Ex post implementation in environments with private goods

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1 Theoretcal Economcs 1 (2006), / Ex post mplementaton n envronments wth prvate goods SUSHIL BIKHCHANDANI Anderson School of Management, Unversty of Calforna, Los Angeles We prove by constructon that ex post ncentve compatble mechansms exst n a prvate goods settng wth mult-dmensonal sgnals and nterdependent values. The mechansm shares features wth the generalzed Vckrey aucton of onedmensonal sgnal models. The constructon mples that for envronments wth prvate goods, nformatonal externaltes (.e., nterdependent values) are compatble wth ex post equlbrum n the presence of mult-dmensonal sgnals. KEYWORDS. Ex post ncentve compatblty, mult-dmensonal nformaton, nterdependent values. JEL CLASSIFICATION. D INTRODUCTION In models of mechansm desgn wth nterdependent values, each player s nformaton s usually one dmensonal. Whle ths s convenent, t mght not capture a sgnfcant element of the settng. For nstance, suppose that agent A s reservaton value for an object s the sum of a prvate value, whch s dosyncratc to ths agent, and a common value, whch s the same for all agents n the model. Agent A s prvate nformaton conssts of an estmate of the common value and a separate estmate of hs prvate value. As other agents care only about A s estmate of the common value, a one-dmensonal statstc does not capture all of A s prvate nformaton that s relevant to every agent (ncludng A). 1 Therefore, t s mportant to test whether nsghts from the lterature are robust to relaxng the assumpton that an agent s prvate nformaton s one dmensonal. Buldng on earler work by Maskn (1992), Jehel and Moldovanu (2001) show that f agents have mult-dmensonal nformaton, nterdependent values, and ndependent sgnals then, Sushl Bkhchandan: sushl.bkhchandan@anderson.ucla.edu An earler verson of ths paper was crculated under the ttle The Lmts of Ex Post Implementaton Revsted. I am grateful to Jeffrey Ely, Benny Moldovanu, Joe Ostroy, Bll Zame, and two anonymous referees for comments that led to substantal mprovements n ths paper. Ths research was supported by the Natonal Scence Foundaton under Grant No. SES A d -dmensonal, d 2, prvate sgnal s A can be mapped, wthout any loss of nformaton, nto a sngle dmenson usng a one-to-one functon f : d. However, agents values wll not be non-decreasng or contnuous n the sgnal f (s A ). Hence, the assumpton of one-dmensonal sgnals s a lmtaton only n conjuncton wth assumptons commonly made n the lterature that a buyer s (one-dmensonal) sgnal s ordered so that a hgher realzaton s more favorable, or that the valuaton s a contnuous functon of the sgnal. Copyrght c 2006 Sushl Bkhchandan. Lcensed under the Creatve Commons Attrbuton- NonCommercal Lcense 2.5. Avalable at

2 370 Sushl Bkhchandan Theoretcal Economcs 1 (2006) unlke n models wth one-dmensonal nformaton, every Bayesan Nash equlbrum s (genercally) neffcent. 2 Jehel et al. (2006) call nto queston the exstence of ex post equlbrum when agents have mult-dmensonal nformaton and nterdependent values. They show that ex post ncentve compatble mechansms do not genercally exst (except, of course, trval mechansms that dsregard the reports of players). 3 We prove an exstence result for non-trval ex post ncentve compatble mechansms n a prvate goods settng when buyers have nterdependent values and multdmensonal sgnals. To reconcle ths wth Jehel et al. s result, we note that ther nonexstence result depends on the assumpton that for any par of outcomes there exst at least two agents who are not ndfferent between that par of outcomes. In a prvate goods envronment, as agents care only about ther own allocaton, ths assumpton s not satsfed. To see ths, consder the sale of one ndvsble object to two buyers, 1 and 2. There are three possble outcomes: a, the good s assgned to buyer, = 1, 2, and a 0, nether gets the good. Buyer 1 s ndfferent between a 2 and a 0 and buyer 2 s ndfferent between a 1 and a 0. There exst pars of outcomes (namely (a 0,a 1 ) and also (a 0,a 2 )) between whch all agents except one s ndfferent. Consequently, preferences over prvate goods are non-generc n the