcular, V may be fnte). The payoff to a bdder wth valuaton v who wns the object wth probablty q and who pays an expected amount x s gven by q v m. We refer to v as bdder s preference or preference type. Let V V V n. The set V s the basc space of uncertanty for ths problem. The behavor of the bdders n the aucton may obvously depend on ther wllngness to pay for the object. It therefore follows that the bdders behavor may also depend on ther belefs about other bdders wllngness to pay, because such belefs convey possbly mportant nformaton about the way other bdders wll behave n the aucton. For the same reason, belefs about belefs are also mportant, and so are belefs about belefs about belefs, and so on, ad nfntum. A complete analyss of the seller s problem therefore requres a model that allows for the specfcaton of the bdders entre nfnte herarchy of belefs about belefs about belefs... about whatever s relevant n the aucton. Such nfnte herarches of belefs may be convenently encoded n what s known as a type space. 2. Type Spaces Bdder s prvate nformaton s captured by ts type θ Θ. The sets of bdders types Θ, N, are assumed to be complete, separable, metrc spaces. For every space X, let (X) denote the space of probablty measures over X. Each type θ Θ s assocated wth a 4

5 preference type bv (θ ) V whch descrbes θ s wllngness to pay for the object, and wth a belef-type b b (θ ) (Θ ) whch s a belef, or a probablty measure, on the space of other bdders types Θ Q j6= Θ j. The space of probablty measures (Θ ) s endowed wth the topology of weak convergence. Each type of each bdder s assumed to know ts own wllngness to pay for the object and ts belefs. Because we focus our attenton n ths secton on a prvate values model, each type θ s preference type bv (θ ) s defned ndependently of θ s belef type b b (θ ).Ths assumpton s relaxed n the next secton. 3 AproductspaceΘ Q N Θ of the players type spaces s called a prvate values type space. Each profle of types θ Θ s called a state of the world. 2.2 The Prvate Values Unversal Type Space Gven the basc space of uncertanty V V V n and the set of bdders N, there exsts 4 a prvate values unversal type space T PV = Y N T PV nto whch every other prvate values type space can be mapped n a belefs-preservng way. That s, for every type space Θ Q N Θ there exsts a unque set of measurable mappngs 5 satsfyng E : Θ T PV, N bv (E (θ )) = bv (θ ) and b b (E (θ )) (A) = b (θ ) E (A) for every measurable set A T PV, where E : Θ T PV s defned by E ³ (θ j ) j6= =(E j (θ j )) j6=. The unversal type set T PV of bdder N s somorphc to the product space V T PV 3 The assumpton that each type knows ts own belef s captured by defnng b (θ ) as a probablty measure over Θ rather than over Θ Θ. The mpled presumpton about the bdders ntrospectve ablty s standard, and s mantaned throughout the paper. 4 The proof of exstence follows from a slght adaptaton of the arguments n Mertens and amr (985), Brandenburger and Dekel (993), and Hefetz (993). 5 whch are n fact also contnuous 5

6 by the mappng τ ³ bv (τ ), b (τ ). Thus n what follows we use the terms T PV and V T PV nterchangeably. Fnally, for the rest of ths secton we drop the superscrpt PV from the notaton whenever there s no rsk of confuson. 2.3 Prors A probablty measure p on a prvate values type space Θ = Q N Θ s called a pror for bdder f bdder s belef-types b (θ ) are the posterors of p. That s, p s a pror for bdder f for every real-valued contnuous functon ϕ : Θ R µ ³ θ ³θ, θ d b (θ ) dp Θ (θ )= ϕ (θ) dp (θ) () Θ Θ ϕ where p Θ s the margnal of p on Θ. A probablty measure p on Θ s called a common pror, or pror for short, f t s a pror for every bdder N. The support of a pror p s called a Harsany-consstent subspace. It s mmedate from the defnton that the set of prors for bdder s convex: If p,p 0 (Θ) are two prors for bdder, then so s αp +( α) p 0 for every α [0, ]. It follows that the set of common prors s also convex. Because every prvate values type space can be embedded n the prvate values unversal type space, no loss of generalty s mpled by restrctng attenton to prors on the prvate values unversal type space T. We thus take the set of prors on the prvate values unversal space, denoted P PV, to be the set of relevant envronments for our study. Defnton. Aprorp (T ) satsfes the Belefs-Determne-Preferences (BDP) property 6 for bdder N f there exsts a measurable subset T p T such that the margnal p T of p on T assgns probablty to T p, p T (T p )=, and no par of dstnct types τ 6= τ 0 n T p hold the same belefs for every two dfferent types τ,τ 0 T p. ˆb (τ ) 6= ˆb (τ 0 ) A pror p that satsfes the belefs-determne-preferences property for bdder s called a BDP pror for bdder. Aprorp that s a BDP pror for every bdder N s called a BDP pror. Because any par of dstnct types τ 6= τ 0 n the prvate values unversal space dffer 6 The noton of belefs-determne-preferences generalzes the one n Neeman (200) and s closely related to Bergemann and Morrs (2003) one-to-one property and to d Aspremont et al. s (2002) noton of no free belefs. Θ 6

