Faculty Research Workng Papers Seres Mechansm Desgn wth Multdmensonal, Contnuous Types and Interdependent Valuatons Nolan Mller John F. Kennedy School of Government - Harvard Unversty John W. Pratt Harvard Busness School Rchard Zeckhauser John F. Kennedy School of Government - Harvard Unversty Scott Johnson Australan Natonal Unversty July 2006 (Revsed September 2006) RWP06-028 The vews expressed n the KSG Faculty Research Workng Paper Seres are those of the author(s) and do not necessarly reflect those of the John F. Kennedy School of Government or Harvard Unversty. Copyrght belongs to the author(s). Papers may be downloaded for personal use only.
Mechansm Desgn wth Multdmensonal, Contnuous Types and Interdependent Valuatons Nolan H. Mller, John W. Pratt, Rchard J. Zeckhauser and Scott Johnson, September 7, 2006 Abstract We consder the mechansm desgn problem when agents types are multdmensonal and contnuous, and ther valuatons are nterdependent. If there are at least three agents whose types satsfy a weak correlaton condton, then for any decson rule and any ε>0 there exst balanced transfers that render truthful revelaton a Bayesan ε-equlbrum. A slghtly stronger correlaton condton ensures that there exst balanced transfers that nduce a Bayesan Nash equlbrum n whch agents strateges are nearly truthful. Keywords: Mechansm Desgn, Interdependent Valuatons, Multdmensonal Types. JEL Classfcaton: C70, D60, D70, D82. Mller (Kennedy School, Harvard Unversty); Pratt (Harvard Busness School); Zeckhauser (Kennedy School, Harvard Unversty); and Johnson (Australan Natonal Unversty, deceased). Please drect correspondence to Rchard Zeckhauser at 79 JFK St, Cambrdge, MA, 02138 or rchard_zeckhauser@harvard.edu. A prevous verson of some results n ths paper crculated under the ttle "Effcent Desgn wth Multdmensonal, Contnuous Types and Interdependent Valuatons." We thank Drew Fudenberg, Jerry Green, Danel Hojman, Matt Jackson, John H. Lndsey II, Rch McLean, Zvka Neeman, Andy Postlewate, Phl Reny, Bll Sandholm, Mchael Schwarz, Ed Shpz, varous semnar partcpants, an assocate edtor and two anonymous referees for helpful comments. Zeckhauser gratefully acknowledges fnancal support of NSF Grant IIS-0428868.
1 Introducton Elctng prvate nformaton to gude socal decsons s a classc problem of economc theory. For the prvate-values case, the poneerng work of Vckrey (1961), Clarke (1971), and Groves (1973) shows that f each agent s preferences depend only on hs own nformaton and f the budget need not balance, externalty payments make honest revelaton a domnant strategy. However, Green and Laffont (1977, 1979) show that domnant strategy mplementaton s generally ncompatble wth the requrement that the transfers balance the budget. If the soluton concept s weakened, postve results are possble. For example, d Aspremont and Gérard-Varet (1979, 1982) show n the prvate-values envronment that f agents belefs about other agents types satsfy a certan condton, whch they call compatblty, then for any effcent decson rule there exst balanced Bayesan ncentve-compatble transfers that mplement t. 1 Later, d Aspremont, Crémer, and Gérard-Varet (1990) show that when there are three or more agents the compatblty condton s genercally true, and hence that for generc dstrbutons of agents types there exsts a Bayesan ncentve-compatble Pareto-optmal mechansm. The mechansm desgn problem has proved more challengng n the case of nterdependent valuatons,.e., when one agent s prvate nformaton affects other agents preferences. Dasgupta and Maskn (2000) study auctons wth nterdependent valuatons and show that a generalzed Vckrey aucton s effcent f bdders types are one dmensonal and satsfy a sngle-crossng property. In general mechansm-desgn problems, postve results have been mostly lmted to the case where agents types take on only fntely many values. Work n ths area ncludes Johnson, Pratt, and Zeckhauser (1990); Matsushma (1990; 1991); and McLean and Postlewate (2004). Crémer and McLean (1985; 1988) study the related queston of when t s possble for the desgner to earn as much proft as he would were he able to observe the agents realzed prvate nformaton the so-called full surplus extracton problem and show that full extracton s possble when agents types are sutably correlated. Aoyag (1998) consders a model wth fnte type sets and nterdependent valuatons and shows that f the dstrbuton of agents types satsfes a dependence condton smlar to ours, then for any decson rule there exsts a balanced, Bayesan ncentve-compatble 1 Unlke ts use n mplementaton theory (see Jackson, 2001), throughout the paper we use mplement to refer to the case where there s an outcome of the game that agrees wth the decson rule. 1
mechansm that mplements t. When types are multdmensonal and contnuous and valuatons are nterdependent, the problem becomes even more dffcult. After ther possblty result for the one-dmensonal case, Dasgupta and Maskn (2000) go on to show that when bdders types are multdmensonal and ndependently dstrbuted there may be no effcent aucton. In a general mechansm desgn framework, Jehel and Moldovanu (2001), henceforth JM, explore the dffcultes of Bayesan ncentve-compatble (BIC) mplementaton of effcent decson rules when types are multdmensonal and contnuous and valuatons are nterdependent. They show that when agents types are ndependently dstrbuted, effcent BIC desgn s possble only when a certan congruence condton relatng the socal and prvate rates of nformaton substtuton s satsfed (JM, p. 1237). In effect, ths congruence condton requres that there be one agent whose relatve preference over any two alternatves remans constant for all values of that agent s nformaton that make the socal planner ndfferent between those alternatves. They then show that when types are multdmensonal the set of payoff functons that satsfy ths condton s non-generc, mplyng that effcent BIC desgn s generally mpossble. 2 The present paper addresses the mechansm desgn problem n envronments n whch agents prvate nformaton s contnuous, multdmensonal, and mutually payoff-relevant (.e., valuatons are nterdependent). However, we relax the JM assumpton that agents prvate nformaton s ndependently dstrbuted. Our prmary nterest s to show that when there are three or more agents and agents types are stochastcally dependent t s possble to desgn a system of budget-balanced transfer payments that nduces agents to (nearly) truthfully reveal ther prvate nformaton and that (nearly) mplements any decson rule. In our frst result (Theorem 1), we show that under a mld dependence condton on the dstrbuton of agents types, whch we call Stochastc Relevance, for any ε>0 there exst budget-balanced transfers such that truthful revelaton s an ε-best response to other agent s truthful announcements. Thus the strategy profle where all agents announce truthfully s a ε-equlbrum. In our second result (Theorem 2), we show that a slghtly stronger verson of Stochastc Relevance, whch we call Unform Stochastc Relevance, ensures that for any 2 Although Jehel and Moldovanu (2001) focuses on the mpossblty of effcent BIC desgn, much of the mportance of the result les n the fact that t mples that robust mechansm desgn usng belef-free concepts such as ex post equlbrum s also mpossble. We return to ths pont n Secton 5. 2
