Mechanism Design with Maxmin Agents: Theory and an Application to Bilateral Trade

Transcription

1 Mechansm Desgn wth Maxmn Agents: Theory and an Applcaton to Blateral Trade Alexander Woltzky Stanford Unversty March 7, 04 Abstract Ths paper develops a general necessary condton for the mplementablty of socal choce rules when agents are maxmn expected utlty maxmzers. The condton combnes an nequalty verson of the standard envelope characterzaton of payo s n quaslnear envronments wth an approach for relatng maxmn agents subectve expected utltes to ther obectve expected utltes under any common pror. The condton s then appled to gve an exact characterzaton of when e cent trade s possble n the blateral tradng problem of Myerson and Satterthwate (983), under the assumpton that agents know each other s expected valuaton of the good (whch s the nformaton structure that emerges when the agents start wth a common pror but are pessmstc about how the other agent mght acqure nformaton before partcpatng n the mechansm). Whenever e cent trade s possble, t may be mplemented by a relatvely smple double aucton format. For helpful comments, I thank Perpaolo Battgall, Aaron Bodoh-Creed, Subr Bose, Gabe Carroll, Aytek Erdl, Ben Golub, Fuhto Koma, Massmo Marnacc, Paul Mlgrom, Suresh Mutuswam, Ilya Segal, Andy Skrzypacz, Joel Sobel, three anonymous referees, and partcpants n several semnars. I thank Jason Huang for excellent research assstance.

2 Introducton Robustness, broadly construed, has been a central concern n game theory and mechansm desgn snce at least the celebrated argument of Wlson (987). The Wlson doctrne s usually nterpreted as callng for mechansms that perform well n a wde range of envronments or for a wde range of agent behavors. However, there s also growng and complementary nterest n robustness concerns on the part of agents nstead of (or n addton to) on the part of the desgner; that s, n askng what mechansms are desrable when agents use robustly optmal strateges. Ths paper pursues ths queston n the case where agents are maxmn expected utlty maxmzers (Glboa and Schmedler, 989) by developng a general necessary condton for a socal choce rule to be mplementable, and by applyng t to gve an exact characterzaton of when e cent trade s possble n the classcal blateral trade settng of Myerson and Satterthwate (983). The necessary condton for mplementaton generalzes a well-known necessary condton n the ndependent prvate values settng, namely that expected socal surplus must exceed the expected sum of nformaton rents left to the agents, as gven by an envelope theorem. That ths condton has any analogue wth maxmn agents s rather surprsng, for two reasons. Frst, the usual envelope characterzaton of payo s need not hold wth maxmn agents. Second, and more mportantly, a maxmn agent s belef about the dstrbuton of opposng types depends on her own type. Ths s also the stuaton wth Bayesan agents and correlated types, where results are qute d erent than wth ndependent types (Crémer and McLean, 985, 988; McAfee and Reny, 99). The dervaton of the necessary condton (Theorem ) address both of these ssues. For the rst, I derve an nequalty verson of the standard envelope condton that does hold wth maxmn agents. The key step s replacng the partal dervatve of an agent s allocaton wth respect to her type wth the mnmum value of ths partal dervatve over all possble belefs of the agent. For the second, I note that, by de nton, a maxmn agent s expected utlty under her own subectve belef s lower than her expected utlty under any belef she nds possble. Ths mples that the sum of agents subectve expected utltes s lower than the sum of ther obectve expected utltes under any possble common pror,

3 whch n turn equals the expected socal surplus under that pror (for a budget-balanced mechansm). Hence, a necessary condton for a socal choce rule to be mplementable s that the resultng expected socal surplus exceeds the expected sum of nformaton rents for any possble common pror; that s, for any pror wth margnals that the agents nd possble. The second part of the paper apples ths necessary condton to gve an exact characterzaton of when e cent blateral trade s mplementable, under the assumpton that the agents know each other s expected valuaton of the good (as well as bounds on the valuatons). As explaned below, ths nformaton structure s the one that emerges when agents have a (unque) common pror on values at an ex ante stage and are maxmn about how the other mght acqure nformaton before enterng the mechansm. In ths settng, the assumpton of maxmn behavor may be an appealng alternatve to the Bayesan approach of specfyng a pror over the set of experments that the other agent may have access to, especally when ths set s large (e.g., conssts of all possble experments) or the agents nteracton s one-shot. Furthermore, the great elegance of Myerson and Satterthwate s theorem and proof suggests that ther settng may be one where relaxng the assumpton of a unque common pror s partcularly appealng. My second man result (Theorem ) shows that the Myerson-Satterthwate theorem sometmes contnues to hold when agents are maxmn about each other s nformaton acquston technology as descrbed above but sometmes not. In the smplest blateral trade settng where the range of possble seller and buyer values s [0; ], the average seller value s c, and the average buyer value s v, Fgure ndcates the combnaton of parameters (c ; v ) for whch an e cent, maxmn ncentve compatble, nterm ndvdually ratonal, and weakly budget balanced mechansm exsts. c c log Above the curve the formula for whch s + c + v c v log + v = v Ths s n lne wth Glboa s exhortaton n hs monograph on decson makng under uncertanty to [consder] the MMEU [Maxmn Expected Utlty] model when a Bayesan result seems to crucally depend on the exstence of a unque, addtve pror, whch s common to all agents. When you see that, n the course of some proof, thngs cancel out too neatly, ths s the tme to wonder whether ntroducng a lttle bt of uncertanty may provde more realstc results, (Glboa, 009, p.69). 3

4 v* c* Fgure : In the blateral trade settng, e cent trade s possble n the regon below the curve and mpossble n the regon above t. the Myerson-Satterthwate theorem perssts, despte the lack of a unque common pror or ndependent types. Below the curve, the Myerson-Satterthwate theorem fals. I call the mechansm that mplements e cent trade for all parameters below the curve n Fgure the ( ) double aucton. It s so called because when a type agent and a type agent trades, the type agent receves a share ( ) of the gans from trade that depends only on her own type and not on her opponent s. The ( ) double aucton has the property that an agent s worst-case belef about her opponent s type s that t s always ether the most favorable value for whch there are no gans from trade or the most favorable value possble. If an agent msreports her type to try to get a better prce, the requrement that her opponent s expected value s xed forces the devator s worst-case belef to put more weght on the less favorable of these values, whch reduces her expected probablty of trade. The share ( ) s set so that ths rst-order cost n terms of belef exactly o sets the rstorder bene t n terms of prce, whch makes the ( ) double aucton ncentve compatble for maxmn agents. Fnally, the ( ) double aucton s weakly budget balanced f and only f ( ) + ( ) for all ; ; that s, f and only f the shares that must be left to the two agents sum to less than one. I also consder several extensons and robustness checks n the blateral trade settng. 4