7 ether by ther belef-type or by ther preference-type, 7 there s no par of dstnct types n T p who hold dentcal belefs but dfferent preferences. In other words, a pror p satsfes the BDP property for bdder f there exsts a p-probablty set V B p where B p (T ), and a functon that maps bdder s belefs to ts wllngness to pay Φ p : Bp V n³ o such that T p s somorphc to the graph bb (τ ), bv (τ ) : τ T p of Φ p. We show that BDP s necessary for full-surplus-extracton. Specfcally, we show that f aprorp permts the extracton of bdder s full surplus p-almost surely, then p s a BDP pror for player. By the revelaton prncple, no loss of generalty s mpled by assumng that the seller employs an ncentve compatble and ndvdually ratonal drect revelaton aucton game or mechansm hq : T [0, ],m : T [0, ] N n whch each bdder s asked to report ts type τ T, and then to partcpate n a lottery n whch t pays an amount m (t), and wns the object wth probablty q (t). A mechansm hq,m N s ncentve-compatble (IC) f every type τ T of every bdder N maxmzes ts expected payoff by truthfully reportng ts type, or (q (τ, τ )ˆv (τ ) m (τ, τ )) dˆb (τ )( τ ) T (q (τ 0, τ )ˆv (τ ) m (τ 0, τ )) dˆb (τ )( τ ) (IC) T for every τ 0 T. A mechansm hq,m N s ndvdually-ratonal f every type τ T of every bdder N prefers to partcpate n the mechansm than to opt out, or (q (τ, τ )ˆv (τ ) m (τ, τ )) dˆb (τ )( τ ) 0. (IR) T Defnton. Aprorp permts the full-surplus-extracton from a set K N of bdders f there exsts an ncentve compatble and ndvdually ratonal mechansm hq,m N that generates an expected payment to the seller that s equal to the full surplus generated by the bdders n K or X m (τ) dp (τ) = max {ˆv (τ )} dp (τ). K K T A pror that permts the full-surplus-extracton from the set K of bdders s called a fullsurplus-extracton (FSE) pror for K. 7 If there were two dstnct types τ 6= τ 0 wth the same preferences and belefs n the prvate values unversal type space, then the belefs-preservng mappngs of type spaces nto the unversal space would not have been unque, n contradcton to the defnton of the unversal type space. T 7

8 Remark. Fx a pror p. In order to extract the full surplus from the bdders n K N n the envronment descrbed by p, np-almost surely every state of the world τ T, the seller must sell the object to the bdder n K who has the hghest wllngness to pay for t at τ at an expected prce that s equal to ths bdder s valuaton of the object. The ndvdual ratonalty constrant mples that the seller cannot sell the object to any bdder n K at τ for a hgher prce, and sellng the object for a lower prce mples a falure to extract the full surplus that s generated by the bdders n K. It therefore follows that the ndvdual ratonalty constrant of any type τ j T j of a bdder n K that wns the object wth a postve probablty under a mechansm hq,m N that extracts the full surplus of the bdders n K must be bndng. Remark 2. Note that the fact that a pror may permt the extracton of the full surplus from the set N of bdders does not mply that t s possble to extract the full surplus of each sngle bdder. For example, f there are two bdders and t s commonly known that bdder s wllngness to pay for the object s strctly lower than that of player 2 s, t s possble to extract the full surplus of the set of bdders {, 2} f and only f t s possble to extract the full surplus of bdder 2 alone. Conversely, t can be shown that f t s possble to extract the full surplus of each sngle bdder, then t s also possble to extract the full surplus of the set N of bdders. Proposton. Aprorp that s a FSE pror for bdder s a BDP pror for bdder. Proof. Suppose that p s a FSE pror for bdder. Lethq,m N be an ncentve compatble and ndvdually ratonal mechansm that extracts the full surplus of bdder. By Remark, bdder must wn the object wth p-probablty under the mechansm hq,m N,and bdder s ndvdual ratonalty constrant must be bndng wth p-probablty under the mechansm hq,m N. Suppose that p s not a BDP pror for bdder. It follows that there exst two dsjont measurable subsets of bdder s types, A,A 0 T, that each have a postve p-probablty and the same range of belefs p T (A ) > 0, p T (A 0 ) > 0, ˆb (A )=ˆb (A 0 ) (T ), but dfferent valuatons. That s, f τ A and τ 0 A 0 are such that ˆb (τ 0 )=ˆb (τ ) then ˆv (τ 0 ) < ˆv (τ ). 8