δ>0 there exst balanced transfers under whch there s a Bayesan Nash equlbrum (BNE) of the announcement game n whch the dstance between agents equlbrum announcements and ther true types s no more than δ,.e., that there s a nearly truthful BNE. Thus our results provde a complement to those of JM. When the dstrbuton of agents types satsfes our dependence assumptons, then ncentve-compatble desgn s possble. Further, our mplementaton results place very few addtonal requrements on agents preferences. 3 In partcular, we do not requre a sngle-crossng property. Mezzett (2004) consders mplementaton of effcent decson rules n a model n whch the socal planner bases transfers on agents reports of both ther types and the utlty they realze from the socal decson. The paper shows that mplementaton of effcent decson rules s generally possble usng a two-stage Groves mechansm. However, snce agents may not realze the utlty from a socal decson untl long after the decson s made, ths framework presupposes, among other thngs, that the planner s able to make long-term commtments to make transfers n the future. Even n crcumstances where the two-stage mechansm s feasble, Mezzett s results apply only to effcent decson rules, whereas the results of ths paper apply to all decson rules. Further, the present paper mposes a more strngent form of budget balance than Mezzett. 4 McAfee and Reny (1992, henceforth MR) consder the full surplus extracton problem n the case of contnuous, multdmensonal, and mutually payoff-relevant types wth stochastcally dependent nformaton. Takng the game played by the agents as gven, they show that t s possble to construct for each agent a fnte menu of partcpaton fee schedules that extracts almost all of the agent s rent from playng the game. However, they do not drectly address the ssue of whch decson rules can be mplemented, the prmary concern of ths paper. For example, wth multdmensonal types and nterdependent values, there s, n general, no ex post effcent aucton mechansm unless addtonal assumptons are made that ensure that the agents multdmensonal nformaton can be summarzed by a one-dmensonal type (Maskn (1992), Dasgupta and Maskn (2000), Krshna (2002)). Therefore, n such envronments, the MR mechansm cannot extract the 3 Specfcally, we requre only that agents drect returns from the center s decson be bounded and sutably smooth. 4 We adopt the standard defnton n the lterature that balanced transfers must sum to zero for any possble choce of actons by the agents. Mezzett s transfers satsfy the weaker requrement that the transfers sum to zero on the equlbrum path when all players play truthful strateges. 3
full nformaton rent (.e., the rents that would be generated f the auctoneer knew the agents types), snce the MR constructon depends on the exstence of an ex post effcent mechansm to whch the partcpaton fees can be appended. The present paper flls ths gap by showng how to construct an ex post effcent mechansm n ths envronment. Appendng the MR mechansm to ours then makes t possble to fully extract the agents surplus. Postve results on ncentve-compatble mplementaton as well as full surplus extracton (e.g., Crémer and McLean (1985, 1988), MR, Aoyag (1998)) rely on constructng a menu of lotteres for each agent such that the agent maxmzes hs expected utlty when he chooses the lottery ntended for hs type. Intutvely, ths s possble whenever learnng an agent s type provdes nformaton about the dstrbuton of the other agents types. Our analyss follows n the same sprt. We captalze on the lterature n statstcal decson theory on strctly proper scorng rules, whch consders how an nformed expert can be nduced to truthfully reveal hs belefs about the dstrbuton of future random events. 5 A scorng rule assgns payoffs to the expert based on hs announced probabltes for varous future events and the event that actually occurs. A strctly proper scorng rule has the property that the decson maker maxmzes hs expected score when he truthfully announces hs belefs about the dstrbuton. Our ncentve-compatblty results rely on payments based on a proper scorng rule to drve agents toward truthful revelaton of ther prvate nformaton. 6 The paper proceeds as follows. Secton 2 presents the model. Secton 3 constructs scorngrule payments that render truthful reportng a Bayesan ε-equlbrum. The basc constructon s adapted n Secton 4 to show that under a slghtly stronger correlaton condton smlarly constructed payments ensure that there s an exact BNE n whch agents strateges are arbtrarly close to truthful. Secton 5 dscusses lmtatons of the approach n the paper, and Secton 6 concludes. All proofs are presented n the Appendx. 5 See Cooke (1991) and the references theren for a dscusson of scorng rules and ther uses. 6 Johnson, Pratt, and Zeckhauser (1992) employs a smlar technque n the case of fnte type and acton spaces. 4
2 The Model Suppose I 3 agents, ndexed by =1,...,I, nteract wth the mechansm desgner, whom we wll call the center. The center s task s to elct agents prvate nformaton n order to choose an alternatve g from a set of alternatves G. Each agent has prvate nformaton or type t T. Agent s type space T s a non-empty, compact, convex subset of d -dmensonal Eucldean space. For each, d s a postve nteger, and d may be dfferent for dfferent agents. We use T = T to denote the product space of the I agents type spaces. Followng the standard notaton we use t =(t 1,...,t I ) for the vector of types, t for the vector of all but agent s type, and t j for all but the types of agents and j. Each agent s utlty s quaslnear n hs drect return from the socal alternatve, g, and money, x, takng the form: u (t, g, x) =V (t, g)+x. Note that agent s drect return from alternatve g depends on all agents types. Hence valuatons are nterdependent. A decson rule g : T G maps a type for each agent to a socal alternatve. For smplcty, we assume that g (t) s sngle valued. For g (t) that s not sngle-valued, our mplementaton result apples to any selecton from g (t), and therefore ths restrcton s wthout loss of generalty. Although we wll mpose a degree of smoothness on g (t), we wll not otherwse restrct t. In partcular, we do not requre that g (t) be effcent. We consder drect mechansms n whch each agent sends a message (announcement) to the center consstng of an element from hs type space. We denote these announcements by a T, and let a, a,anda j refer respectvely to the full announcement vector, the announcement vector leavng off agent, and the announcement vector leavng off agents and j. The remander of the mechansm conssts of a transfer functon x (a) for each and a decson rule g (a), wththe standard nterpretaton that the agents announce a, socal alternatve g (a) s realzed and transfer x (a) s made to agent. An announcement strategy for agent s a functon s ( ) :T T that specfes agent s announcement n the message game as a functon of hs nformaton. We wll use the notaton s ( ) to refer to a strategy for agent and s (t ) to denote to the announcement agent makes under strategy s ( ) when hs type s t. Thus, s ( ) s an element of a functon space, whle s (t ) resdes 5
n d -dmensonal Eucldean space. We wll use τ to denote the dentty functon on T,.e., agent s truthful strategy. Denote the vector of transfer rules to all agents by x (a), whch we call a transfer scheme. Atransferschemesbalanced f ts transfers sum to zero for all possble announcement vectors: P I =1 x (a) =0for all a. If a decson rule s mplemented by a balanced transfer scheme, t requres no outsde subsdy. Snce our mechansm s essentally the same for any decson rule and depends on the decson rule only through the drect return functon, we ntegrate the decson rule nto the drect return functon, and then wrte V (t, g (a)) as v (t, a). If there exsts a transfer scheme that satsfes a partcular soluton concept wth payoffs v (t, a), then those transfers mplement g (a) under that soluton concept. We make the followng assumpton regardng v (t, a): Assumpton 1 (Smooth Drect Returns): For each, expected drect returns are (jontly) twce contnuously dfferentable n a and t. 7 Assumpton 1 s not nnocuous snce t mples restrctons on the contnuty of the underlyng decson rule, g (a), and on the set of possble decsons, G. Nevertheless, contnuty seems to be a reasonable restrcton n any stuaton that s approprately modeled usng contnuous types. Further, dscontnuous decson rules can often be approxmated by contnuous ones, and the results below would generalze to the case of decson rules that can be approxmated by contnuous rules. Snce v (t, a) s contnuous and T s compact, Assumpton 1 mples that drect returns are bounded. Let M 0 denote the bound. That s, for any, t and a, E {v (t, a) t } M. Types are dstrbuted accordng to commonly known pror dstrbuton F (t) wth support T. Let f (t j t ) be the densty of agent j s prvate nformaton condtonal on agent s prvate nformaton, t,andletf (t t ) be the densty of all other agents prvate nformaton condtonal on agent s prvate nformaton. We mpose two assumptons on agents belefs, a smoothness condton and a correlaton condton. 7 In order to ensure the functon s dervatves exst on the boundary of the doman, we assume v (a, t) s defned and twce contnuously dfferentable on an open set contanng T 2. An alternatve approach would be to apply a unque extenson theorem such as Proposton 7.5.11 n Royden (1988, p. 149). We adopt the same approach n Assumpton 2. 6