5 One observaton here s that f the average types of the two agents do not have gans from trade wth each other (e.g., f the par (c ; v ) les below the 45 lne n Fgure ), then e cent trade can be mplemented wth a mechansm even smpler than the ( ) double aucton, whch I call a reference rule. A reference rule works by settng a reference prce p and specfyng that trade occurs at prce p f ths s acceptable to both agents, and that otherwse trade occurs (when e cent) at the reservaton prce of the agent who refuses to trade at p. The ntuton for why references rules are ncentve compatble wth maxmn agents f (and only f) c v s smple and s gven n the text. Ths paper ons a growng lterature on games and mechansms wth maxmn agents, or wth agents who follow robust decson rules more generally. In contrast to much of ths lterature, the current paper shares the followng mportant features of classcal Bayesan mechansm desgn: () the mplementaton concept s (partal) Nash mplementaton; () the only source of uncertanty n the model concerns exogenous random varables, namely other agents types; and () for Theorem, the model admts the possblty of a unque common pror as a specal case. Several recent papers derve qute permssve mplementablty results wth maxmn agents by relaxng these assumptons, n contrast to the relatvely restrctve necessary condton of Theorem. Bose and Darpa (009), Bose and Mutuswam (0), and Bose and Renou (03) relax () by consderng dynamc mechansms that explot the fact that maxmn agents may be tme-nconsstent. A central feature of ther approach s that agents cannot commt to strateges, so they do not obtan mplementaton n Nash equlbrum. Ther approach also reles on takng a partcular poston on how maxmn agents update ther belefs, an ssue whch does not arse here. D Tllo, Kos, and Messner (0) and Bose and Renou (03) relax () by assumng that agents are maxmn over uncertan aspects of the mechansm tself. Ths lets the desgner extract the agents nformaton by ntroducng bat provsons nto the mechansm. The mechansms consdered n these four papers are undoubtedly nterestng and may be appealng n partcular applcatons. However, they arguably rely on a more thoroughgong commtment to maxmn behavor than does the current paper (agents must be tme-nconsstent, or must be maxmn over endogenous random varables). Even f one accepts ths commtment, t stll seems natural to ask what s possble n the more standard 5

6 case where () and () are sats ed. De Castro and Yannels (00) relax () by assumng that agents belefs are completely unrestrcted, and nd that e cent socal choce rules are then always mplementable. Ths s consstent wth Theorem, as wth completely unrestrcted belefs agents can always expect the worst possble allocaton, whch mples that the necessary condton of Theorem s vacuously sats ed. For example, n the blateral trade settng, e cent trade s always possble, as agents are always certan that they wll not trade and are therefore wllng to reveal ther types. The model of Lopomo, Rgott, and Shannon (009) sats es (), (), and (), but consders agents wth ncomplete preferences as n Bewley (986) rather than maxmn preferences. There are two natural versons of ncentve compatblty n ther model, whch bracket maxmn ncentve compatblty n terms of strength. Lopomo, Rgott, and Shannon show that the stronger of ther notons of ncentve compatblty s often equvalent to ex post ncentve compatblty (whereas maxmn ncentve compatblty s not), and that full extracton of nformaton rents s genercally possble under the weaker of ther notons, and s sometmes possble under the stronger one. Fnally, Bodoh-Creed (0) sats es (), (), and () and consders maxmn agents. Bodoh-Creed s man result provdes condtons for an exact verson of the standard envelope characterzaton of payo s to apply wth maxmn agents, based on Mlgrom and Segal s (00) envelope theorem for saddle pont problems. As dscussed below, these condtons are not sats ed n my settng. Bodoh-Creed also does not derve a general necessary condton for mplementablty lke Theorem, and hs treatment of applcatons (ncludng blateral trade) focuses not on e cency but on revenue-maxmzaton usng full-nsurance mechansms, as n Bose, Ozdenoren, and Pape (006). The paper proceeds as follows. Secton presents the model. Secton 3 gves general necessary condtons for mplementaton. Secton 4 apples these condtons to characterze when e cent blateral trade s possble. Secton 5 contans addtonal results n the blateral trade settng. Secton 6 concludes. The appendx contans omtted proofs and auxlary results. 6

7 Model Agents and Preferences: A group N of n agents must make a socal choce from a bounded set of alternatves Y R n. Each agent has a one-dmensonal type ; = R, and agents have quaslnear utlty. In partcular, f alternatve y = (y ; : : : ; y n ) s selected and a type agent receves transfer t, her payo s y + t : Agent s type s her prvate nformaton. In addton, each agent has a set of possble belefs about her opponents types, where s an arbtrary nonempty subset of ( ), the set of Borel measures on (throughout, probablty measures are denoted by, and the correspondng cumulatve dstrbuton functons are denoted by F ). Each agent evaluates her expected utlty wth respect to the worst possble dstrbuton of her opponents types among those dstrbutons n ; that s, the agents are maxmn optmzers. Mechansms: A drect mechansm (y; t) conssts of a measurable allocaton rule y :! Y and a measurable and bounded transfer rule t :! R n. Gven a mechansm (y; t), let U ^ ; ; U ^ ; ; = y ^ ; + t ^ ; ; = E hu ^ ; ; ; U ( ) = nf U ; ; : Thus, U ^ ; ; s agent s utlty from reportng type ^ aganst opposng type pro le gven true type, U ^ ; ; s agent s expected utlty from reportng type ^ aganst belef gven true type, and U ( ) s agent s worst-case expected utlty from reportng her true type. 3 The assumpton that utlty s multplcatve n and y s for smplcty. One could nstead assume that utlty equals v (y; ) + t for some absolutely contnuous and equd erentable famly of functons fv (y; )g, as n Mlgrom and Segal (00) or Bodoh-Creed (0). 3 The term worst-case s only used heurstcally n ths paper, but the meanng s generally that f mn U ^ ; ; exsts, then a mnmzer s a worst-case belef; whle f the mnmum does not 7