space of socal choce settngs consdered by Jehel et al.; ther defnton of genercty requres the presence of externaltes. Therefore, even f buyers have mult-dmensonal sgnals, the possblty of exstence of nontrval ex post ncentve compatble sellng mechansms n generc prvate goods models s not precluded. We prove an exstence result for ex post ncentve compatble mechansms for the sale of an ndvsble (prvate) good to n buyers wth mult-dmensonal sgnals and nterdependent values. The assumptons of the model are not non-generc n prvate goods envronments. In the constructed mechansm, the rule for decdng whether buyer 1, say, should be assgned the object s as follows. (The mechansm s llustrated n Fgure 1 for the case of n = 2 buyers.) Fx the other buyers sgnals at some realzaton. Partton buyer 1 s set of possble sgnal realzatons nto equvalence classes or ndfference curves such that buyer 1 s reservaton value s constant on an ndfference curve. These ndfference curves are completely ordered by buyer 1 s value. If a generalzaton of the sngle-crossng property s satsfed then there exsts a pvotal ndfference curve for buyer 1 wth the property that t s ex post ncentve compatble to award the object to buyer 1 f and only f buyer 1 s sgnal realzaton s on an ndfference curve that s greater than the pvotal one. On the pvotal ndfference curve, as llustrated n Fgure 1, the maxmum of the other buyers values s equal to buyer 1 s value. If buyer 1 wns, the prce pad by hm s equal to hs value on the pvotal ndfference curve; f he loses he pays nothng. Ths mechansm s non-trval and can be extended to multple objects when buyer preferences over objects are subaddtve. 2 See also Harstad et al. (1996), who obtan suffcent condtons under whch an effcent allocaton s attaned by common aucton forms. McLean and Postlewate (2004) show that effcent Bayesan mplementaton s possble when sgnals are correlated. 3 In a recent workng paper, Mezzett and Parreras (2005) obtan suffcent condtons for exstence of dfferentable ex post mechansms.

3 Theoretcal Economcs 1 (2006) Ex post mplementaton 371 Buyer 1 s second sgnal, s 12 Pvotal ndfference curve Buyer 1 s ndfference curves V 1 (,s 2 ) Buyer 2 s ndfference curves V 2 (s 2, ) Buyer 2 s sgnal fxed at s 2 V W > V P > V L Buyer 1 wns on ths ndfference curve Buyer 1 loses on ths ndfference curve V 2 (s 2,s 1 )<V W V 1 (s 1,s 2 )=V W V 2 (s 2,s 1 )=V P V 2 (s 2,s 1 )>V L V 1 (s 1,s 2 )=V P V 1 (s 1,s 2 )=V L Buyer 1 s frst sgnal, s 11 FIGURE 1. Indfference curves n buyer 1 sgnal space, s 1 = (s 11,s 12 ) The mechansm shares the feature wth the generalzed Vckrey aucton of onedmensonal nformaton models that the prce pad by the wnnng buyer s equal to ths buyer s value at the lowest possble sgnal (.e., on the pvotal ndfference curve) at whch ths buyer would just wn. Thus, ex post equlbra n aucton models wth one-dmensonal sgnals are robust n that non-trval ex post equlbra exst even when buyers have mult-dmensonal sgnals. In a mult-dmensonal sgnal settng the pvotal ndfference curve for a buyer conssts of a contnuum of ths buyer s sgnal realzatons whereas n a one-dmensonal settng there s exactly one pvotal sgnal realzaton for ths buyer. Consequently, when the hghest two buyer values are close to each other no buyer s sgnal s above hs pvotal ndfference curve. Ths ensures that the subset of buyers sgnals n whch one buyer gets the object does not share a common boundary wth the (dsjont) subset of buyers sgnals n whch another buyer gets the object. 4 The socal cost of ncentve compatblty n our model s that the object s retaned by the auctoneer and gans from trade are not realzed when the hghest two buyer valuatons are close to each other. There are only prvate goods n our model. Thus, n envronments wth prvate goods, nformatonal externaltes (.e., nterdependent values) alone do not preclude the exstence of ex post equlbrum n the presence of mult-dmensonal sgnals. One needs consumpton externaltes or publc goods, n addton to nformaton external- 4 It s precsely the exstence of such a common boundary that s used by Jehel et al. to show the nonexstence of ex post ncentve compatble mechansms n a settng n whch auctons are non-generc.