Assumpton 2 (Smooth Condtonal Dstrbutons): For each and j 6=, condtonal denstes f (t j t ) are jontly contnuous n t j and t and contnuously dfferentable n t. Smlarly, f (t t ) are jontly contnuous n t and t and contnuously dfferentable n t. We assume that the agents prvate nformaton s not ndependently dstrbuted, whch departs from the JM model. Specfcally, our nformatveness assumpton, whch we call Stochastc Relevance, s that the condtonal dstrbuton of the center s nformaton be dfferent for dfferent values of each agent s prvate nformaton. Assumpton 3 (Stochastc Relevance): For each, there exsts an agent j 6= such that for any dstnct types t and t 0 there exsts t j T j such that: f (t j t ) 6= f t j t 0. Let k k R denote the Eucldean norm, ktk R = ³ P k (t k) 2 1/2,wheretk denotes the k th component of t,andletk k 2 denote the L 2 norm, kfk 2 = ³ R 1/2. f 2 ds We wll wrte fj ( t ) when we wsh to denote agent s belefs about the dstrbuton of t j consdered as a functon. Lemma 1 follows as an mmedate consequence of Assumptons 2 and 3. Lemma 1: Assumptons 2 and 3 mply that for each, for any δ>0 there exsts μ>0 such that: t t 0 R δ mples 0 fj ( t ) f j t 2 μ. Taken together, Assumptons 2 and 3 and Lemma 1 mply that f (t j t ) and f (t j t 0 ) dffer on an open subset of T j and that f j ( t ) and f j ( t 0 ) are close together (as functons n L 2)fand only f t s close to t 0. Thus they capture the dea that types should have smlar belefs f and only f they are close together. 8 8 Although t would add sgnfcant notatonal burden, Stochastc Relevance could be relaxed to allow for the case 7
3 Exstence of Nearly Bayesan Incentve Compatble Transfers We begn by consderng the queston of whether there exst transfers that make the truth nearly a best response, provded that all other agents announce truthfully. Consderng ths queston allows us to llustrate our constructon n the smplest settng. In the next secton, we go on to show that a smlarly constructed payments establsh that there s a nearly truthful BNE of the game. We begn wth the noton of ε-bayesan Incentve Compatblty. 9 Transfer scheme x (a) s ε-bayesan Incentve Compatble (ε-bic) f for any, t,anda : E {v (t, t,t )+x (t,t ) t } E {v (t, t,a )+x (t,a ) t } ε. That s, f for each agent, announcng truthfully s an ε-best response to the other agents truthful announcements. As dscussed earler, the mechansm we propose draws on the decson-theoretc lterature on proper scorng rules. In partcular, we employ the quadratc scorng rule. Suppose that agent j s usng the truthful announcement strategy, s j ( ) =τ j, and player s beng scored based on how well he predcts agent j s announced type. The quadratc score assgned to type t j when agent announces a s gven by: Z Q (t j a )=2f (t j a ) f (t j a ) 2 dt j. T j Lemmas 2 and 3 establsh basc propertes of the quadratc scorng rule that wll be used n the subsequent analyss. Lemma 2: For any agent, chooseanagentj accordng to Assumpton 3, and suppose agent j truthfully announces hs type, s j ( ) =τ j. Truthful revelaton unquely maxmzes agent s expected quadratc score: Z t =argmax Q (t j a ) f (t j t ) dt j. a T T j where agent s belefs about the jont dstrbuton of a group of agents types depends on t even though the margnal dstrbuton for any other agent s type does not. Aoyag (1998) presents such a condton (Assumpton 2) for the fnte case. 9 ε-bayesan Incentve compatblty appears, for example, n d Aspremont and Gérard-Varet (1982). 8
As Selten (1988) notes, the proof that truthful revelaton unquely maxmzes the expected quadratc score also shows that the expected loss from agent s announcng a 6= t nstead of hs true type t s equal to the square of the L 2 -dstance between agent s belefs when hs type s a andwhenhstypest. Lemma 3 explots ths property. Lemma 3: For any agent, choose an agent j accordng to Assumpton 3. For any δ > 0 there exsts ε>0 such that the expected quadratc score for the dstrbuton of agent j s type from announcng a 6= t wth ka t k R δ s at least ε worse than announcng truthfully: ka t k R δ mples Z Z Q (t j a ) f (t j t ) dt j Q (t j a ) f (t j a ) dt j ε. T j T j Lemma 2 establshes that f the agents care only about the transfer, truthful announcement s agent s unque best response when other agents tell the truth. Lemma 3 ensures that there s no sequence of announcements far away from the truth whose expected scores converge to the expected score of the truth. Ths s needed n order to establsh a unform lower bound on the loss from an announcement that s far from truthful. Our frst man results shows that there exst ε-bic, balanced transfers. The ntuton s that n choosng whether to announce hs true type or some other type the agent weghs the effects of lyng on the expected transfer and on the expected drect return. If transfers are based on the quadratc scorng rule, then tellng the truth maxmzes the agent s expected transfer. However, snce announcng truthfully does not necessarly maxmze the expected drect return, the agent may have an ncentve to devate from truth-tellng, sacrfcng expected transfer n order to enjoy a personally superor socal alternatve. Of course, the agent s wllngness to do so depends on how quckly the expected transfer declnes relatve to the ncrease n expected drect return. By scalng up the scorng-rule based payments to the agent, the center can ncrease the mportance of the transfer loss relatve to the drect return gan, makng anythng but a small devaton from the truth unproftable. 9
Theorem 1: Under Assumptons 1-3, for any decson rule and any ε>0 there exst ε-bic, balanced transfers. The essence of the proof s to dvde agent s announcements nto two groups those that are wthn δ of the truth and those that are not. Under the quadratc scorng rule, the expected transfer s maxmzed by tellng the truth. Thus announcements that are wthn δ of the truth yeld a smaller expected transfer but a possbly larger drect return. However, by choosng δ suffcently small we ensure that the drect return gan from any announcement wthn δ of the truth must be less than ε. On the other hand, Assumptons 2 and 3 ensure that the loss n expected transfer from movng from a truthful announcement to one that s more than δ from the truth must be unformly bounded away from zero, and thus scalng up the transfers ncreases the mnmum loss n transfer from an announcement at least δ from the truth. Snce drect returns are bounded, a suffcent scalng of the transfers ensures that the gan n drect return gan cannot outwegh the transfer loss, and thus that announcements that are at least δ from the truth must nvolve a total expected utlty loss of at least ε. The transfers are balanced usng a permutaton constructon. That s, f agent 1 s gven ncentves to report truthfully by comparng hs announcement to that of agent 2, then the transfer to agent 1 can be funded by a thrd agent (e.g., agent 3) wthout affectng any agent s ncentve to report truthfully. Repeatng ths process for all agents balances the budget. Thus, whle three or more agents are needed n order to balance the budget, f budget balance s not a concern, ε-bic transfers exst wth only two agents. 4 Exstence of a Nearly Truthful Bayesan Nash Equlbrum Theorem 1 establshes that compensatng agents usng a suffcently large scalng of the quadratc scorng rule renders truthful revelaton an ε-best response, provded that the other agents announce truthfully. Although ths dea has some ntutve appeal and makes the role of the quadratc scorng rule transparent, requrng agents to play merely ε-best responses rather than exact best responses begs the queston of whether ths lmted ratonalty s necessary or merely a convenence. To address ths concern, we next argue that, under reasonable condtons, payments based on a scalng 10