8 A dstngushng feature of ths paper s the noton of ncentve compatblty employed, whch I call maxmn ncentve compatblty. A mechansm s maxmn ncentve compatble (MMIC) f arg max nf U ^ ; ; for all ; N: () ^ I restrct attenton to MMIC drect mechansms throughout the paper. The motvaton for dong so s provded n Appendx B, where I prove the approprate verson of the revelaton prncple. In partcular, I show that when agents are maxmn optmzers, any Nash mplementable socal choce rule s MMIC. 4 The man assumpton underlyng ths result s that an agent cannot commt to randomze, meanng that mxed strateges are evaluated accordng to the maxmn crteron realzaton-by-realzaton. If nstead agents could commt to randomze, then restrctng attenton to MMIC mechansms n the sense of () would no longer be wthout loss of generalty. Assumng that agents cannot commt to randomze seems as natural as the alternatve n most contexts, and s also the more common assumpton n the lterature on mechansm desgn wth maxmn agents (e.g., Bose, Ozdenoren, and Pape, 006; de Castro and Yannels, 00; Bodoh-Creed, 0; D Tllo, Kos, and Messner, 0). In addton to MMIC, I also consder the followng standard mechansm desgn crtera. Ex Post E cency (EF): y () arg max yy P y for all. Interm Indvdual Ratonalty (IR): U ( ) 0 for all. Ex Post Weak Budget Balance (WBB): Ex Post Strong Budget Balance (SBB): P t () 0 for all. P t () = 0 for all. E cency s self-explanatory. Interm ndvdual ratonalty s mposed wth respect to agents own worst-case belefs; n addton, all results n the paper contnue to hold wth ex post ndvdual ratonalty (XPIR) (.e., U ( ; ; ) 0 for all ; ). exst (whch s possble, as U ^ ; ; may not be contnuous n ), then a lmt pont of a sequence that attans the n mum s a worst-case belef. 4 The soluton concept throughout the paper s thus Nash equlbrum (ths s made explct n Appendx B and s captured mplctly by () n the text). In partcular, there s no strategc uncertanty or hgher order ambguty (as n Ahn (007)). 8 The

9 ex post verson of budget balance seems approprate, snce there s no unque common pror. The d erence between weak and strong budget balance s that wth weak budget balance the mechansm s allowed to run a surplus. An allocaton rule y s maxmn mplementable f there exsts a transfer rule t such that the mechansm (y; t) sats es MMIC, IR, and WBB. 3 Necessary Condtons for Implementaton The most general result n the paper s a necessary condton for maxmn mplementaton, whch generalzes a standard necessary condton for Bayesan mplementaton wth ndependent prvate values. In partcular, n an ndependent prvate values envronment wth common pror dstrbuton F, t s well-known that an allocaton rule y s Bayesan mplementable only f the expected socal surplus under y exceeds the expected nformaton rents that must be left for the agents n order to satsfy ncentve compatblty. It follows from standard arguments that ths condton may be wrtten as X y () d X ( F ( )) y ; d ; () where y ; = E [y ( ; )] (recall that s the measure correspondng to cdf F ). I wll show that a smlar condton s necessary for maxmn mplementaton, despte the lack of ndependent types or a unque common pror. Intutvely, the requred condton wll be that () holds for all dstrbutons F wth margnals that the agents nd possble, wth the mod caton that, on the rght-hand sde of (), the expected nformaton rents under F are replaced by the expectaton under F of type s mnmum possble nformaton rent. I begn by formalzng these concepts. Gven a measure (), let S denote ts margnal wth respect to S, for S N. Let be the set of measures () such that, for all N, the margnals and are parwse ndependent and. Some examples may clarfy ths de nton. 9

10 If n =, then = (where ). Suppose the set of each agent s possble belefs takes the form of a product = Q 6= for some sets of measures ( ) (n partcular, each agent beleves that her opponents types are ndependent). Then = Q N T 6= If n >, t s possble that s empty. For nstance, take the prevous example wth T 6= = ; for some.. Fnally, let ~y ( ) = nf y ; : Thus, ~y ( ) s the smallest allocaton that type may expected to receve. The followng result gves the desred necessary condton. Theorem An allocaton rule y s maxmn mplementable only f, for every measure, X y () d X ( F ( )) ~y ( ) d : (3) Comparng () and (3), () says that the obectve expected socal surplus under F must exceed the obectve expected nformaton rents, whereas (3) says that the obectve socal surplus must exceed the obectve expectaton of (a lower bound on) the agents subectve nformaton rents, re ectng the fact that agents subectve expected allocatons are not derved from F. In addton, () must hold only for the true obectve dstrbuton F (.e., the true common pror dstrbuton), whle (3) must hold for any canddate obectve dstrbuton F (.e., any dstrbuton n ). Wth these mod catons, Theorem shows how a standard necessary condton for Bayesan mplementaton carres over to maxmn mplementaton. Furthermore, Theorem generalzes ths condton, as n the case of a unque ndependent common pror, t follows that = for all, = fg, and ~y ( ) = y ;, so (3) reduces to (). In Secton 4, Theorem s appled to characterze when e cent blateral trade s mplementable. The rst step n provng Theorem s dervng an nequalty verson of usual envelope characterzaton of payo s, Lemma. Lemma s related to Theorem of Bodoh-Creed 0