9 In partcular, for every type τ A there exsts a type τ 0 A 0 such that ˆb (τ 0 )=ˆb (τ ) but ˆv (τ 0 ) < ˆv (τ ). It follows that (q (τ, τ )ˆv (τ ) m (τ, τ )) dˆb (τ )( τ ) T (q (τ 0, τ )ˆv (τ ) m (τ 0, τ )) dˆb (τ )( τ ) T = (q (τ 0, τ )ˆv (τ ) m (τ 0, τ )) dˆb (τ 0 )( τ ) T > (q (τ 0, τ )ˆv (τ 0 ) m (τ 0, τ )) dˆb (τ 0 )( τ ) T 0. The frst nequalty follows from the (IC) constrant for type τ ; the followng equalty follows from the fact that ˆb (τ 0 )=ˆb (τ ); the next strct nequalty follows from the fact that ˆv (τ 0 ) < ˆv (τ ) and that q (τ 0, τ )=for p-almost every type τ 0 A 0 ; and the last nequalty follows from the (IR) constrant for type τ 0. It therefore follows that bdder s ndvdual ratonalty constrant s not bndng for p-almost every type τ A. A contradcton. Conversely, t can be shown that f p s a BDP pror for bdder, then for every ε>0 t s possble to extract bdder s full surplus up to ε (McAfee and Reny, 992); f, n addton, bdder has only fntely many types, then t s possble to extract bdder s full surplus (Crémer and McLean, 988). 2.4 Genercty In fnte dmensonal spaces, genercty s often dentfed wth full Lebesgue measure. A set that has Lebesgue measure zero s consdered nongenerc, small, or atypcal. A set that has full Lebesgue measure s consdered generc, large, or typcal. The stuaton n nfntedmensonal spaces s more complcated. Unlke the Lebesgue measure n a fnte-dmensonal Eucldan space R k, whch s spread unformly across the space, n nfnte-dmensonal spaces there s no (sgma-addtve) measure that flls up the space. For example, n an nfntedmensonal separable Banach space, any open ball of radus r > 0 contans an nfnte sequence of dsjont open balls of radus r, so f a translaton-nvarant measure were to 4 assgn a postve measure to these balls, then the r-ball would be assgned an nfnte measure for any r>0. 8 Therefore, probabltes or measures n nfnte-dmensonal spaces are not satsfactory devces for determnng whether events are typcal or not. An appealng noton of smallness n nfnte-dmensonal spaces s based on the observaton that an event E n a fnte-dmensonal Eucldan space R k has Lebesgue measure zero 8 Furthermore, confnng attenton to full-support quas-nvarant measures, whch preserve null-sets under translatons (such as the Gaussan measures on the Eucldean spaces), s unhelpful ether. Under farly general condtons, f there does not exst a non-trval full support nvarant measure on an nfnte-dmensonal space, then nether does there exst such a quas-nvarant measure (see, e.g., Yamasak 985). 9

10 f and only f there exsts a postve measure µ on R k such that E and all ts translatons {x + y : x E},y R k, have µ-measure zero. Chrstensen (974) and Hunt, Sauer, and York (992) have reled on ths observaton to propose a noton of smallness that concdes wth full Lebesgue measure n fnte dmensonal spaces and that extends naturally to nfnte-dmensonal spaces. They defned a Borel subset of a complete metrc topologcal vector space to be shy f there exsts a postve measure µ on the space such that the set and all ts translatons have µ-measure zero, and called the complement of a shy set prevalent. They showed that shy sets satsfy all the requrements one would expect from small or neglgble events. In partcular, a subset of a shy set s shy, every translaton of a shy set s shy, a countable unon of shy sets s shy, and no open set s shy. Anderson and ame (200) have adapted Chrstensen (974) and Hunt et al. s (992) defnton to the case n whch the relevant parameter set s a convex subset C of a topologcal vector space X. Because we are nterested n determnng the genercty of the set of FSE prors relatve to the space of prors on the unversal type space, ths s the defnton whch s approprate for our purpose. It turns out that for our analyss t s not necessary to rely on Anderson and ame s general defnton of shyness, but rather on a smpler and stronger noton called fnte shyness. Let λ H denote the Lebesgue measure on a fnte-dmensonal subspace H X. Defnton. (Anderson and ame, 200) A unversally measurable 9 subset E C s fntely shy n C X f there exsts a fnte-dmensonal subspace H X such that λ H (C + p) > 0 for some p X and λ H (E + x) =0for every x X. An arbtrary subset F X s fntely shy n C f t s contaned n a fntely shy unversally measurable set. Anderson and ame (200) show that f a set E s fntely shy n C then t s also shy n C. AsubsetY C s sad to be prevalent n C f ts complement C\Y s shy n C. 0 Example. Anderson and ame (200). Everywhere dfferentable concave functons are fntely shy n the cone of all concave functons. Example 2. Stnchcombe (200). Both the subspaces of purely atomc measures and purely non-atomc measures are fntely shy n the space of all measures (n any topology n whch they are Borel, or more generally, unversally measurable). 9 A subset E X s unversally measurable f t s measurable wth respect to the completon of every regular Borel probablty measure on X. 0 In ther defnton, Anderson and ame requred the convex subset C X to be completely metrzable, but as they menton n a footnote, the defnton makes sense even wthout ths requrement, whch s needed only for establshng some enhanced propertes of shyness and prevalence (e.g., f E s prevalent n F and F s prevalent n G then E s prevalent n G). Ths addtonal requrement s not needed for establshng the basc propertes of shy sets, namely that a subset of a shy set s shy, that every translaton of a shy set s shy, that a countable unon of shy sets s shy, and that no open set s shy. 0