of the quadratc scorng rule can be used to nduce a BNE n whch agents strateges are arbtrarly close to the truth. For a fxed transfer scheme x (a), a BNE of the announcement game s a vector of strateges (s 1 ( ),...,s I ( )) such that for each and t : s (t ) arg max a E t {v (t, a,s ( )) + x (a,s ( )) t }. We endow the space of announcement strateges wth the sup norm: ks ( ) ŝ ( )k sup =sup t à d! 1/2 X (s n (t ) ŝ n (t )) 2. n=1 For δ>0, we call an announcement strategy, s ( ), δ-truthful f ks ( ) τ k sup δ. That s, a δ-truthful announcement strategy s one n whch the agent s announcement s always wthn dstance δ of hs true type. We say that a transfer scheme δ-mplements a decson rule n BNE f under those transfers there exsts a BNE n whch all agents strateges are δ-truthful. 10 Note that the concept of δ-mplementaton n BNE allows for the exstence of BNE that are not δ-truthful. Let C denote the space of contnuous announcement strateges for agent. For δ>0, let C (δ) be the space of contnuous, δ-truthful announcement strateges for agent : C (δ ) n o s ( ) C : ks ( ) τ k sup δ. In the usual way, let C (δ) denote the product space C (δ), and let C (δ) denote the product space of C j (δ) for all agents except, each endowed wth the approprate product topology. The key step n constructng a δ-truthful BNE s ensurng that a verson of Stochastc Relevance remans true even when agents announcements are only δ-truthful. In order to ensure ths we strengthen stochastc relevance as follows: Assumpton 4 (Unform Stochastc Relevance): There exsts φ>0 such that for each, there exsts an agent j 6= such that for any dstnct types t and t 0 there exsts an open ball 10 We may, on occason, refer to sngle announcements as δ-truthful f for a partcular t, s (t ) t R δ or to strategy profles as beng δ-truthful f each ndvdual strategy s δ-truthful. 11
θ j ³t,t 0 T j wth radus φ such that f (t j t ) 6= f ³ t j t 0 for all t j θ j ³t,t 0. Stochastc Relevance (Assumpton 3) mples that, for any dstnct types t and t 0, f (t j t ) and ³ f t j t 0 dffer on an open set of types for agent j. Unform Stochastc Relevance (Assumpton 4) strengthens Stochastc Relevance by requrng that there be a lower bound on the sze of the ³ open set on whch f (t j t ) and f t j t 0 dffer that s ndependent of the partcular par of types t and t 0 that s chosen. It s straghtforward to show that, by vrtue of compactness, Assumpton 3 mples the exstence of such a unform bound provded that t and t 0 are bounded away from each other,.e., as long as there exsts δ>0 such that kt t 0 k R δ. Thus,totheextentthatUnform Stochastc Relevance s stronger than Stochastc Relevance, t only restrcts the behavor of belefs as types t and t 0 become (arbtrarly) close together. Snce agent s belefs are contnuous n t,ast and t 0 become very close, the two types belefs must also become very close. Unform Stochastc Relevance rules out the case n whch as t 0 converges to t the Lebesgue measure of the set of t j where ther assocated belefs dffer, {t j T j f (t j t ) 6= f (t j t 0 )}, converges to zero. In other words, under Unform Stochastc Relevance t cannot be that as t 0 approaches t, f t j t 0 approaches f (t j t ) by becomng equal ³ to t on an ever-larger set of t j. Seen n ths way, t s clear that many of the famles of belefs economsts typcally consder satsfy Unform Stochastc Relevance. For example, belefs where t j s dstrbuted normally wth mean t satsfy Unform Stochastc Relevance. For an example of a famly of belefs that does not satsfy Unform Stochastc Relevance, consder T = T j =[0, 1] 2. Suppose that f (t j t ) s unformly dstrbuted on a dsk centered at t j = t and havng radus 1/10 (for t sutably dstant from the boundary of T j ). Consder t =(1/2, 1/2). Let λ ( ) denote Lebesgue measure. Snce ³n ³ o lm kt 0 (1/2,1/2)k λ t j T j f (t j t ) 6= f t j t 0 0 =0, R these belefs volate Unform Stochastc Relevance. Ths s because f (t j t ) and f (t j t 0 ) are equal on ther common support, and as t 0 converges to t, the supports of f (t j t ) and f (t j t 0 ) converge as well. 11 11 On the other hand, f f (t j t ) s dstrbuted as a cone wth a crcular base of radus 1/10 and peak at t,these 12
The exstence of a unform lower bound on how often the belefs of two dfferent types of agent dffer s mportant when agents strateges are permtted to be δ-truthful (as n Theorem 2) rather ³ than exactly truthful (as n Theorem 1). If f (t j t ) and f t j t 0 ³ are equal except for a very small set of t j, then t s possble that, even though f (t j t ) and f t j t 0 dffer, the dstrbuton of agent j s announcements resultng from a partcular δ-truthful strategy (e.g., the δ-truthful strategy that ³ s constant over the set of t j where f (t j t ) and f t j t 0 dffer) s the same for t and t 0. Lemma 4 shows that Unform Stochastc Relevance ensures that dfferent types t and t 0 have dfferent belefs about the dstrbuton over a set of dscrete events comprsed of groups of announcements for some agent j 6=, and that ths dfference remans even f agent j dstorts hs announcement slghtly. 12 Lemma 4: Assumpton 4 mples that there exsts a δ > 0 such that for any 0 <δ<δ and any n o agent, there s an agent j 6= such that T j contans a fnte set of dsjont balls B j = b j 1,...,bj M wth radus greater than δ such that for any t, t 0 wth t 6= t 0 there s at least one bj m such that ³ f (t j t ) 6= f t j t 0 for all t j b j m. 13 The key dstncton between Assumpton 4 and Lemma 4 s that Assumpton 4 asserts that for any t and t 0 theresanopenball(whchmaydependont and t 0 ) over whch the assocated belefs of dfferent types dffer, whle Lemma 4 establshes the exstence of a fnte set of balls such that no matter whch t and t 0 are chosen ther assocated belefs dffer over at least one ball. To see the role that Lemma 4 wll play n the proof of Theorem 2, consder two dstnct types t and t 0 for agent. By Lemma 4, let b j m be the ball n agent j s announcement space satsfyng Lemma 4 that dstngushes these types. If agent j announces truthfully, types t and t 0 assgn dfferent probabltes to event t j b j m, and so a scorng rule based on whether agent j s announcement s n b j m can be used to truthfully elct whether agent s type s t or t 0. The lower bound on the sze of the balls n B j ensures that there s a partton of events that belefs would satsfy Unform Stochastc Relevance snce the set of ponts where the denstes of f (t j t ) and f t j t 0 are equal remans small (.e., has Lebesgue measure zero) even as t and t 0 become arbtrarly close together. 12 Lemma 4 s also useful for a more techncal reason. When s j ( ) C j (δ), playerj s announcement can be constant over an open nterval. Hence, even though t j has a densty, the dstrbuton of j s announcements can have pont masses. Whle vrtually the same theory of proper scorng rules apples ether to dscrete or contnuous dstrbutons, to our knowledge there s no theory of proper scorng rules for mxed dstrbutons. Therefore we move to usng a scorng rule for the dstrbuton over dscrete events. 13 The exstence of Lemma 4 s B j clearly mples Unform Stochastc Relevance. Hence Unform Stochastc Relevance holds f and only f such a B j exsts. 13