11 (0), whch gves an exact characterzaton of payo s usng Mlgrom and Segal s (00) envelope theorem for saddle pont problems. The d erence comes because the maxmn problem () admts a saddle pont n Bodoh-Creed but not n the present paper; one reason why s that Bodoh-Creed assumes that y ; s contnuous n and (hs assumpton A8), whch may not be the case here. 5 For example, e cent allocaton rules are not contnuous, so Bodoh-Creed s characterzaton need not apply for e cent mechansms. I dscuss below how Theorem may be strengthened f () s assumed to admt a saddle pont. 6 Lemma In any maxmn ncentve compatble mechansm, U ( ) U ( ) + ~y (s) ds for all : (4) Proof. As Y s bounded, Theorem of Mlgrom and Segal (00) may be appled to the problems nf U ^ ; ; and max^ nf U ^ ; ; to show that U ( ) s absolutely contnuous. Hence, U ( ) s d erentable almost everywhere, and U ( ) = U ( ) + R U 0 (s) ds. Gven a pont of d erentablty 0 ;, x a sequence k convergng to 0 from above. Then U k U 0 nf U 0 ; ; k nf = nf U 0 ; ; 0 + k nf U 0 ; ; 0 + = nf k 0 k 0 ~y 0 : y 0 ; U 0 nf k 0 0 ; ; 0 y 0 ; y 0 ; nf U nf U 0 ; ; 0 0 ; ; 0 Hence, U ( k ) U ( 0 ) k 0 ~y 0, and therefore U 0 0 ~y 0. The result follows as U ( ) = 5 A careful readng of Bodoh-Creed (0) reveals that some addtonal assumptons are also requred for the exstence of a saddle pont, such as quasconcavty assumptons. Bodoh-Creed (04) provdes an alternatve dervaton of hs payo characterzaton result usng addtonal contnuty assumptons n leu of assumptons guaranteeng the exstence of a saddle pont. Nether set of assumptons s sats ed n the current settng. 6 The present approach of boundng U ( ) also bears some resemblence to Segal and Whnston (00), Carbaal and Ely (03), or Kos and Messner (03), wth the d erence that the d culty here s not relatng U ( ) to R U 0 () d, but rather relatng U 0 () to (bounds on) y ;.

12 U ( ) + R U 0 (s) ds. The remanng step n the proof of Theorem relates the bounds on subectve expected utltes n (4) to the obectve socal surplus on the left-hand sde of (3). The key reason why ths s possble s that a maxmn agent s subectve expected utlty s a lower bound on her expected utlty under any probablty dstrbuton she nds possble. Hence, the sum of agents subect expected utltes s a lower bound on the sum of ther obectve expected utltes under any measure, whch n turn s a lower bound on the obectve expected socal surplus under (f weak budget balance s sats ed). Proof of Theorem. Suppose mechansm (y; t) sats es MMIC, IR, and WBB. For any measure ( ), ntegratng (4) by parts yelds Recall that U ( ) d U ( ) + ( F ( )) ~y ( ) d : U ( ) ( y () + t ()) d for all : Combnng these nequaltes mples that, for every measure = ( ), or ( y () + t ()) d d U ( ) + ( F ( )) ~y ( ) d ; ( y () + t ()) d U ( ) + ( F ( )) ~y ( ) d : (5) Note that every measure s of the form ( ) for each. for every, summng (5) over yelds X ( y () + t ()) d X Fnally, P U ( ) 0 by IR and P U ( ) + X If the maxmn problem () admts a saddle pont R ( F ( )) ~y ( ) d : Thus, t () d 0 by WBB, so ths nequalty mples (3). ^ ( ) ; ( ) (where MMIC mples

13 that ^ ( ) = ), then, lettng y ( ) = y ; ( ) be type s expected allocaton under her worst-case belef ( ), Theorem 4 of Mlgrom and Segal (00) or Theorem of Bodoh-Creed (0) mples that (4) may be strengthened to U ( ) = U ( ) + y (s) ds for all : (6) The same argument as n the proof of Theorem then mples that the necessary condton (3) may be strengthened to X y () d X ( F ( )) y ( ) d : Thus, f the maxmn problem admts a saddle pont then a necessary condton for maxmn mplementaton s that, for every measure, the expected socal surplus under exceeds the expectaton under of the sum of the agents subectve nformaton rents (.e., the nformaton rents under the worst-case belefs ( )). Unfortunately, I am not aware of su cent condtons on the allocaton rule y alone that ensure the exstence of a saddle pont. Applyng the Debreu-Fan-Glcksberg xed pont theorem to (), a su cent condton on the mechansm (y; t) s that y and t are contnuous n and U ^ ; ; s quasconcave n ^ and quasconvex n. A well-known necessary condton for an e cent allocaton rule to Bayesan mplementable s that an ndvdually ratonal Groves mechansm runs an expected surplus (Makowsk and Mezzett, 994; Wllams, 999; Krshna and Perry, 000). Ths follows because the standard envelope characterzaton of payo s mples that the nterm expected utlty of each type n any e cent and Bayesan ncentve compatble mechansm s the same as her nterm expected utlty n a Groves mechansm. However, ths result does not go through wth maxmn ncentve compatblty, even f () admts a saddle pont. Ths s because the envelope characterzaton of payo s wth MMIC, (6), depends on types expected allocatons under ther worst-case belefs ( ), and these belefs n turn depend on transfers as well 3

14 as the allocaton rule. In partcular, dstnct e cent and MMIC mechansms that gve the same nterm subectve expected utlty to the lowest type of each agent need not gve the same nterm subectve expected utltes to all types, n contrast to the usual payo equvalence under Bayesan ncentve compatblty. 7 Indeed, I show n Secton 4 that, n the context of blateral trade, the e cent allocaton rule may be maxmn mplementable even f all IR Groves mechansms run expected de cts for some measure. On the other hand, the condton that an IR Groves mechansm runs an expected surplus s also su cent for an e cent allocaton rule to be Bayesan mplementable, because, followng Arrow (979) and d Aspremont and Gérard-Varet (979), lump-sum transfers of the form h ( ) may be used to balance the budget ex post wthout a ectng ncentves. Ths result also does not carry over wth maxmn ncentve compatblty, as these transfers can a ect agents worst-case belefs and thereby a ect ncentves. Ths ssue makes constructng desrable MMIC mechansms challengng, and ths paper does not contan postve results on maxmn mplementaton outsde of the blateral trade context (where, however, a full characterzaton s provded). 4 Applcaton to Blateral Trade In ths secton, I show how Theorem can be appled to obtan a full characterzaton of when e cent blateral trade or blateral publc good provson s possble when agents know each other s expected valuaton of the good. The key assumptons n ths secton are that n = and Y = f0 = (0; 0) ; = (; )g (e.g., fno trade; tradeg, fno provson; provsong), and that n addton every measure sats es E [ ] = for some ; (note that, as n the general case, can be negatve). 8 E [ ] The results n ths secton actually only requre the weaker assumpton that for all ; the ntuton s that, wth maxmn agents, only bounds on how bad an agent s belef can be are bndng. 9 However, Secton 5. shows that the 7 Ths pont was already noted by Bodoh-Creed (0). 8 The assumpton that les n the nteror of s wthout loss of generalty: f ;, then there would be no uncertanty about agent s value, and e cent trade could always be mplemented wth a Groves mechansm. 9 Wthout a bound on how bad belefs can be, de Castro and Yannels s (00) theorem shows that e cent 4