11 2.5 FSE Prors are Non-Generc In ths secton we show that the set of FSE prors, denoted F, sfntely shy n the set of prors on the prvate values unversal type space, P. Postve multples of prors n P consttute a convex cone of (postve) measures. Takng the dfferences of pars of such measures yelds the vector space of sgned measures that are generated by P, denoted M. We assume that the topologcal vector space M s endowed wth a topology that satsfes the followng two propertes: () the mappngs (p, p 0 ) p + p 0 (α, p) αp are contnuous for every par of prors p, p 0 P and scalar α R; and(2)asubseta R s Borel f and only f for every par of prors p, p 0 P the one-dmensonal set of weghted averages {αp +( α) p 0 : α A} (2) s a Borel subset of M. These two propertes are satsfed for a large varety of topologes on M, ncludng the topology of weak convergence and the topology of the total varaton norm, but not for extremely strong topologes such as the totally dsconnected topology n whch every subset of M s open. The result below apples to any metrc topology on M whch satsfes the two propertes above and that s also at least as strong as the topology of weak convergence. We start wth two lemmata. Lemma. Let f,f 2 B be two dfferent BDP prors for bdder, andletf f 2 = f + f be the Jordan decomposton of the sgned measure f f 2 on T, where f + and f are two mutually sngular postve measures on T. Then both f + and f are postve multples of BDP prors for player. Proof. Because f and f 2 satsfy (), or µ ϕ (τ, τ ) d b (τ )( τ ) df T (τ )= T T T ϕ (τ) df (τ) (3) for every player N and every contnuous ϕ : T R, so does f f 2. Because f f 2 = f + f and both f +,f are mutually sngular postve measures, both f +,f satsfy (3) f as well. Therefore, +, f (T ) are common prors. f + f We next show that f + f +, f are BDP prors for player. Because f f k,k=, 2 are BDP prors for player, there exst subsets T f k T such that f k T f k =, In partcular, f a topology of strategc proxmty as dscussed n the ntroducton belongs to ths range of topologes, then our result mples that the shy subset of FSE prors cannot be open n that topology. Hence, only the set of NFSE prors remans a potental canddate for beng open and dense n such a topology.

12 the projecton T f k of T f k on T s the graph of a functon Φ f k : B f k S (where B f k s the projecton of T f k on (T )), and for every τ =(...,τ j,...) T f k and every player j 6=, the margnal b j (τ j ) T of τ j on T assgns probablty to T f k. Ths means that f T f T f 2 6= then for every τ =(...,τ j,...) T f T f 2 and every player j 6=, the margnal b b j (τ j ) T of τ j on T assgns probablty to T f T f 2. It then follows that the graphs of Φ f and Φ f 2 concde on B f B f 2 almost surely accordng to both f T and f T 2, because on T f T f 2 n eachoftheprorssanaverageofj s belefs bbj (τ j ):τ =(...,τ j,...) T f T 2o f.sof we defne Φ : B f B f 2 S by ½ Φ f (b Φ (b )= ) b B f Φ f 2 (b ) otherwse then both f T and f T 2 assgn probablty to the graph of Φ. Ths mples that the margnal of the sgned measure f f 2 on T assgns measure zero to the complement of the graph of Φ. So ths must also be true for the margnals of the fnte, postve measures f + and f on B S. In other words, both the margnals of and f f on T assgn probablty to the graph of Φ, whch means that f + f + and f f are BDP prors. Lemma 2. The set B of BDP prors for bdder s a Borel subset of the space of prors P. Proof. If the lemma obtans when P s equpped wth the topology of weak convergence, t also obtans for any stronger metrc topology. It s therefore enough to proceed assumng that P s equpped wth the topology of weak convergence. By defnton, a pror p P s a BDP pror f and only f the margnal of p on T = V (T ) s concentrated on a measurable graph Φ p : B p V. Ths s expressble by countably many condtons, n the followng way. Snce V s separable, there s a countable collecton {A n } n of subsets of V whch s closed under complements and fnte unons and generates the Borel sgma-feld of V. Hence there are also countably many parttons {Γ m } m of V to fntely many dsjont subsets n o N m A nm k {A n } n. Smlarly, Snce (T ) s separable, there exsts a countable collecton ª k= Y l of subsets of (T l ) whch s closed under complements and fnte unons and generates the Borel sgma-feld of (T ). Hence, there are also countably many parttons n {Λ r } r of (T ) to fntely many dsjont subsets n Y lr k o L r k= Y l ª l. f + f + 2

13 The margnal of p on T = V (T ) s concentrated on the graph of Φ p f and only f n o N m for every partton Γ m = A nm k of S N[ m p k= k= ³ A nm k (Φ p ) ³ A nm k T = Intutvely, as the parttons (Γ m ) m of V get fner, the unon of the rectangles A nm ³ k (Φ p ) A nm k approxmates ncreasngly well the graph of Φ p n o. N m Now, for each partton Γ m = A nm n ³ o N k of V, (Φ p ) m A nm k s a partton of k= k= (T ), that can be approxmated arbtrarly well (n terms of the probabltes assgned to the partton members by the margnal of p on (T )) by parttons n {Λ r } r. Hence, the margnal of p on T = V (T ) s concentrated on a measurable graph from (T ) to n o N m V f and only f for every natural number q and for each partton Γ m = A nm k of n o k= L r V there exsts a partton Λ r = of (T ) wth L r = N m and N[ m p k= Y lr k ³ A nm k k= Y lr k T q Formally, therefore, the set F of FSE prors s \ \ \ [ N[ m ³ p P : p A nm k N m q r k= Y lr k T q whch s a Borel subset of the space of prors P. Theorem. The set B of BDP prors for bdder s fntely shy n the space P of prors on the unversal type space. Proof. Let g P be a non-bdp pror for player, and let c P such that c, g are mutually sngular. Consder the one-dmensonal subspace of M H = {α (g c) :α R}. By (2), Lebesgue measure λ H s well defned on H. We have that α (g c) +c = αg + ( α) c P f and only f α [0, ] and hence λ H (P c) = > 0. Moreover, H \ {0} conssts entrely of sgned measures α (g c),α6= 0whose margnals on T are not concentrated on a graph of some measurable functon Φ : (T ) S, because the margnal of g s not concentrated on such a graph by assumpton, and g c. 3