dstngushes any two types even f j s allowed to dstort hs announcements usng a δ-truthful strategy (for δ<δ ). To see how, let b j m be the ball n B j to whch t and t 0 assgn dfferent probabltes to the event t j b j m. Let r m be the radus of b j m,andletˆb j m be the ball wth the same center and radus r m δ. If j s strategy s δ-truthful, then the set of types that announce a j ˆb j m must be contaned n b j m. Hence whenever s j ( ) C j (δ), typest and t 0, assgn dfferent probabltes to the event a j ˆb j m condtonal on s j ( ). To see why, consder Fgure 1, whch llustrates the one-dmensonal case. Suppose that b j m = ht j δ, t+ j + δ. Accordng to Lemma 4, the denstes for some t s are drawn n such a way that they don t cross over ths regon. Now, look at the smaller event, h ˆbj m = t j,t+ j b j m. Note that snce types can only dstort ther announcements by δ or less, f t j δ-truthfully announces a j ˆb j m,thent j b j m. However, snce the denstes for these values of t are ranked over the entre set b j m, condtonal on s j ( ) C j (δ), two dstnct types whose denstes do not cross over b j m cannot assgn the same probablty to j s announcement beng n ˆb j m. In Fgure 1, the heavy black lnes on the horzontal axs ndcate the set of types t j that make announcements n ˆb j m for some hypothetcal s j ( ) C j (δ). Lookng at the shaded regons above the t j n ths set, the denstes for the varous values of t do not cross. Thus for any two t whose denstes do not cross over b j, the one wth the hgher densty must assgn hgher probablty to the event a j ˆb j m for any possble announcement strategy s j ( ) C j (δ). If, as asserted by Unform Stochastc Relevance and Lemma 4, there s a fnte set of balls B j such that for each possble par of types there s one ball over whch ther denstes do not cross, then we can use the sets ˆb j 1,...,ˆb j M along wth ˆb j 0 T j\ ˆb j m (.e., everythng else ) as a partton of events that dstngushes every par of types for every possble strategy s j ( ) C j (δ). That s, condtonal on s j ( ), dfferent types t have dfferent belefs about the dstrbuton over the events n ˆB nˆbj o j j = 0, ˆb 1,...,ˆb j M. Thus, transfers based on the quadratc scorng rule appled to the events n ˆB j (condtonal on s j ( )) are strctly proper. If agent only cared about the transfer, hs best response to s j ( ) under these transfers would be to announce hs true type. When agent also cares about the drect return from the socal choce, basng transfers on a suffcently large scalng of the quadratc scorng rule ensure that agent s best response s close to truthful. Theorem 2 establshes that there s a transfer scheme that δ-mplements any decson rule n 14
densty f(t j t ) for varous values of t. t j + - δ - + t j t j ndcates the set of t j where a ( t ) t, t + j j j j t j + + δ t j Fgure 1: δ-truthful strateges assgn dfferent probabltes to n h o a j t j,t+ j. BNE. ThemantoolemployedsSchauder sfxed pont theorem (see Zedler, 1985). 14 Theorem 2: Under assumptons 1, 2, and 4, for any decson rule and any δ>0 there exst balanced transfers that δ-mplement that decson rule n BNE. The ntuton for the proof begns by notng that each agent s payoff s a lnear combnaton of hs drect return and transfer. Thus, the stuaton where the transfers are multpled by a large (postve) constant s one where the agent puts small relatve weght on hs drect return, whch s smlar to the case where the agent puts zero weght on hs drect return. When agents care only about ther transfers, transfers based on the quadratc scorng rule ensure that truthful revelaton s a strct equlbrum. If we knew that ths equlbrum changed smoothly wth the relatve weght put on agents drect returns, then games wth nearby payoffs would have a nearby equlbrum. Thus, games n whch the relatve weght on transfers was very hgh would have an equlbrum n whch agents strateges were nearly truthful. Unfortunately, ths smooth dependence property, whch s related to lower hem-contnuty of the equlbrum correspondence, does not hold n general. Nevertheless, by explotng the fact that the truthful equlbrum of the transfers-only game s strct, we are able to show that when agents strateges are nearly truthful, nearby games satsfy 14 Merowtz s (2003) uses a slghtly dfferent verson of Schauder s Theorem to prove a general exstence result for equlbra n Bayesan games wth nfnte type and acton spaces. 15
the requrements for the applcaton of Schauder s fxed pont theorem, and thus that games where agents put small relatve weght on ther drect returns have a nearly truthful equlbrum. 15 Theorem 2 establshes the exstence of an equlbrum n whch agents play nearly truthful strateges. The queston remans whether, under the transfers that nduce the δ-truthful equlbrum, other equlbra exst as well and, f so, whether those equlbra are also δ-truthful. In general, there s no reason to rule out such equlbra. Gven that agents ncentves are prmarly drven by ther desre to maxmze the transfer they receve and that these transfers are determned by how well each agent predcts the announcements of the other agents, t s easy to magne that there could be equlbra n whch all agents permute ther announcements n such a way that announcements are no longer close to truthful but stll predct other agents announcements well. There s, however, one crcumstance n whch t s possble to establsh that all equlbra must be nearly truthful. Ths s the case n whch the center receves a sgnal of ts own that s stochastcally related to the agents types. 16 In effect, for each agent, the center s nformaton plays the role of the agent j whose nformaton s used to score agent. Snce no agent s behavor can affect the dstrbuton of the center s nformaton, the expected payment from transfers based on the quadratc scorng rule appled to the center s nformaton s unquely maxmzed by tellng the truth regardless of the other agents strateges. The argument n Theorem 2 then establshes that for any δ>0 there exsts a δ-truthful equlbrum. Further, snce a suffcently large scalng of the payments ensures that all best responses are δ-truthful, all equlbra must be δ-truthful. Returnng to the case where the center does not receve an nformatve sgnal, whle Theorem 2 establshes that agents strateges are nearly truthful, from the perspectve of the socal planner our real nterest s not whether agents are tellng the truth, but rather whether the resultng socal choce rule s close to that mpled by the planner s desred rule, and whether realzed socal welfare s close to the planner s desred welfare level. These desrable propertes follow, however, because transfers are balanced and agents payoffs are assumed to be contnuous condtonal on the socal choce functon (Assumpton 1). 15 The key step s to establsh that agents best responses to any δ-truthful opponents strateges are unque, whch requres condtonng transfers on the other agents strateges. 16 If the center s nformaton has a densty, then the center s nformaton must satsfy the player j role n Assumpton 3. 16
MR shows that when agents types are correlated, for any game the center can extract from each agent nearly all of the rents that agent earns by partcpatng n the game. 17 Ther mechansm offers agents a fnte menu of partcpaton fee schedules such that, when the agent selects hs preferred schedule and then plays the game, he s left wth a rent that, though postve, s arbtrarly small. Whle MR shows that for a gven game, a partcpaton fee schedule can ensure that agents nterm partcpaton constrants can be satsfed at (nearly) no cost to the center, they do not address the queston of whether, for a gven decson rule, a game exsts that mplements that decson rule. In partcular, the center cannot extract the full nformaton rent (.e., the rent that would be generated f the center could observe agents types and make ex post effcent decson) unless there exsts a mechansm that mplements the ex post effcent decson rule. Pror to ths paper, there have been no results that show the general exstence n the standard mechansm desgn framework of an ex post effcent mechansm when agents have multdmensonal, contnuous types and nterdependent valuatons. 18 To the extent that ex post effcent mechansms have been shown to exst, these results typcally requre addtonal assumptons on the form of agents drect returns functons, e.g., sngle crossng. The results n ths paper do not mpose any restrctons on drect returns functons beyond smoothness (Assumpton 1). We show that f agents types are correlated, then any decson rule, ncludng the ex post effcent decson rule, can be mplemented arbtrarly closely (.e., δ-truthfully). Provded that belefs satsfy MR s condton (*), our result, coupled wth the MR result, establshes that the center can extract (approxmately) the full nformaton rent and satsfy agents nterm partcpaton constrants by frst offerng agents a menu of partcpaton fees and then runnng our scorng-rule based system. 17 Ther requred condton (*) s strctly stronger than our Assumpton 3. 18 As dscussed earler, Mezzett s (2004) mechansm operates n a slghtly dfferent framework than the standard models and uses a weaker form of budget balance. 17