15 equalty assumpton E [ ] = s approprate f agents have a unque common pror wth mean ( ; ) and support ; ; at an ex ante stage and may acqure addtonal nformaton pror to enterng the mechansm. I therefore adopt the equalty assumpton for consstency wth ths nterpretaton. The model remans general enough to cover the classcal blateral trade and blateral publc good provson settngs, as follows. Blateral Trade: Agent s the seller and agent s the buyer. They can trade (y = ) or not (y = 0). The seller s type s tmes her value of retanng the obect (or equvalently her cost of provdng t). The buyer s type s hs value of acqurng the obect. t ( ; ) s the prce receved by the seller. t ( ; ) s Publc Good Provson: tmes the prce pad by the buyer. Agents and can ether share the cost C R of provdng a publc good (y = ) or not (y = 0). An agent s type s her bene t from the good, net of a benchmark payment of C. t ( ; ) s to C. tmes what agent pays for the good n addton The specal case of ths model where =, = ;, wll be useful for provdng ntuton for the results. I call ths the algned supports case. Wth algned supports, the least favorable type of agent does not have (strct) gans from trade wth any type of agent, and the most favorable type of agent has (weak) gans from trade wth every type of agent. For example, algned supports n the blateral trade settng means that the sets of possble values of the seller and the buyer concde. Two specal knds of dstrbutons wll play an mportant role n the analyss. Let be the Drac measure on, so that s for sure. 0 that s, l ; h Thus, l ; h Let l ; h corresponds to the possblty that agent s value be the Bernoull measure on l and h satsfyng E l ;h [ ] = ; s gven by = l wth probablty h and h = h wth probablty l. h l corresponds to the possblty that agent s value may take on only value l or h. The results to follow requre and l ; h l for certan values of l ; h. trade s always mplementable. 0 In the nformaton acquston nterpretaton of Secton 5., 0 corresponds to the possblty that agent may acqure no new nformaton about her value before enterng the mechansm. In the nformaton acquston nterpretaton of Secton 5., l ; 0 h corresponds to the possblty that agent may observe a bnary sgnal of her value before enterng the mechansm, where the bad sgnal lowers her expected value to l and the good sgnal rases her expected value to h. 5

16 I now turn to the characterzaton result. A prelmnary observaton s that f no type of some agent has gans from trade wth the average type of agent, then e cent trade s always possble. In ths case, e cent trade can be mplemented by smply gvng the entre surplus to agent : certanty of no-trade s then a worst-case belef for every type of agent, so truthtellng s trvally optmal for agent, and truthtellng s optmal for agent by the usual Vckrey-Clarke-Groves logc. Proposton Assume that + 0 and for some f; g. Then e cent trade s mplementable (wth strong budget balance). Proof. See Appendx A. In lght of Proposton, the man result of ths secton consders the non-trval case where some type of each agent has gans from trade wth the average type of the other agent (.e., + > 0 for = ; ). I also assume that some type of each agent does not have strct gans from trade wth the average type of the other agent (.e., + 0 for = ; ). Ths s an analogue of the overlappng supports assumpton of the Myerson-Satterthwate theorem. Put together, the assumptons that + > 0 and + 0 for = ; say that the ntervals and are su cently wde or su cently well-algned. For example, these assumptons hold wth algned supports. The result s the followng. Theorem Assume that + > 0 and + 0, and that ; for all [ ; ], for = ;. + + mn ; + Then e cent trade s mplementable f and only f + mn ; +! Proof. See Appendx A. + mn ;! + mn ;! log + + mn ;! log + + mn ;!! : (*) Theorem shows that, under mld restrctons, e cent blateral trade between maxmn agents s possble f and only f Condton (*) holds. In other words, the Myerson-Satterthwate Note that ; =, so we have. 6

17 mpossblty result holds wth maxmn agents f and only f Condton (*) fals. As wll become clear, Condton (*) says precsely that, n the ( ) double aucton, ( ) + ( ) for all ;. To understand the economc content of Condton (*), I dscuss three aspects of the condton. Frst, what does Condton (*) mply for comparatve statcs and other economc results? Second, where does Condton (*) come from? And, thrd, why s Condton (*) necessary and su cent condton for mplementaton, whle Theorem only gves a necessary condton? To see the mplcatons of Condton (*), rst set asde the rst term n each of the products on the left-hand sde (.e., the +mnf ; g + and +mnf ; g + terms). These terms vansh wth algned supports, and may be vewed as adustments that are needed f there are some types that are ether sure to trade (f > < ) or sure to not trade (f ). Next, note that each of the remanng products s of the form log ( + x), whch x s decreasng n x. In partcular, ncreasng makes Condton (*) harder to satsfy; a very rough ntuton s that ncreasng makes agent more con dent that he wll trade, whch makes shadng hs report to get a better transfer more temptng. Another observaton s that Condton (*) always holds when + 0; that s, when the average types of each agent do not have strct gans from trade wth each other (e.g., ths s why the curve n Fgure les above the 45 lne). Ths follows because, usng the nequalty log + x the left-hand sde of Condton (*) s at least x, +x = + + : Ths s consstent wth Proposton 4 below, whch shows that e cent trade s mplementable wth reference rules when + 0. In partcular, the parameters for whch e cent trade s mplementable wth general mechansms but not wth reference rules are precsely those that satsfy Condton (*) but would volate Condton (*) f the log + x terms were approxmated by x. Ths gves one measure of how restrctve reference rules are. +x To see (heurstcally) where Condton (*) comes from, suppose that supports are algned and that the worst-case belef of a type agent who reports type ^ s ^ ;, the belef that mnmzes the probablty that strct gans from trade exst (among belefs wth 7