14 However, λ H (B + x) =0for every x M. In fact, H (B + x) s ether empty or a sngleton. Indeed, assume by contradcton that f + x = h = α (g c) f 2 + x = h 2 = α 2 (g c) where h,h 2 H, f,f 2 B and α >α 2. Then By lemma, n the Jordan decomposton f f 2 =(α α 2 ) g (α α 2 ) c f f 2 = f + f f +,f are non-negatve multples of BDE prors for player. However, snce both (α α 2 ) g and (α α 2 ) c are mutually sngular postve measures, by the unqueness of the Jordan decomposton we must have f + = (α α 2 ) g f = (α α 2 ) c But ths s mpossble, snce (α α 2 ) g s a postve multple of an non-bde pror for player. Corollary. The set F of Full-Surplus-Extracton prors for player s fntely shy n the set P of all common prors on the unversal space T. Proof. By proposton we have F B, so the corollary follows from the defnton of fntely shy sets. Remark 3. Inspecton of the argument presented n ths secton reveals that t also mples that the set F f of fnte-support FSE prors for player s fntely shy n the space of all fnte-support prvate values prors P f on the unversal space T. The proofs apply verbatm. Fnally, we have shown n ths secton that the set of prors on the unversal type space that permts the extracton of the players full surplus s shy. The queston of whether or not t s genercally possble to approxmate full surplus remans open. We conjecture, but have been so far unable to prove, that for every ε>0, both the sets of prors n whch t s possble to extract up to ε of total surplus, and no more than ε below total surplus are large n the sense that nether s shy. 2 2 A partal answer to ths queston s provded by Neeman (200), who for the case of publc good provson, descrbes an example n whch, f belef do not determne preferences, then the probablty that the publc good can be provded decreases to zero wth the number of players whle effcency requres that the publc good be provded wth probablty. Ths result may be nterpreted as mplyng that the total surplus that can be extracted from the players converges to zero at the same tme that the total surplus that could be generated by the players remans unformly bounded away from zero. 4

15 3 Implementaton wth Interdependent Valuatons In ths secton, we demonstrate how the results obtaned n the prevous secton for an aucton problem wth prvate values, can be generalzed to any mechansm desgn problem wth nterdependent types. Specfcally, we ask whether a gven decson rule, whch s a mappng from players types nto outcomes s genercally mplementable. Whenever possble, we rely on the notaton used n the prevous secton. Let N = {,..., n} be a fnte set of players, and X a measurable set of outcomes. The players preferences over outcomes depend on the state of nature k K. Thespaceofstates of nature K s our basc space of uncertanty. It s assumed to be a complete, separable, metrc space, that s endowed wth ts Borel σ-feld. When the state of nature s k K, the outcome x X prevals, and player receves a monetary transfer m, her payoff s gven by where u (x, k)+m u : X K R s a Borel measurable functon. The players are assumed to be expected utlty maxmzers. 3. Type Spaces For every player N, the set of player s types Θ s assumed to be a complete, separable, metrc space. Every type θ Θ s assocated wth a probablty measure on the space of states of nature K and the other players types Θ = Q j6= Θ j. The space of probablty measures (K Θ ) s endowed wth the topology of weak convergence. Wth a slght abuse of notaton, we say that Θ (K Θ ). Ths formulaton, whch mples that the uncertanty of θ Θ s about K Θ but not about Θ, captures the dea that each type has a suffcently developed ntrospectve ablty to determne ts own belef. Type θ s belef type b (θ ) (Θ ) s the margnal θ Θ of the probablty measure θ on the other players types Θ. Type θ s preference type bv (θ ) s any verson of the expected payoff functons ³ U x; θ, θ : X Θ R, θ Θ that satsfes Θ U ³ δ ³θ, θ ; θ, θ ³ dθ Θ = u δ ³θ, θ, κ dθ K Θ for every measurable decson rule δ : Θ X. ThetypespacestheproductΘ Q N Θ of the players type sets. Each θ Θ s called a state of the world. 5