5 Lmtatons of Quaslnear Mechansm Desgn Ths paper employs the quaslnear mechansm desgn framework, and as such t suffers from the well-known lmtatons of the approach. 19 These nclude, frst, that the transfers needed to nduce (near) truth-tellng may be very large, and thus for small δ our mechansm may be nfeasble f agents face lmted lablty constrants. Second, the quaslnear framework assumes that agents are rsk neutral wth respect to ther transfers. If agents are rsk averse over ther transfers, then t wll not generally be possble to (nearly) mplement any decson rule wth budget balance. However, f the center s nterested n nducng (nearly) truthful revelaton wthout budget balance, then redefnng the transfers n terms of utltes nstead of monetary amounts wll accomplsh ths goal. Recently, Neeman (2004) and Hefetz and Neeman (2006) have launched another lne of crtcsm aganst the lterature on mechansm desgn wth correlated nformaton. They argue that although the correlaton requrements employed n the lterature appear rather reasonable, they have the common feature that an agent s belefs unquely determne hs preferences, whch they term the BDP property. Stochastc relevance, as emboded n Assumptons 3 and 4 of ths paper, mples the BDP property. Hefetz and Neeman (2006) show that the BDP property s a non-generc property of the unversal type space. Thus, whle correlaton seems lke a reasonable assumpton, the set of BDP belefs s, n a sense, small. Another lne of crtcsm regardng Bayesan mechansm desgn s that Bayesan equlbrum s belef-based. As such, ncentve compatble mechansms are hghly senstve to the nformaton structure of the problem. MR observes that such dependence casts doubt on whether such results teach us much about real-world asymmetrc nformaton problems. Ths crtcsm has led to the search for robust mechansms that do not depend on agents belefs about others nformaton, usually employng the concept of ex post equlbrum. In lght of ths, the JM result on the generc mpossblty of effcent BIC desgn wth ndependently dstrbuted types also mples the mpossblty of ex post ncentve compatblty. Our results provde a counterpont to JM by showng that BIC desgn s possble n ther envronment f the ndependence assumpton s relaxed. However, our BIC mechansm s not ex post ncentve compatble. Indeed, Jehel et al. (2006) show 19 Crémer and McLean (1988) dscuss the lmted lablty and rsk neutralty assumptons n the context of ther full extracton result. 18
that only constant decson rules are mplementable n ex post equlbrum n generc mechansm desgn problems wth multdmensonal, contnuous types and nterdependent valuatons. 6 Concluson Ths paper extends the mechansm desgn lterature to show that when agents types are contnuous, multdmensonal, and mutually payoff relevant, that ncentve-compatble mplementaton of any decson rule s possble provded that agents types satsfy one of our rather mld correlaton condtons. Thus we provde a complement to the JM mpossblty result for the case of ndependent nformaton. Our results also complement MR by showng that there s an ex post effcent mechansm n the multdmensonal, contnuous, mutually payoff-relevant case. Whle we show the exstence of transfers that nduce a δ-truthful BNE, we have not consdered the queston of whether there exst transfers that render the exact truth a BNE. Ths s a techncally dauntng task; t remans an open queston. The scorng-rule based approach we adopt has the advantage of beng smpler than the approaches commonly adopted n the mechansm desgn lterature. Stochastc relevance (as emboded n Assumpton 3 or 4) requres verfyng only that dstrbutons are dfferent for dfferent types, whch s substantally easer than verfyng the compatblty condton of d Aspremont and Gérard-Varet (1979; 1982), the lnear ndependence condton of Crémer and McLean (1985; 1988), or the generalzaton of the Crémer-McLean condton found n MR, each of whch must hold for all pror dstrbutons for each agent s type. Beyond ts advantage of smplcty, stochastc relevance s also slghtly weaker than any of these condtons. 20 The scorng-rule-based payments used n our mechansm are also relatvely easy to construct and our proofs provde a blueprnt for dong so. Our approach represents an advance over exstng methods, whch generally prove the exstence of a mechansm but provde lttle or not gudance on how t can be constructed. 21 20 To be far, the task of full surplus extracton s more demandng than (nearly) truthful mplementaton, and so whle these papers employ strcter condtons they also acheve stronger results. Our condton s very smlar to that employed by Aoyag (1998) n the fnte case. 21 Frequently, such approches rely on a lnear systems approach to demonstrate exstence. See d Aspremont, Crémer, and Gérard-Varet (1990) for a survey of the use of ths method. 19
References [1] Aoyag, M. (1998): Correlated Types and Bayesan Incentve Compatble Mechansms wth Budget Balancedness, Journal of Economc Theory, 79, 142-151. [2] Berge, C. (1979): Topologcal Spaces: ncludng a treatment of mult-valued functons, vector spaces, and convexty. Translated by E.M. Patterson. Dover Publcatons, Inc., Mneola, NY. [3] Clarke, E. (1971): Multpart Prcng of Publc Goods, Publc Choce, 8, 19-33. [4] Cooke, Roger M. (1991): Experts n Uncertanty: Opnon and Subjectve Probablty n Scence. New York: Oxford Unversty Press. [5] Crémer, J., and R. McLean (1985): Optmal Sellng Strateges Under Uncertanty for a Dscrmnatng Monopolst When Demands Are Interdependent, Econometrca, 53, 345-361. [6] (1988): Full Extracton of Surplus n Bayesan and Domnant Strategy Auctons, Econometrca, 56, 1247-1257. [7] d Aspremont, C., and L.-A. Gérard-Varet (1979): Incentves and Incomplete Informaton, Journal of Publc Economcs, 11, 25-45. [8] (1982): Bayesan Incentve Compatble Belefs, Journal of Mathematcal Economcs, 10, 83-103. [9] (1998): Lnear Inequalty Methods to Enforce Partnershps Under Uncertanty: An Overvew, Games and Economc Behavor, 25, 311-336. [10] d Aspremont, C., J. Crémer and L.-A. Gérard-Varet (1990): Incentves and the Exstence of Pareto-Optmal Revelaton Mechansms, Journal of Economc Theory, 51, 233-254. [11] Dasgupta, P., and E. Maskn (2000): Effcent Auctons, Quarterly Journal of Economcs, 115, 341-388. [12] Green, J. and J.-J. Laffont (1977): Characterzaton of Satsfactory Mechansms for the Revelaton of Preferences for Publc Goods, Econometrca, 45, 427-438. [13] Green, J. and J.-J. Laffont (1979): Incentves n Publc Decson Makng. Amsterdam: North Holland. [14] Groves, T. (1973): Incentves n Teams, Econometrca, 41, 617-663. [15] Hefetz, A. and Z. Neeman (2006): On the Generc (Im)Possblty of Full Surplus Extracton n Mechansm Desgn, Econometrca, 74, 213-233. [16] Jackson, M. (2001): A Crash Course n Implementaton Theory, Dscusson Paper, Caltech, forthcomng n Socal Choce and Welfare. [17] Jehel, P. and B. Moldovanu (2001): Effcent Desgn wth Interdependent Valuatons, Econometrca, 69, 1237-1259. [18] Jehel, P., M. Meyer-ter-Vehn, B. Moldovanu and W. Zame (2006): The Lmts of Ex-Post Implementaton. forthcomng, Econometrca. 20