18 E [ ] = ). U ^ ; ^ ; = 0. Then Suppose also that the mechansm s ex post ndvdually ratonal, so that U ^ ; ^ ; ; = + ^ + ^ + t ^ ; + + ^! + ^ (0) ; where +^ s the probablty of trade (.e., the probablty that +^ = under ^ ; ) and + t ^ ; s type s payo n the event that trade occurs. Assumng that t s d erentable, the rst-order condton for truthtellng to be optmal t = + + t ; : + Solvng ths d erental equaton for t ; yelds t ; = + + " k log + + # ; where k s a constant of ntegraton. The constant that keeps transfers bounded as! s k = + log, whch gves t ; = ( ) + ; where ( ) = + log + +! : Now, lettng t ( ; ) = ( ) ( + ) for all ; ; so that the resultng mechansm s an ( ) double aucton as descrbed n the ntroducton, t may be ver ed that weak budget balance holds for all ( ; ) f and only f t holds for ; (and t also may be ver ed that ^ ; s ndeed a worst-case belef). Therefore, 8

19 e cent trade s mplementable f and only f t ; + t ; 0; or equvalently + + 0; or + : Ths s precsely Condton (*) (n the algned supports case). In other words, Condton (*) says that the shares of the socal surplus that must be left to the hghest types n the ( ) double aucton sum to less than one. Fnally, why does the su cent condton for mplementablty that + match the necessary condton from Theorem? Recall that the necessary condton s that expected socal surplus exceeds (a lower bound on) expected nformaton rents (.e., (3) holds) for any dstrbuton. A rst observaton s that t su ces to compare the socal surplus and nformaton rents under the crtcal dstrbuton ; ;, as ths dstrbuton may be shown to mnmze the d erence between the left- and rght-hand sdes of (3). Lettng be the probablty that = under ;, the expected socal surplus under ; ; n the algned supports case equals + ; as under ; ; there are strct gans from trade only f = and =. Next, the expected (lower bound on) agent s nformaton rent under ; equals ~y ( ) d + ( ) ~y ( ) d ; {z } =0 9

20 whch may be shown to equal + = + : The explanaton for the appearance of the term here s that ths s the fracton of the socal surplus that must be left to type n an MMIC mechansm when type s subectve expected allocaton s ~y ( ) (n partcular, the bound on agent s subectve nformaton rents gven by ntegratng ~y ( ) s tght n the current settng), and the last equalty follows by algned supports. Combnng these observatons, the necessary condton from Theorem reduces to ; whch s equvalent to +. The approach taken to constructng the ( ) double aucton s qute d erent from standard approaches n Bayesan mechansm desgn. In partcular, the approach here s to post type s worst-case belef to be ; (the belef the mnmzes the probablty that strct gans from trade exst); solve a d erental equaton comng from ncentve compatblty for t ;, whch gves the formula for ( ); and then verfy that ; s ndeed type s worst-case belef n the ( ) double aucton. In contrast, a standard approach mght be to use an AGV mechansm. However, as argued above, standard arguments for why usng such mechansms s wthout loss of generalty do not apply wth maxmn agents. Indeed, I close ths secton by showng that n some cases e cent trade s mplementable n an ( ) double aucton, but not n any AGV mechansm. Wth maxmn agents, the natural de nton of an AGV mechansm s a mechansm where, for all ;, t ( ; ) = E ( ) [ y ( ; )] + h ( ) for some worst-case belef ( ) arg mn U ; ; and some lump-sum transfer functon h :! R. Suppose that > 0 for = ; and Condton (*) and the 0

21 assumptons of Theorem hold (t may be checked that these assumptons are mutually consstent). Theorem then mples that an ( ) double aucton mplements e cent trade, but I clam that e cent trade s not mplementable n any AGV mechansm. To see ths, rst note that ndvdual ratonalty of type mples that h 0 for = ;, as otherwse one would have U ( ) U ; ; = h < 0. Next, WBB mples that t ( ; ) + t ( ; ) = E ( ) [ y ( ; )] + E ( ) [ y ( ; )] + h ( ) + h ( ) 0: On the other hand, EF, E [ ] =, and > 0 mply that, for all ;, E [ y ( ; )] + E [ y ( ; )] = Pr (y ( ; ) = ) E [ : y ( ; ) = ] + Pr (y ( ; ) = ) E [ : y ( ; ) = ] > 0: Pr (y ( ; ) = ) + Pr (y ( ; ) = ) Therefore, h ( ) + h ( ) < 0, whch s nconsstent wth h 0 for = ;. A closely related pont s that f + > 0 and Condton (*) and the assumptons of Theorem hold, e cent trade s mplementable even though every IR Groves mechansm runs an expected de ct for some measure, n contrast to the results of Makowsk and Mezzett (994), Wllams (999), and Krshna and Perry (000) for Bayesan mechansm desgn. Ths follows because IR agan mples that h 0 for = ; for a Groves mechansm gven by t ( ; ) = y ( ; ) + h ( ) ; so the expected de ct of such a mechansm under the measure s equal to t ( ; ) + t ( ; ) = + + h ( ) + h ( ) > 0:

22 5 Further Results on Blateral Trade Ths sectons presents addtonal results on blateral trade wth maxmn agents. Secton 5. descrbes how the assumpton that agents know each other s expected valuaton may be nterpreted n terms of nformaton acquston. Secton 5. explores the robustness of Theorem to alternatve models of ambguty averson. Secton 5.3 characterzes when e cent trade s possble wth reference rules, a partcularly smple and appealng class of mechansms. 5. Informaton Acquston Interpretaton The assumpton that agents know the mean and bounds on the support of the dstrbuton of each other s value emerges naturally when agents share a unque common pror at an ex ante stage but are uncertan about the nformaton acquston technology that one s opponent can access pror to enterng the mechansm. Ths secton provdes the detals of ths argument. Consder the followng extenson of the model. Each agent s ex post utlty s ~ y + t ; where ~ R s her realzed ex post value. There s an ex ante stage at whch the agents belefs about the ex post values ~ ; ~ are gven by a (unque) common product measure ~ on ; ; wth mean ( ; ) (the common pror). Before enterng the mechansm, each agent observes the outcome of a sgnalng functon ( experment ) :! M (where M s an arbtrary message set), whch s nformatve of her own ex post value but ndependent of her opponent s. Agent s nterm value, (whch corresponds to her type n the man model), s then her posteror expectaton of ~ after observng the outcome of her experment. gven by That s, after observng outcome m, agent s valuaton for the good s E ~ h ~ ~ = m. (7) Note that the ssue of updatng ambguous belefs does not arse n ths model. In par-

23 tcular, each agent knows her own sgnalng functon, so the updatng n (7) s completely standard. However, the followng observaton shows that the man model can be nterpreted as resultng from each agent s beng maxmn about her opponent s sgnalng functon at the nterm stage (.e., after she observes her own sgnal). h Remark If a measure s the dstrbuton of = E ~ ~ ~ = m under ~ for some experment, then E [ ] = and supp (where supp denotes the support of ). The fact that E [ ] = s the law of terated expectaton. The fact that supp follows because ~ ; wth probablty under ~. Thus, assumng that agent nds some partcular set of measures satsfyng E [ ] = and supp possble amounts to assumng that he nds t possble that agent may have access to some subset of all possble experments. 3 Wth ths nterpretaton, the assumpton that means that agent nds t possble that agent acqures no nformaton about her value before enterng the mechansm (beyond the common pror), whle the assumpton that l ; h means that agent nds t possble that agent observes a bnary sgnal of her value, where the bad realzaton lowers her expectaton of ~ to l and the good realzaton rases her expectaton of ~ to h. 5. Less Extreme Ambguty Averson Although the maxmn expected utlty (MMEU) model s perhaps the best-studed model of ambguty averson, ts emphass on agents worst-case belefs strkes some researchers as extreme (see Glboa (009) or Glboa and Marnacc (0) for dscusson). therefore consders the robustness of Theorem to less extreme models. Ths secton In partcular, I consder the epslon contamnaton model axomatzed by Kopylov (008) and the varatonal preferences model axomatzed by Maccheron, Marnacc, and Rustchn (006). Both of 3 More precsely, the set of possble measures s ontly determned by the set of experments agent may have access to and the pror. For example, every measure such that E [ ] = and supp s the dstrbuton of for some experment f and only f the pror puts probablty on agent s ex post value beng ether or (see, for example, Theorem of Shmaya and Yarv (009) or Proposton of Kamenca and Gentzkow (0)). 3

24 these models nest MMEU as a specal case and allow for a tractable analyss of the robustness of Theorem as one moves away from MMEU. I nd that Theorem may fal to be robust to epslon contamnaton preferences (dependng on the pror that s contamnated wth ambguty averson), but s robust to varatonal preferences. 5.. Epslon Contamnaton To model agents wth epslon contamnaton preferences, assume there s a (unque) common pror CP compatblty s = CP CP and a constant " [0; ] such that the noton of ncentve arg max ^ ( ") U ^ ; CP ; + " mn U ^ ; ; : (8) Note that " = 0 corresponds to Bayesan IC and " = corresponds to maxmn IC. I show that the concluson of Theorem may change dscontnuously as " decreases from, dependng on the pror CP. The ntuton s that ths apparently small change n belefs radcally changes the ncentves of types for whom certanty of no-trade s a worst-case belef (.e., types < ). In partcular, n the MMEU model, ncentve compatblty s unusually easy to satsfy for types <, as these types do not expect to trade n the worst-case and are therefore wllng to reveal ther nformaton. In contrast, n the epslon contamnaton model for " <, ncentve compatblty s, n a sense, unusually hard to satsfy for <, as these types now condton on the " probablty event that the opponent s type s gven by the common pror CP, and thus behave as f they are Bayesan expected utlty maxmzers. One mght therefore expect the Myerson-Satterthwate mpossblty result to apply for any " < whenever a type < and a type < have gans from trade wth each other, whch occurs when + < 0. However, n the epslon contamnaton model types > do not condton on the opponent s type beng gven by CP, and f e cent trade among these types can be mplemented wth a surplus, ths surplus can then be used to subsdze trade among types <. Nonetheless, f CP puts su cently small weght on types >, ths subsdy cannot be large enough n expected terms (accordng to CP ) 4

25 to allow the < trades to trade e cently, at least f the ex post verson of ndvdual ratonalty s mposed. The followng result formalzes ths ntuton. Mantan the assumptons of Theorem, so that f + < 0 then Theorem mples that e cent trade s possble wth maxmn agents. The followng result shows that, for some pror CP, ths concluson fals n the epslon contamnaton model for any " <. Proposton If + < 0 and + > 0 for = ;, then there exsts a common pror CP wth postve densty on ; ; such that e cent trade s not mplementable wth ex post IR n the epslon contamnaton model for any " <. Proof. See Appendx A. 5.. Varatonal Preferences When agents have varatonal preferences, the approprate noton of ncentve compatblty s arg max for some functon c : ( )! R +. preferences f mn ^ ( ) U ^ ; ; + c (9) Note that varaton preferences concde wth MMEU 8 < 0 f c = : f = I show that Theorem holds for a class of varatonal preferences that extends far beyond the specal case of MMEU. In partcular, consder the smple class of varatonal preferences where 9 = ; : c = a E [ ] for some a > 0: Ths class admts the model of Secton 4 as the lmtng case where a!. followng result shows that n fact Theorem holds for varatonal preferences of ths type for any a. Ths ndcates a sgn cant degree of robustness of Theorem to consderng varatonal preferences that are less extreme than MMEU. The proof proceeds by showng that f a, an agent s worst-case belef wth varatonal preferences (.e., the mnmzng 5 The