16 The prvate-values settng of the prevous secton s a partcular case of the formulaton descrbed n ths secton. If a type θ s a product probablty θ = θ K θ Θ ³ then the expected payoff functons U x; θ, θ are ndependent of θ, and can therefore be denoted U ( x; θ ). In the settng of the sngle object prvate values aucton consdered n the prevous secton, outcomes x =(x,...,x n ) are gven by vectors that descrbe the probablty wth whch each bdder or player wns the object. Bdders payoffs are lnear n the probablty wth whch they wn the object and are ndependent of both other bdders types and the probabltes wth whch other bdders wn the object. Hence, f we let denote the vector that has n the -th place and 0 everywhere else, then for every vector x, U (x; θ )=x U ( ; θ ). Moreover, because U ( ; θ ) descrbes θ s payoff when t wns the object for sure, whch we denoted n the prevous secton by bv (θ ), every type s preferences can be completely descrbed by ts preference type bv (θ ). To further llustrate the defnton, consder now the case of a sngle object pure common value aucton wth two bdders. In ths case k K s the true value of the object. Suppose that bdder knows the value k wth certanty; that bdder 2 has no prvate nformaton, only a belef whch specfes the probabltes p k of the potental values of k; and that all of ths s common knowledge among the bdders. Then bdder 2 has a sngle type θ 2. Bdder s types have the form 3 θ k = δ k δ θ2, k K, respectvely. Bdder s preference and belef types are therefore gven by bv θ k = U x; θ k, θ 2 = x k b b θ k = δ θ2 The unque type θ 2 of bdder 2 s a probablty measure over K Θ, whch assgns probablty p k to the combnaton k, θ k for k K, and zero probablty to any other combnaton n K Θ. The preference and belef types of θ 2 are gven by bv 2 θ2 = U2 x; θ2,θ k = x2 k b b2 θ2 θ k = pk Thus bdder 2 s preference type depends non-trvally on bdder s type θ k. 3.2 The Unversal Type Space Gven the basc space of uncertanty K and the set of players N, there exsts a unversal type space T = Y N T 3 δ denotes the unt-mass probablty measure. 6

17 nto whch every other type space T can be unquely mapped n a belefs-preservng way (Mertens and amr, 985; Brandenburger and Dekel, 993; and Hefetz, 993). That s, for every type space Θ there exsts a unque set of measurable mappngs 4 (E : Θ T ) N satsfyng E (θ )(A) =θ E (A) for every measurable A K T, where E : K Θ K T s defned by ³ E k,(θ j ) j6= = ³k,(E j (θ j )) j6= It turns out that n the unversal type space for player, T, s somorphc wth (K T ) (and not just wth a subset of t) for every N. We therefore refer to T and (K T ) nterchangeably. The prvate-values unversal type space T PV descrbed n the prevous secton s a subset of the unversal type space T that s presented here n the specal case n whch K = Q N V. It s the subset of T n whch t s commonly known that each player s types τ T, N, have a product form as follows τ = τ V τ V T. The defnton of a common pror on the unversal space s the same as n secton 2.3 above. The space of envronments of nterest s the set of prors P on the unversal type space T. 3.3 EDR Prors are Non-Generc In secton 2 we have consdered prvate-values envronments of a partcular knd, n whch the prvate preferences of a player could be represented by a one-dmensonal valuaton. We now proceed to the general (quas-lnear) setup defned at the begnnng of ths secton. Defnton. Aprorp permts the mplementaton of a decson rule δ : T X f there exsts an ncentve compatble and ndvdually ratonal drect revelaton mechansm δ,(m ) N where m : T R denotes the payment to player as a functon of the players types, that mplements δ. 5 A pror that permts the mplementaton of every decson rule s called an 4 whch are n fact also contnuous. 5 A drect revelaton mechansm δ, (m ) N s ncentve compatble f (U (δ (τ, τ )) + m (τ, τ )) dˆb (τ )( τ ) (U (δ (τ 0, τ )) + m (τ 0, τ )) dˆb (τ )( τ ) T T for every player N, and player s types τ,τ 0 T. It s ndvdually-ratonal f (U (δ (τ, τ )) + m (τ, τ )) dˆb (τ )( τ ) 0 T for every player N, and player s types τ T. 7

18 EDR pror. Defnton. A pror p permts the full extracton of the players surplus relatve to a decson rule δ : T X f there exsts an ncentve compatble and ndvdually ratonal drect revelaton mechansm δ, (m ) N that mplements δ wth payment functons that leave each type of each player wth zero surplus. 6 Remark 4. As n the prevous secton, the revelaton prncple mples that n the two defntons above, no loss of generalty s entaled by restrctng attenton to drect revelaton mechansms. Remark 5. The prevous secton was devoted to nvestgatng the possblty of the extracton of the players full surplus relatve to the ex-post effcent allocaton rule n the context of a sngle object prvate values aucton. Remark 6. Aprorp that permts full extracton relatve to a decson rule δ also permts the mplementaton of δ but the opposte need not be true. For example, the second prce aucton mplements the ex-post effcent allocaton rule n a sngle object aucton envronment wth prvate values, but as we have seen n the prevous secton, t s not genercally the case that t s possble to extract the full surplus of the bdders relatve to the ex-post effcent allocaton rule. Hence the statement that t s mpossble to mplement a gven decson rule s stronger than the statement that t s mpossble to extract the players surplus relatve to ths rule. Recently, Aoyag (998) and d Aspremont, Crémer, and Gérard-Varet (2002) showed that n models wth at least 3 players and a fxed fnte number of types for each player that s larger than or equal to 2, t s genercally possble to mplement every decson rule. 7 In contrast, we show below that fnte-support EDR prors are BDP prors. Snce the set B f of fnte-support BDP prors s fntely shy wthn the set P f of all fnte-support prors, we conclude that the set E f of fnte-support EDR prors s non-generc wthn P f. In order to establsh ths result, we mpose the mld assumpton that for every player there exsts an outcome x 0 that f mplemented, generates a payoff of 0 for player regardless of player s type. Lettng the players opt out of the mechansm ensures the exstence of such outcomes. Proposton 2. A fnte support EDR pror s a BDP pror. 6 That s, the payment functons (m ) N are such that m (τ, τ ) dˆb (τ )( τ )= U (δ (τ, τ )) dˆb (τ )( τ ) T T for every player N, and player s types τ T. 7 Ther noton of mplementaton does not requre ndvdual ratonalty but rather budget-balance. However, every budget balanced mechansm can be transformed nto an ndvdualy ratonal mechansm by addng asuffcently large constant to each player s payment functon m. 8