4 372 Sushl Bkhchandan Theoretcal Economcs 1 (2006) tes, for generc non-exstence. Ex post equlbrum has been employed mostly n aucton models wth prvate goods; 5 hence, these models are robust to relaxng the assumpton of one-dmensonal sgnals. The mechansm descrbed above s condtonally effcent n that whenever t allocates the object, t s to the buyer wth the hghest valuaton. Furthermore, t s more effcent than any other condtonally effcent mechansm. However, ths mechansm need not be effcent, subject to ncentve constrants, and a constraned effcent mechansm need not be condtonally effcent. Contrast ths to one-dmensonal sgnal models, where constraned, condtonal, and frst-best effcency are attaned n the same mechansm because ncentve constrants do not bnd. The paper s organzed as follows. A model wth two buyers s presented n Secton 2, together wth prelmnary results. An exstence result for ex post ncentve compatble mechansms n a model wth two buyers and one prvate good s proved n Secton 3. Ths result s generalzed to n buyers, and possble extensons to models wth many buyers and many objects are explored n Secton A of the appendx. A suffcent condton and a necessary condton for constraned effcency s provded n Secton 4. Secton 5 concludes. 2. THE MODEL The man dea can be seen n a model wth two buyers, = 1, 2, and one ndvsble object. The nformaton state s denoted s. Each buyer observes a d 2 dmensonal prvate sgnal s = (s 1,s 2,...,s d ) about the nformaton state. The doman of s s S = [0,1] d. Wthout loss of generalty, the buyers nformaton jontly determnes the nformaton state wth s = (s 1,s 2 ). 6 Buyer s reservaton value for the object n nformaton state s = (s,s j ) s V (s,s j ). Buyers have quaslnear utlty. If buyer gets the object n state s and pays t, then hs utlty s V (s ) t ; f he does not get the object and pays t, hs utlty s t. The outcome n whch buyer, = 1, 2, s allocated the object s denoted a, and the outcome n whch no buyer gets the object s a 0. A (determnstc) mechansm conssts of an allocaton rule h and payment functons ˆt, = 1, 2. The allocaton rule h : S {a 0,a 1,a 2 } s a functon from the buyers reported sgnals to an outcome; the payment functon ˆt : S { } s a functon from the buyers reported sgnals to a monetary payment by buyer. A mechansm (h, ˆt ) s ex post ncentve compatble f for, j = 1, 2, j, V (s,s j )1 {h(s,s j )=a } ˆt (s,s j ) V (s,s j )1 {h(s,s j )=a } ˆt (s,s j ) s, s, s j, (1) where 1 A s the ndcator functon of the event A. In other words, at each nformaton state f buyer j truthfully reports hs sgnal then buyer can do no better than truth- 5 See, for example, Crémer and McLean (1985), Ausubel (1999), Dasgupta and Maskn (2000), Perry and Reny (2002), and Bergemann and Välmäk (2002). 6 An nformaton state wll also be denoted as s = (s,s j ), where, j {1,2}, j.