26 measure n (9)) concdes wth her worst-case belef n the MMEU model, so the ( ) double aucton remans ncentve compatble wth varatonal preferences. Proposton 3 Suppose the assumptons of Theorem are sats ed and Condton (*) holds, so that the ( ) double aucton mplements e cent trade wth maxmn agents. Then the ( ) double aucton also mplements e cent trade when agents have varatonal preferences wth a. Proof. See Appendx A. 5.3 E cent Trade wth Reference Rules A common ust caton for ntroducng concerns about robustness nto mechansm desgn s that these concerns may argue for the use of smpler or otherwse more ntutvely appealng mechansms. The ( ) double aucton ntroduced n the prevous secton s smple n some ways (for example, t s arguably more standard than other mechansms that have appeared n the lterature on mechansm desgn wth maxmn agents), but t does nvolve a carefully chosen transfer rule. In ths secton, I pont out that e cent trade can also be mplemented n an extremely smple class of mechansms, whch I call reference rules, n the case where the average types of the two agents do not have gan from trade wth each other (.e., when + 0). Reference rules also have the advantage of satsfyng strong strong rather than weak budget balance. I de ne a reference rule as follows. De nton A mechansm (y; t) s a reference rule f 8 9 < f + 0 = y ( ; ) = : 0 f + < 0 ; 6

27 and there exst transfers t R for = ; such that t = t and 8 >< t ( ; ) = >: t f t ; t ; + 0 f < t ; t ; + 0 f t ; < t ; f + < 0 9 >= >; : 4 Wth a reference rule, agents trade wth reference transfers (t t ) f they are both wllng to do so. Otherwse, the agent who s unwllng to trade at the reference transfers receves her reservaton transfer, and the other agent receves the full gans from trade. For example, n the classcal blateral trade settng (where c s the seller s cost, v s the buyer s value, and p s the prce), a reference rule corresponds to settng a reference prce p, tradng at prce p f c p v, and otherwse tradng at whchever value s closer to p (.e., prce s v f c v < p ; prce s c f p < c v). Reference rules clearly satsfy EF, (ex post) IR, and SBB, so an IC reference rule mplements e cent trade. The followng result characterzes when IC reference rules exst; that s, when e cent trade s mplementable wth reference rules. Proposton 4 Assume that for = ;. Then e cent trade s mplementable wth reference rules f and only f ether. + 0 (the average types do not have strct gans from trade), or. + 0 (every par of types has gans from trade). Proof. See Appendx A. The ntuton for Proposton 4 s partcularly easy to see n the classcal blateral trade settng. The ntuton for why reference rules are ncentve compatble when c v and p [v ; c ] s captured n Fgure. Observe that every buyer wth value v c may be certan that no gans from trade exst, as he may beleve that the dstrbuton of seller values s the degenerate dstrbuton on c. Hence, certanty of no-trade s a worst-case belef for these buyers, and they are therefore wllng to reveal ther nformaton. In contrast, buyers 4 Note that ths de nes t ( ; ) for all ;, because f < t and < t then + < t t = 0. 7

28 buyers certan of no trade prce ndependent of buyer s report v* p* c* prce ndependent of seller s report sellers certan of no trade Fgure : Reference rules wth p [v ; c ] are ncentve compatble. wth value v > c do beleve that gans from trade exst wth postve probablty. But t s optmal for these buyers to reveal ther values truthfully as well: msreportng some ^v > c does not a ect the prce regardless of the seller s value (as prce equals c f c > p and equals p f c p ), and msreportng some ^v c agan gves payo 0 n the worst-case (as certanty that the seller s value equals c would agan be a worst-case belef). Therefore, truthtellng s optmal for every buyer type, and the argument for sellers s symmetrc. The ntuton for why reference rules are not ncentve compatble when c < v may be seen n Fgure 3. Suppose the reference prce p s greater than c. Consder a buyer wth value v (c ; p ). If he reports hs value truthfully, then whenever he trades under the reference rule he does so at prce v, whch gves hm payo 0. Suppose he nstead shades hs report down to some ^v (c ; v). Then whenever he trades the prce s ^v, whch gves hm a postve payo, and n addton he expects to trade wth postve probablty (snce ^v > c ). Hence, he wll shade down. The same argument shows that n any reference rule a seller wth c (p ; v ) shades up. Fgure 3 shows that a consequence of ths argument s that a reference rule cannot be IC for both agents when c < v, regardless of where the reference prce p s set. The term reference rule s taken from Erdl and Klemperer (00), who recommend the use of such mechansms n mult-unt auctons. They hghlght that reference rules perform well n terms of agents local ncentves to devate, a d erent crteron from what I consder here. Reference rules also bear some resemblance to the downward exble prce mechansm of Börgers and Smth (0). Ther mechansm starts wth a xed prce p whch the seller may then lower to any p 0 < p, whereupon the partes decde whether to 8

29 buyers devate sellers devate c* p* v* buyers devate c* v* p* sellers devate p* c* v* Fgure 3: When c < v, no reference rule s ncentve compatble. trade at p 0. 6 Concluson Ths paper contrbutes to the study of mechansm desgn where agents follow robust decson rules, n partcular where agents are maxmn expected utlty maxmzers. I establsh two man results. Frst, I gve a general necessary condton for the mplementablty of a socal choce rule, whch generalzes the well-known condton from Bayesan mechansm desgn that expected socal surplus must exceed expected nformaton rents. Ths condton nvolves both a mod caton of the usual envelope characterzaton of payo s and a smple but mportant conceptual connecton between maxmn agents subectve expected utltes and the obectve expected socal surplus under a common pror. Second, I apply ths result to gve a complete characterzaton of when e cent blateral trade s possble, when agents know lttle beyond each other s expected valuaton of the good (whch s the nformaton structure that results when agents are maxmn about the experment that the other may access before partcpatng n the mechansm). Somewhat surprsngly, the Myerson- Satterthwate mpossble result sometmes contnue to hold wth maxmn agents, despte the lack of a unque common pror or ndependent types. When nstead e cent trade s possble, t s mplementable wth a relatvely smple double aucton format, the ( ) double aucton; sometmes, t s also mplementable wth extremely smple reference rules. 9