[19] Johnson, S., J. Pratt and R. Zeckhauser (1990): Effcency Despte Mutually Payoff-Relevant Prvate Informaton: The Fnte Case, Econometrca, 58, 873-900. [20] Krshna, V. (2002): Aucton Theory. London: Academc Press. [21] Maskn, E. (1992): Auctons and Prvatzaton," n H. Sebert (ed.), Prvatzaton, Kel: Insttut fur Weltwrtschaften der Unverstät Kel, 115-136. [22] Marsden, J. and M. Hoffman (1993): Elementary Classcal Analyss, W.H. Freeman and Co., New York. [23] Matsushma, H.(1990): Domnant Strategy Mechansms wth Mutually Payoff-Relevant Informaton and wth Publc Informaton, Economcs Letters, 34, 109-112. [24] (1991): Incentve Compatble Mechansms wth Full Transferablty, Journal of Economc Theory, 54, 198-203. [25] McAfee, P. and P. Reny (1992): Correlated Informaton and Mechansm Desgn, Econometrca, 60, 395-421. [26] McLean, R. and A. Postlewate (2004): Informatonal Sze and Effcent Auctons, Revew of Economc Studes, 71, 809-827. [27] Merowtz, A. (2003): On the Exstence of Equlbra to Bayesan Games wth Non-Fnte Type and Acton Spaces, Economc Letters, 78, 213-218. [28] Mezzett (2004): Mechansm Desgn wth Interdependent Valuatons: Effcency, Econometrca, 72, 1617-1626. [29] Neeman, Z. (2004): The Relevance of Prvate Informaton n Mechansm Desgn, Journal of Economc Theory, 117, 55-77. [30] Royden, H. (1988): Real Analyss, Macmllan Publshng Company, New York. [31] Selten, R. (1998): Axomatc Characterzaton of the Quadratc Scorng Rule, Expermental Economcs, 1, 43-61. [32] Vckrey, W. (1961): Counterspeculaton, Auctons, and Compettve Sealed-Tenders, Journal of Fnance, 16, 8-37. [33] Zedler, E. (1985): Nonlnear Functonal Analyss and ts Applcatons I: Fxed Pont Theorems, Sprnger-Verlag, New York. Appendx: Proofs Proof of Lemma 1: Suppose not. Then there exsts δ>0 such that for all n there exsts a sequence of pars of types (t n,t 0 n ) such that t n t 0 n R δ and kf j ( t n ) f j ( t 0 n )k 2 1/n. 21
By compactness, ths sequence has a convergent subsequence. Let the lmt be (t,t 0 ), and note that t 6= t 0. Wehavethatlm kf j ( t ) f j ( t 0 )k 2 =0, and: µz 1/2 lm fj 0 ( t n ) f j tn = lm fj (t j t n ) f j tj t 0 2 n dtj 2 = = = µz lm µz µz µz n mn 1, fj (t j t n ) f j tj t 0 2 o 1/2 n dt j ³ n lm mn 1, f j (t j t n ) f j tj t 0 2 o 1/2 n dt j n mn 1, lm f j (t j t n ) f j tj t 0 2 o 1/2 n dt j n mn 1, f j (t j t ) f j tj t 0 2 o 1/2 dt j n where the fact that mn 1, f j (t j t n ) f j (t j t 0 n ) 2o 1 allows us to apply Lebesgue s Domnated Convergence Theorem n movng from the second to thrd lne. Snce µz n mn 1, fj o 1/2 (t j t ) f j tj t 0 2 dt j =0, f (t j t ) f (t j t 0 ) =0for almost all t j. Snce f (t j t ) s contnuous, f (t j t )=f (t j t 0 ) for all t j. However, ths volates Assumpton 3. ProofofLemma2: R The result s standard. Ths proof follows Selten (1998). Let Υ (a t )= Q (tj a ) f (t j t ) dt j be agent s expected transfer when s j ( ) =τ j. Substtutng n the defnton of the quadratc scorng rule, we have: Z Ã Z! Υ (a t )= 2f (t j a ) f (t j a ) 2 dt j f (t j t ) dt j. T j T j Rearrangng Υ (a t ) yelds: Υ (a t )= Z f (t j t ) 2 dt j T j Z (f (t j a ) f (t j t )) 2 dt j. T j Hence: Z Υ (a t ) Υ (t t )= (f (t j a ) f (t j t )) 2 dt j, T j whch s zero when a = t and strctly negatve otherwse. Proof of Lemma 3: From Lemma 1, for any δ there exsts a μ > 0 such that f (t j a ) f (t j t ) 2 μ for all a and t wth a t δ. Usng the notaton from the 22
proofoflemma2, Z Z Q (t j a ) f (t j t ) dt j Q (t j a ) f (t j a ) dt j T j T j = Υ (a t ) Υ (t t ) Z = (f (t j a ) f (t j t )) 2 dt j T j = 2 kf (t j a ) f (t j t )k 2 μ 2. Lettng ε = μ 2 completes the proof. Proof of Theorem 1: Consder agent, and suppose all other agents announce truthfully, s j ( ) =τ j, j 6=. Thus for the remander of the proof we can replace a j wth t j n the expresson for agent s payoff. Snce expected drect returns are contnuous n announcements and T s compact, expected drect returns are unformly contnuous. Hence for any ε>0 there exsts δ>0 such that t a R δ mples E (v (t, t,a ) t ) E (v (t, t,t ) t ) ε. Gven δ, lett 0 = {(t,a ) T T : t a R δ} be the set of type-announcement pars that are at least δ apart. For each, choosej () accordng to Assumpton 2. Defne the payments to agent accordng to the quadratc scorng rule: x (t,a )=2f t j() a ZT j() f t j() a 2 dtj(). (1) Snce the expected quadratc score s unquely maxmzed at a = t and T 0 s compact, by Lemma 3thereexstsanˆε >0 such that for any (t,a ) T 0 : E ª ª x tj(),a t E x tj(),t t ˆε. (2) Next, scale the payments to agent accordng to x t j(),a : x 2 M +1 tj(),a = x tj(),a. (3) ˆε Consder the expected utlty reaped by a truthful announcement as compared to announcng a wth (t,a ) T 0. E v (t, t,a)+x ª tj(),a t E v (t, t,t )+x ª tj(),t t = E {v (t, t,a ) v (t, t,t ) t } + E x ª tj(),a x tj(),t t < 2 M + 2 M +1 " # x tj(),a x tj(),t ˆε ˆε < 2 M + 2 M +1 < 0. ˆεˆε Hence under payment scheme x tj(),a, announcng truthfully earns a hgher payoff than any announcement such that (a,t ) T 0. And, by the choce of δ, announcements a such that (a,t ) / T 0 have lower expected transfers than truthful announcements and have expected drect 23
returns that exceed those of truthful announcement by less than ε. Hence truthful announcement s an ε-best response. Snce s chosen arbtrarly, payments can be constructed that make the truth an ε-best response for all agents. To balance the budget, for each agent choose an agent κ () / {, j ()} wth the understandng that κ () wll fund s transfer. Let K = {k κ (k) =} be the set of all agents whose transfers funds. Agent s net transfer s therefore: x (t,a )=x X tj(),a tj(k),a k. (4) k K x k Snce agent s announcement does not affect the terms after the summaton, hs ncentves are not affected, and transfer scheme x (t, a) s ε-bic and balances the budget. ProofofLemma4: For each choose an approprate j accordng to Assumpton 4. Lay a d -dmensonal rectangular grd over T j by dvdng each of the d dmensons nto ncrements of sze β>0. Thus, the grd dvdes T j nto hypercubes wth sdes of length β. The maxmum dstance between any two ponts n a hypercube s β d (.e., the dstance n R d between (0, 0,...,0) and (β,β,...β)). Choose β such that β d <φ. Consder two dstnct types t and t 0 ³ for agent. By Assumpton 4, for any t and t 0, there exts a ball θ j t,t 0 wth radus φ such that ³ ³ f (t j t ) 6= f t j t 0 for all t j θ j ³t,t 0. Let c be the center of θ j t,t 0. As llustrated n Fgure 2, the maxmum dstance between c and any pont n the same hypercube s r m β d <φ, and thus the hypercube contanng c s contaned n θ j ³t,t 0. 22 Snce the sde length of each hypercube s β, there s³ a ball of radus β/3 wthn ths hypercube (llustrated by the dotted crcle) such that f (t j,t ) 6= f t j,t 0 for all t j wthn the ball. Snce there are a fnte set of grd elements, takng one such ball for each grd element defnes a fnte set of dsjont ³ balls B j such that for each dstnct t and t 0 there s at least one ball on whch f (t j t ) and f t j t 0 are not equal. ProofofTheorem2: The proof employs transfers based on a large scalng of the quadratc scorng rule. However, rather than workng wth K as the (large) scalng appled to the transfers, yeldng payoffs u (a, t) =v (t, a) +K x (a,a j ), we wll nstead work wth the equvalent formulaton n whch γ =1/K and payoffs aregvenbyũ (a, t) =γ v (t, a)+x (a,a j ). In ths formulaton, the agent s utlty functon depends contnuously on γ. The game wth γ =0s one n whch agent cares only about the transfer, and the game wth γ postve but small (whch corresponds a large scalng of the transfers) can be thought of as a slghtly perturbed verson of the γ =0game. The proof explots the fact that f the condtons for applcaton of Schauder s fxed pont theorem are satsfed when γ =0and best response correspondences are sngle-valued and sutably contnuous, then they are also satsfed for small-but-postve values of γ. Wthout loss of generalty, assume δ δ as specfed n Lemma 4. 23 For each choose a player j satsfyng Lemma 4, and let ˆB nˆbj o j j = 0, ˆb 1,...,ˆb j M, denote the partton of agent j s 22 For c on the boundary between several hypercubes, all such hypercubes are contaned n θ j (t,t 0 ), andanysuch hypercube can be used for the remander of the argument. 23 If not, use δ n the followng constructon. Snce t establshes that there s a δ -truthfulbne,thsbnesalso δ-truthful for δ>δ. 24