5 Theoretcal Economcs 1 (2006) Ex post mplementaton 373 fully report hs sgnal. 7 If a mechansm (h, ˆt ) satsfes (1) then the allocaton rule h s mplementable and ˆt s sad to mplement h. Clearly, ex post ncentve compatblty mples that ˆt (s,s j ) = ˆt (s,s j ) f h(s,s j ) = h(s,s j ); otherwse, f say ˆt (s,s j ) < ˆt (s,s j ) then buyer has an ncentve to msreport s at nformaton state (s,s j ). In effect, the payment functon ˆt maps {a 0,a 1,a 2 } S j to { }. The followng characterzaton of ex post mplementablty s due to Chung and Ely (2003). LEMMA 1 (Chung and Ely 2003). An allocaton rule h s mplementable f and only f for each, a k, and s j, j, there exst transfers ˆt (a k,s j ) { } such that h(s ) argmax a k {V (s,s j )1 {h(s,s j )=a } ˆt (a k,s j )}. Wthout loss of generalty we assume that ˆt (a 0,s j ) = ˆt (a j,s j ) for all s j. 8 Thus, buyer s monetary payment depends only on whether or not buyer s assgned the object and on buyer j s reported sgnal, j. We restrct attenton to mechansms n whch a buyer pays nothng f he does not get the object; that s, ˆt (a k,s j ) = 0 f a k a. From Lemma 1 t s clear that ths does not decrease the set of mplementable allocaton rules. Wth ths restrcton on monetary payments we may wrte ˆt (a k,s j ) t (s j ) f a k = a 0 otherwse. If h(s,s j ) a for all s then let t (s j ) =. For mechansms n whch losng buyers pay nothng, the requrement of ex post ncentve compatblty,.e., condton (1), s rewrtten as follows. For = 1, 2, j, V (s,s j ) t (s j ) 1 {h(s,s j )=a } V (s,s j ) t (s j ) 1 {h(s,s j )=a } s, s, s j. (2) The functon t (s j ) s buyer s payment condtonal on gettng the object. One may thnk of t (s j ) as a personalzed prce at whch the object s avalable to buyer at nformaton states (,s j ). Let t = (t 1,t 2 ). Thus, any mplementable allocaton rule may be mplemented wth personalzed prces. One may also ask what type of personalzed prces mplement some allocaton rule. To ths end, defne a par of personalzed prce functons t (s j ), t j (s ) to be admssble f n each nformaton state at most one buyer s value exceeds hs personalzed prce: V (s,s j ) > t (s j ) = V j (s j,s ) t j (s ) s,s j. 7 Ex post ncentve compatblty s the same as unform equlbrum of d Aspremont and Gérard-Varet (1979) and unform ncentve compatblty of Holmström and Myerson (1983). 8 If, say, ˆt (a 0,s j ) < ˆt (a j,s j ) for some s j, then from Lemma 1 we see that h(s,s j ) a j for any s. Therefore, lettng ˆt (a j,s j ) ˆt (a 0,s j ), the transfers ˆt (a 0,s j ), ˆt (a j,s j ), ˆt (a,s j ) satsfy the argmax condton of Lemma 1 at s j.

6 374 Sushl Bkhchandan Theoretcal Economcs 1 (2006) Defne an allocaton rule supported by admssble personalzed prces (t 1,t 2 ): a 1 f V 1 (s 1,s 2 ) > t 1 (s 2 ) h(s 1,s 2 ) a 2 f V 2 (s 2,s 1 ) > t 2 (s 1 ) otherwse. a 0 (3) As t s admssble, h s a well-defned allocaton rule. Usng (2) t s easly verfed that (h,t ) s ex post ncentve compatble. A mechansm (h,t ) s non-trval f there exst at least two dstnct outcomes, each of whch s n the range of h at a postve (Lebesgue) measure of nformaton states. Any par of personalzed prces mplements an ex post mplementable allocaton rule. However, the mechansm may be trval. Exstence of a non-trval ex post ncentve compatble mechansm s proved n the next secton. 3. EXISTENCE We prove that under reasonable assumptons, a non-trval ex post ncentve compatble mechansm exsts n the model descrbed n the prevous secton. An extenson of ths result to n buyers s straghtforward and sketched out n Secton A of the appendx. The possblty of non-trval mechansms when many objects are to be allocated s explored n Secton B of the appendx. We assume that hgher sgnals correspond to better news. That s, players reservaton values do not decrease as buyer sgnals ncrease. 9 In order to smplfy the proofs, we assume also that buyers reservaton values are contnuous. ASSUMPTION 1. For = 1, 2, V s (a) non-decreasng and (b) contnuous. The assumpton that V s non-decreasng n s j can be dropped provded one assumes that V s ncreasng n s. The next assumpton s a generalzaton of the snglecrossng property. 10 ASSUMPTION 2. For, j = 1, 2, for any s j we have V (s,s j ) V (s,s j ) V j (s j,s ) V j (s j,s ) s > s. As buyer s sgnal ncreases from s to s, the ncrease n s value s not less than the ncrease n buyer j s value. That s, buyer s value s at least as responsve as buyer j s value to changes n buyer s sgnal. In models wth one-dmensonal sgnals, Assumpton 2 s a verson of the sngle-crossng property that s a suffcent condton for exstence of an effcent mechansm n such models (see Maskn 1992). The next assumpton rules out the unnterestng case where the effcent rule s trval. As any trval rule s ex post ncentve compatble, f Assumpton 3 s volated then the effcent rule s ex post ncentve compatble. 9 The followng termnology regardng monotoncty of a functon f : n s adopted. For x, x n, x > x denotes that x s at least as large as x n every co-ordnate and x x. If f (x ) f (x ) whenever x > x then f s non-decreasng. If f (x ) > f (x) whenever x > x then f s ncreasng. 10 An equvalent assumpton s that for each s j, V (s,s j ) V j (s j,s ) s a non-decreasng functon of s.