19 Proof. Suppose that p s not a BDP pror for player. Thenplayer has two types τ,τ 0 T that each have a postve p-probablty, the same belefs ˆb (τ )=ˆb (τ 0 ) b (T ), but dfferent preference types ˆv (α 0 ) 6= ˆv (α ). That s, there exst a profle of other players types τ T such that b (τ ) > 0 andanoutcome x X such that (wthout loss of generalty) U ( x; τ, τ ) >U ( x; τ 0, τ ) 0. Defne the decson rule δ : T X by ½ x τ =(τ 0 δ (τ) =, τ ) x 0 otherwse Consder any system of monetary transfers m : T R, N and suppose that the mechansm δ, (m ) N s ncentve compatble. In partcular, for type τ 0 X (U (δ (τ 0, τ );τ 0, τ )+m (τ 0, τ )) b ( τ ) τ T X (U (δ (τ, τ );τ 0, τ )+m (τ, τ )) b ( τ ) τ T or U ( x; τ 0, τ ) b ( τ )+ X m (τ 0, τ X ) b ( τ ) m (τ, τ ) b ( τ ). τ T τ T Therefore, f nstead of truthfully reportng ts type, τ reports t s type τ 0, then ts expected payoff s X (U (δ (τ 0, τ );τ, τ )+m (τ 0, τ )) b ( τ ) τ T = U ( x; τ, τ ) b ( τ )+ X m (τ 0, τ ) b ( τ ) τ T > U ( x; τ 0, τ ) b ( τ )+ X m (τ 0, τ ) b ( τ ) τ T X m (τ, τ ) b ( τ ) τ T = X (U (δ (τ, τ );τ, τ )+m (τ, τ )) b ( τ ) τ T 9

20 n contradcton to the presumed ncentve compatblty of δ, (m ) N. It follows that the decson rule δ cannot be mplemented, so p s not an EDR pror. Corollary 2. The set of fnte support EDR prors s fntely shy n the set of prors wth fnte support P f. 3.4 Implementaton of Effcent Decson Rules As explaned n the prevous subsecton, f players belefs determne ther preferences, then not only can a seller extract the full surplus, but any socal choce functon can be mplemented. The fact that, as we have shown n the prevous subsecton, players belefs do not genercally determne ther preferences, mples that not all socal choce functons can be mplemented. Ths stll leaves open the queston of whether or not a specfc decsonrule can be genercally mplemented. An answer to ths queston can be provded for the case of effcent decson rules. Implementaton of a decson rule requres that the mechansm desgner be able to nduce players to reveal both the preference and belef components of ther types. A player s belefs about other players types can always be fully extracted at a cost by standard arguments (see e.g. d Aspremont and Gerart-Varet, 979). Thus a mechansm desgner may generally face a trade-off betweenthecostandbeneft of extractng a player s belef. However, f the decson rule to be mplemented s effcent, then as shown by Bergemann and Morrs (2003), the players belefs can be extracted at no cost. It therefore follows that t s possble to provde a precse characterzaton of whether a gven socal choce functon s (nterm) mplementable on arbtrary fnte type spaces n terms of condtons that arse when lookng at standard mplementaton n an envronment wth stochastcally ndependent types (for detals, see Bergemann and Morrs, 2003). The gst of ths characterzaton s that after a player s belef type has been costlessly extracted, then f the player s belefs do not determne ts preferences, the player has to be gven some rent n order to nduce t to reveal ts payoff type truthfully, n a smlar way to the rent players have to be gven when ther types are stochastcally ndependent. 4 Related Lterature Followng an example n Myerson (98) that showed that a seller n an aucton may be able to explot the presence of correlaton among bdders to extract the bdders full surplus, Crémer and McLean (985, 988) showed that a monopolstc seller can genercally extract the full surplus of rsk neutral consumers and bdders, respectvely, n models wth a fxed number of types. 8 McAfee and Reny (992) constructed a smlar aucton to the one descrbed by Crémer and McLean that (approxmately) extracts the full surplus of the bdders when the number of bdders types s uncountably large, but dd not explctly address the 8 Crémer and McLean (988, Appendx B) have ndcated how some of ther results can be generalzed to allow for a contnuum of types. 20