υ j (t,t' ) φ c r m β(d ) ½ T j Fgure 2: Assumpton 4 mples Lemma 4 announcement space descrbed above. 24 Let p s j( ) ³ˆbj Z a {t j s j (t j ) ˆb j } f (t j a ) dt j be the probablty of event ˆb j f agent s type s a, condtonal on j s announcement strategy. If agent j plays strategy s j ( ), let the transfer to agent be x s j( ) (a,a j ), whch s based on the quadratc scorng rule appled to the events n ˆB j accordng to: x s j( ) (a,a j )=2p s j( ) ³ˆbj (a j ) a MX p s j( ) ³ˆbj m a 2. Note that for any s j ( ) C j (δ), by Lemma 4, transfers x s j( ) (a,a j ) represent a strctly proper scorng rule, and hence for each,, s j ( ) C j (δ), andt, agent s expected transfer s maxmzed by announcng truthfully (a = t ). For any s j ( ), foreachvalueoft agent chooses a to maxmze: Z γ v (t, s (t ),a ) f (t t ) dt + x ZT s j( ) (a,a j ) f (t j t ) dt j. (5) T j ³ˆbj Snce p s j( ) a s contnuous n s j ( ) (whch mples that x s j( ) (a,a j ) s contnuous n s j ( )) and v (t, a) s contnuous n t and a, (5) s contnuous n s ( ), and n partcular n s j ( ). Further, our assumptons ensure that (5) s contnuously dfferentable n a. Notethatthetransfersare constructed for each s j ( ) n order to render truthful reportng a best response when γ =0(and m=1 24 We denote the player as j rather than j () for notatonal convenence. 25
thus, as we show below, they render nearly truthful reportng a best response when γ s postve but suffcently small). Let BA (t,s ( ),γ ) denote agent s best announcements,.e., announcements that maxmze (5) gven hs type, the other agents strateges, and the weght places on hs drect return. BA (t,s ( ),γ ) may be mult-valued. Let BR (s ( ),γ ) denote the best response operator for agent, parameterzed by γ. That s, for a gven value of γ, BR (s ( ),γ ) maps s ( ) to bestacton correspondences BA (t,s ( ),γ ). Let BR(s ( ),γ) = (BR 1 (s 1 ( ),γ 1 ),...,BR I (s I ( ),γ I )) denote the best response operator for all agents, where γ =(γ 1,...,γ I ). Let C E (δ) be the unformly equcontnuous set of δ-truthful strateges for player defned by: C E (δ) = n s ( ) C (δ) : ψ >0, t,t 0 T, t 0 t 00 ³ o <ψmples s ³t 0 s t 00 R < 2ψ. R We defne C E (δ) and CE (δ) n the usual way. Snce C E (δ) s unformly bounded, closed, and equcontnuous, the Arzela-Ascol Theorem mples that t s compact. 25 The remander of the proof shows that for γ suffcently small, BR (s ( ),γ) s a contnuous map from C E (δ) nto tself, and thus by Schauder s Fxed Pont Theorem has a fxedpont. Suchafxed pont s a δ-truthful BNE of the announcement game. 26 Step 1: Note that for each, T s a compact, convex subset of a fnte-dmensonal Eucldean space wth a non-empty nteror, and that agent s objectve functon (5) s jontly contnuous n t, a, s ( ), andγ. Thus for any t, γ,ands ( ) C E (δ), agent s best response exsts and by the Theorem of the Maxmum (Berge, 1997) BA (t,s ( ),γ ) s upper hem-contnuous n t and γ. Step 2: Next, we show that for γ suffcently small, for any s ( ) C E (δ) and any a BA (t,s ( ),γ ), ka t k R δ. Suppose that for some s ( ) and t, for any γ > 0 there exsts a BA (t,s ( ),γ ) and ka t k > δ. Take a sequence γ = 1/n, and let a n BA (t,s ( ), 1/n). By compactness, a n has a convergent subsequence. Let a be the lmt pont, and note that ka t k δ. Let W s j( ) (a,t )= R T j x s j( ) (a,a j ) f (t j t ) dt j. Snce BA (t,s ( ),γ ) s upper-hemcontnuous, a BA (t,s ( ), 0), whch contradcts that a = t s the unque maxmzer of W s j( ) (a,t ). Hence for γ suffcently small, all best responses are wthn δ of beng truthful. Step 3: Next, we argue that for γ suffcently small, BA (t,s ( ),γ ) s sngle-valued. Fx s ( ). Snce these transfers are strctly proper, for any t, a = t maxmzes W s j( ) (a,t ) strctly and unquely. Snce a = t maxmzes W s j( ) (a,t ), a = t satsfes the frst-order necessary condtons: D a W s j( ) (t,t )= 0, whered a denotes the gradent vector wth respect to the components of a and 0 denotes the d -dmensonal zero (column) vector. 27 Ths mples that W s j( ) (a,t ) <W s j( ) (t,t )+D a W s j( ) (t,t ) (a t ), whch establshes that W s j( ) (t,t ) s locally strctly concave n a at a = t. By contnuty, W s j( ) (a,t ) s locally strctly concave n 25 See Marsden and Hoffman (1993), p. 273. 26 Usng a slghtly dfferent verson of Schauder s theorem, Merowtz (2003) proves the exstence of a BNE n general Bayesan games wth nfnte type and actons spaces. 27 Ths also holds true for t on the boundary of T snce a = t s the unque global maxmzer of W s j ( ) (a,t ), and would be even f t were not constraned to come from T. Snce the relevant functons are defnedoveranopen set contanng T 2 (see footnote 7), t must be that D a W s j ( ) (t,t )= 0 for t on the boundary of T. 26
a for a near t,assw s j( ) (a,t )+γ E (v (a, t) s j ( ),t ) for γ suffcently small (snce v (a, t) has bounded frst and second dervatves). Let ρ s j( ) > 0 be the largest ρ such that W s j( ) (a,t ) s locally strctly concave n a for all t and a n the ntersecton of T and the open ball wth center at t and radus ρ. Let ρ be the mnmum of all ρs j( ). To establsh that ρ s strctly postve, suppose not. In ths case there exsts a sequence of strateges s n j ( ) CE j (δ) such that ρsn j ( ) 0. Snce Cj E (δ) s compact, ths sequence has a convergent subsequence. Let s j ( ) CE j (δ) denote the lmt pont. We have then that there exsts t such that W s j ( ) (a,t ) W s j ( ) (t,t ) for all a n a neghborhood of t. However, ths contradcts that a = t s the unque maxmzer of W s j ( ) (a,t ) at t. Hence ρ > 0. By contnuty, for γ suffcently small, W s j( ) (a,t )+γ E (v (a, t) s j ( ),t ) s locally strctly concave for all t and all a wth ka t k ρ /2. The argument of the prevous paragraph establshes that for γ suffcently small, all best responses are wthn ρ /2, and hence for γ suffcently small that BA (t,s ( ),γ ) s sngle valued. Step 4: Next, we show that there exsts γ > 0 such that BR (s ( ),γ ) s a contnuous map from C E (δ) to C (δ). Let γ be the largest γ such that BA (t,s ( ),γ ) s sngle valued for all s ( ) C E (δ) and all t. Compactness of C E (δ) ensures that γ > 0. If not, then there exsts a sequence γ n,t n,sn ( ) wth γ n 0 such that BA t n,s γ n ( ),γ n s mult-valued. Hence BA t n,s γ n ( ),γ n t >ρ /2. By compactness, we can wthout loss of generalty assume that γ n,t n,sn ( ) converges. Denote the lmt t,s ( ), 0. Snce BA (t,s ( ),γ ) s upper hem-contnuous, we have that lm BA t n,s n ( ),γ n = BA t,s ( ), 0, and thus that for any a BA t,s ( ), 0, a t >ρ /2. However, ths contradcts the concluson of the prevous paragraph. For the remander of the proof, assume that γ <γ. Snce γ <γ, BA (t,s ( ),γ ) s upper hem-contnuous and sngle-valued n t for any s ( ) C E (δ), ths mples that BA (t,s ( ),γ ) s a contnuous functon of t, and hence that BR (s ( ),γ ) maps to contnuous functons of t for γ suffcently small: BR (s ( ),γ ):C E (δ) C (δ). Further, snce (5) depends contnuously on s ( ) and γ, the Theorem of the Maxmum also establshes that BR (s ( ),γ ) s contnuous n s ( ) and γ (snce BR (s ( ),γ ) s upper hem-contnuous and sngle-valued). Step 5: Fnally, we argue that for γ suffcently small, BR (s ( ),γ ) maps C E (δ) to CE (δ). We use BR C E (δ),γ to denote the set of strateges that are best responses to some strategy n C E (δ). To establsh equcontnuty, we must show that for any ψ>0, andanyt,t 0 T : t 0 t 00 <ψmples sup R s ( ) BR (C E (δ),γ ) ³ s ³t 0 s t 00 R < 2ψ. (6) However, note that BA (t,s ( ), 0) = t for any s ( ) C E (δ), and thus that: ³ s t 0 t 00 <ψmples sup R s ( ) BR (C E (δ),0) s ³t 0 t 00 R <ψ. (7) Hence (6) s satsfed when γ =0. Snce BR (s ( ),γ ) s contnuous n γ, (6) s also satsfed for γ suffcently small. Let γ be such that (6) holds for γ <γ. A compactness argument smlar to the one used above to show that γ > 0 establshes that γ > 0. For the remander of the proof, consder only γ γ = mn{γ,γ }. For such γ, 27
BR (s ( ),γ ) maps C E (δ) to CE (δ). Step 6: Next, we apply Schauder s Fxed-Pont Theorem. Schauder s fxed pont theorem says that a contnuous operator that maps a nonempty, compact, convex subset of a Banach space nto tself has a fxed pont. 28 The precedng argument establshes that C E (δ) s such a subset and that BR (s ( ),γ) s a contnuous operator that maps C E (δ) nto tself when γ γ for all. Hence, BR (s ( ),γ) has a fxed pont. The fxed pont of the best-response mappng s a δ-truthful BNE of ths game, and thus a δ-truthful BNE of the game where payoffs aregvenby u (a, t) =v (t, a) +K x (a,a j ) for K =1/γ. Transfers are balanced usng the same type of permutaton as was employed n the proof of Theorem 1. 28 See Corollary 2.13 n Zedler (1985). 28