7 Theoretcal Economcs 1 (2006) Ex post mplementaton 375 ASSUMPTION 3. For each buyer, there exsts a postve measure of nformaton states at whch ths buyer s valuaton s strctly greater than the other buyer s valuaton. We construct an admssble par of personalzed prces under Assumptons 1 and 2. If, n addton, Assumpton 3 holds then the allocaton rule mplemented by ths par of admssble prces s non-trval. Fx buyer j s sgnal at some level s j. The doman of s, j, s the unt cube n d + and each buyer s valuaton s non-decreasng n s. Therefore, wth buyer j s sgnal fxed at s j, the maxmum of ether buyer s reservaton value as a functon of buyer s sgnal s attaned when s = 1, where 1 denotes the pont (1,1,...,1) n d +. Smlarly, the mnmum of ether buyer s value as a functon of s s attaned at s = 0 (0,0,...,0). Defne S (λ,s j ), the set of sgnals of buyer that lead to the same reservaton value for buyer as the sgnal ŝ = λ1, where λ [0,1]. That s, S (λ,s j ) {s S V (s,s j ) = V (λ1,s j )} 0 λ 1. Thus, for a fxed s j, buyer s sgnal space s parttoned nto equvalence classes or ndfference curves, S (λ,s j ), one for each λ [0,1], wth V (λ 1,s j ) V (λ1,s j ) whenever λ > λ. Whle buyer s value (as a functon of s ) s constant on hs ndfference curve S (λ,s j ), buyer j s value s, n general, not constant on ths set. The maxmum of buyer j s value on buyer s ndfference curve S (λ,s j ) s V m and the maxmum s acheved at j (λ,s j ) max V j (s j,s ), (4) s S (λ,s j ) s m j (λ,s j ) arg max V j (s j,s ). s S (λ,s j ) Thus, V m j (λ,s j ) = V j (s j,s m j (λ,s j )) and V (s m j (λ,s j ),s j ) = V (λ1,s j ). As S s compact and V (,s j ) s contnuous, S (λ,s j ) s compact. Ths, together wth the contnuty of V j (s j, ) mples that V m j (λ,s j ) exsts. The contnuty of the valuatons (Assumpton 1b) mples that V m j (λ,s j ) s contnuous n λ and s j. As V j s non-decreasng n s (Assumpton 1a), V m j (λ,s j ) s non-decreasng n λ. Fgure 1 depcts ndfference curves of buyers 1 and 2 n buyer 1 s (two-dmensonal) sgnal space, keepng buyer 2 s sgnal fxed at some value s 2. By Assumpton 1a, ndfference curves are negatvely sloped. However, () the ndfference curves need not be convex; () ndfference curves may touch the axes; and () the maxmum value for buyer 2 n buyer 1 s ndfference curve may be attaned at more than one pont. Thck ndfference curves are not ruled out, unless we strengthen Assumpton 1a to requre that buyers valuatons be (strctly) ncreasng n sgnals. At any other value of buyer 2 s sgnal, s 2 s 2, buyer 1 s ndfference curves n s 1 space are dfferent from, and may ntersect wth, the ones depcted n Fgure The two sets of buyer 1 ndfference curves for two dfferent values of buyer 2 sgnals do not ntersect f buyer 1 s valuaton s separable n s 1 and s 2,.e., V 1 (s 1,s 2 ) = u (s 1 ) + v (s 2 ).