21 ssue of genercty. McAfee et al. (989), Johnson et al. (992), and Brusco (998) have establshed related results n more specfc contexts. For a general formulaton of ths result, whch allows for a contnuum of multdmensonal, mutually payoff relevant, agents types, see Johnson et al. (2002). Recently, Aoyag (998) and d Aspremont et al. (2002) have used a smlar argument to the one used by Crémer and McLean (985, 988) to show that t s genercally possble to mplement any decson rule n models wth fnte type spaces. 9 A number of authors have argued that the condtons that are mposed n order to obtan these full-rent-extracton results, whle standard n many applcatons, are nevertheless very strong. Crémer and McLean (988) suggested that full rent extracton s not robust to the ntroducton of rsk averson or lmted lablty constrants, and emphaszed the dependence of these results on the common pror assumpton. Followng ther suggeston, Robert (99) showed that for any gven aucton mechansm, when agents are rsk averse or face lmted lablty constrants, the functon that relates the common pror to the seller s proft and to total surplus (and hence also to the sum of nformaton rents captured by the agents) s contnuous n the pror. Snce t s known that agents do obtan postve nformaton rents n ndependent envronments, Robert concluded that full nformaton rent extracton also fals n nearly ndependent envronments wth rsk averse agents or agents that face lmted lablty constrants. More recently, Laffont and Martmort (2000) have establshed the contnuty of the mechansm s outcome functon also for envronments wth rsk-neutral agents who are not constraned by lmted lablty, but who may form collusve coaltons. Intutvely, the reason that full rent extracton fals under these crcumstances s that the aucton mechansms that extract the full buyers rent rely on lotteres whose varance ncreases to nfnty at ndependence. Thus, n nearly ndependent envronments, mechansms that rely on such lotteres volate the buyers lmted lablty or partcpaton constrants. Because these lotteres also prescrbe payments to and from agents that strongly depend on the actons of other agents, mechansms that rely on such lotteres are hghly susceptble to colluson among the agents, and fal n nearly ndependent envronments where these payments are large. Fnally, ths paper makes a contrbuton to the growng lterature about robust mechansm desgn that has stemmed out of Robert Wlson s vew that further progress n game theory depends on succesve reducton n the base of common knowledge requred to conduct useful analyses of practcal problems (Wlson, 987). 20 As shown by Neeman (200), full-surplus extracton hnges on the fact that t s commonly beleved that a player s belef determnes, or predcts wth certanty, the player s preferences. Once ths assumpton s relaxed, the full surplus of the players cannot be extracted. Ths paper presents a model n whch t s shown that t s genercally ncorrect to assume that the desgner and players mantan such common belef assumptons. 9 What turns out to be the necessary and suffcent condton for mplementaton of every decson rule was frst ntroduced by d Aspremont and Gérard-Varet (982). d Aspremont et al. (2002) demonstrate that ths condton (condton B) s strctly weaker than Aoyag s (998) strct regularty condton whch have been shown to be suffcent for mplementaton of every decson rule. 20 See, e.g., Bergemann and Morrs (2003), Chung and Ely (2004), Neeman (200,2003), Wensten and Yldz (2004) and the references theren. 2

22 4. The Relatonshp to Crémer and McLean (988) Crémer and McLean (988) showed that wthn the set of models wth a fxed fnte number of types n 2 for each player (or equvalently, wthn the set of prors that are supported on a fxed fnte number of types n 2 for each player ), thesetofprorsthatpermt full-surplus extracton from any bdder s generc. How come we get the opposte result when we consder the set of prors that are supported on all fnte numbers of types, or wth arbtrary fnte supports? Themanreasonsthatthesetofprorsthataresupportedonafxed fnte number of types s not closed under averagng. For example, the average of the common prors that are represented by the two matrces τ 2 =(v 2,b 2 ) τ 2 = ³ṽ 2, 2 b τ =(v,b ) a b τ = ³ṽ, b c d τ 0 2 =(v 2,b 0 2) τ 0 2 = τ 0 =(v ³,b 0 ) a 0 b 0 τ 0 = ṽ, b 0 c 0 d 0 ³ ṽ 2, b 0 2 (where a + b + c + d = a 0 + b 0 + c 0 + d 0 =) s not the pror that s represented by the matrx ³ τ 00 2 =(v 2,b 00 2) τ = ṽ 2, b 2 τ 00 =(v,b 00 ) (a + 2 a0 ³ ) (b + 2 b0 ) τ 00 = ṽ, but rather the followng pror τ 2 =(v 2,b 2 ) τ 2 = ³ṽ 2, 2 b b 00 2 a 2 (c + c0 ) 2 (c + c0 ) τ 0 2 =(v 2,b 0 2) τ 0 2 = τ =(v,b ) 2 b 0 0 τ = ³ṽ, b 2 2 τ 0 =(v,b 0 ³ ) a0 2 b0 τ 0 = ṽ, b c0 2 d0 ³ ṽ 2, b 0 2 whch s a pror that s supported on 8 rather than 4 states. In partcular, the average of the common prors that are represented by the two matrces s not v 2 =0 ṽ 2 = v =0 0 2 ṽ = 0 2 v =0 4 ṽ = 4 v 2 =0 ṽ 2 = v =0 0 2 ṽ = 0 2 v 2 =0 ṽ 2 =