8 376 Sushl Bkhchandan Theoretcal Economcs 1 (2006) Ex post ncentve compatblty mposes the followng necessary condton. If buyer 1, say, s allocated the object at nformaton state (s 1,s 2 ), then he should also be allocated the object at any nformaton state (s 1,s 2) such that V 1 (s 1,s 2) > V 1 (s 1,s 2 ). Otherwse, buyer 1 would have an ncentve to report s 1 nstead of s 1 at the nformaton state (s 1,s 2). That s, an mplementable allocaton rule must be weakly monotone. 12 We construct an ex post ncentve compatble mechansm n whch, for each value of s j, there exsts a λ j (s j ) [0,1] such that buyer wns f and only f hs sgnal s n an ndfference curve (wth an ndex) greater than λ j (s j ). Clearly, ths allocaton rule satsfes weak monotoncty. Call S (λ j (s j ),s j ) the pvotal ndfference curve for buyer at s j. Any s n the pvotal ndfference curve s a pvotal sgnal for buyer. Buyer s personalzed prce s defned to be V m j (λ j (s j ),s j ), the maxmum of buyer j s value n buyer s pvotal ndfference curve (and ths s usually equal to buyer s value on the pvotal ndfference curve). 13 These personalzed prces are admssble f V (λ1,s j ) V m j (λ,s j ) s non-decreasng n λ. Ths s shown n the next lemma. LEMMA 2. If Assumptons 1 and 2 are satsfed then for any s j and 1 λ > λ > 0, V (λ 1,s j ) V m j (λ,s j ) V (λ1,s j ) V m j (λ,s j ). PROOF. To smplfy notaton, we wrte s m j (λ ), s m j (λ) for s m j (λ,s j ), s m j (λ,s j ). Let λ m [0,1] be such that λ m s m j (λ ) S (λ,s j ). To see that λ m exsts, defne f (x ) V (xs m j (λ ),s j ), where x [0,1], and note that f (1) = V (s m j (λ ),s j ) = V (λ 1,s j ) V (λ1,s j ) V (0,s j ) = f (0). By Assumpton 1b, f (x) s a contnuous functon of x, and therefore there exsts λ m such that f (λ m ) = V (λ m s m j (λ ),s j ) = V (λ1,s j ). (As shown n Fgure 2, λ m s m j (λ ) s on the lne jonng s m j (λ ) to the orgn.) Hence, V (λ1,s j ) V m j (λ,s j ) = V (λ m s m j (λ ),s j ) V m j (λ,s j ) V (λ m s m j (λ ),s j ) V j (s j,λ m s m j (λ )) V (s m j (λ ),s j ) V j (s j,s m j (λ )) = V (λ 1,s j ) V m j (λ,s j ) where the frst nequalty follows from the fact that λ m s m j (λ ) S (λ,s j ) and (4), and the second nequalty from Assumpton 2. For λ [0,1], defne g j (λ;s j ) V (λ1,s j ) V m j (λ,s j ). 12 See Bkhchandan et al. (2006) for condtons under whch weak monotoncty s also suffcent for ncentve compatblty. 13 In Fgure 1, V 1 (s 1,s 2 ) = V P s the pvotal ndfference curve for buyer 1 when buyer 2 s sgnal s at s 2. On ths ndfference curve, the hghest valuaton of buyer 2 s also V P, whch s buyer 1 s personalzed prce at s 2.

9 Theoretcal Economcs 1 (2006) Ex post mplementaton 377 s 2 Buyer s ndfference curves Buyer j s ndfference curves λ >λ s m j (λ ) λ m s m j (λ ) V j (s j, )=V m j (λ,s j ) S (λ,s j ) V j (s j, )=V m j (λ,s j ) S (λ,s j ) V j (s j, )=V m j (s j,λ m s m j (λ )) s 1 FIGURE 2. Proof of Lemma 2. Lemma 2 mples that g j (λ;s j ) s a non-decreasng functon of λ. The contnuty of V and of V m j mples that g j (λ;s j ) s contnuous (n λ). Thus, the followng s well defned: 1 f g j (1;s j ) < 0 λ j (s j ) max{λ [0,1] g j (λ;s j ) = 0} f g j (1;s j ) 0 g j (0;s j ) 0 f g j (0;s j ) > 0, where 0 s a negatve number arbtrarly close to 0. Hence, V (λ1,s j ) > V m j (λ,s j ) f and only f λ > λ j.14 Defne V m j (0,s j ) = V m j (0,s j ). Then, as V m j (λ,s j ) s non-decreasng n λ, we have Let V (λ1,s j ) > V m j (λ j,s j ) f and only f V (λ1,s j ) > V m j (λ,s j ) f and only f λ > λ j. (5) t (s j ) V m j (λ j,s j ) (6) be buyer s personalzed prce as a functon of s j. 15 Theorem 1 shows that the followng allocaton rule s non-trval and mplementable: buyer wns f and only f hs valuaton exceeds the personalzed prce defned n (6). 14 Hereafter, the dependence of λ j on s j s usually suppressed to smplfy the notaton. 15 Note that f λ j [0,1) then V (λ j 1,s j ) = V m j (λ j,s j ) and therefore t (s j ) = V (λ j 1,s j ). If λ j = 0 then V (0,s j ) > V m j (0,s j ) = t (s j ) and f λ j = 1 then V (1,s j ) V m j (1,s j ) = t (s j ).

10 378 Sushl Bkhchandan Theoretcal Economcs 1 (2006) THEOREM 1. The personalzed prces t = (t1,t 2 ) defned n (6) are admssble. The mechansm (h,t ), where h s supported by t, s non-trval and ex post ncentve compatble. PROOF. Suppose that the nformaton state s (s,s j ). Let λ be defned by V (λ 1,s j ) = V (s,s j ). Note that (4) mples V 2 (s 2,s 1 ) V m 12 (λ 1,s 2 ) and V 1 (s 1,s 2 ) V m 21 (λ 2,s 1 ). (7) Suppose that V 1 (s 1,s 2 ) > t1 (s 2) = V12 m (λ 12,s 2). By (5), λ 1 > λ 12 and V 1(s 1,s 2 ) = V 1 (λ 1 1,s 2 ) > V12 m (λ 1,s 2 ). Hence, (7) mples that V21 m (λ 2,s 1 ) > V 2 (s 2,s 1 ) = V 2 (λ 2 1,s 1 ). From (5) we have λ 21 λ 2 and, therefore, V 2 (s 2,s 1 ) = V 2 (λ 2 1,s 1 ) V21 m (λ 21,s 1) = t2 (s 1). An dentcal argument mples that f, nstead, V 2 (s 2,s 1 ) > t2 (s 1), then V 1 (s 1,s 2 ) t1 (s 2). Thus, t s admssble. Defne an allocaton rule a 1 f V 1 (s 1,s 2 ) > t1 (s 2) h (s 1,s 2 ) a 2 f V 2 (s 2,s 1 ) > t2 (s 1) otherwse. a 0 Clearly, the mechansm (h,t ) s feasble and ex post ncentve compatble. To complete the proof, we show that (h,t ) s non-trval. Let nformaton state s 1 = (s1 1,s 2 1) be such that V 1(s1 1,s 2 1) > V 2(s1 1,s 2 1 ). Assumpton 3 guarantees that there exsts a postve measure of such nformaton states. By Assumpton 2, V 1 (1,s2 1) > V 2(s2 1,1) and by Assumpton 1a, V 2 (s2 1,1) = V 12 m (1,s 2 1). Thus, V 1(1,s2 1) > V 12 m (1,s 2 1), whch mples λ 12 (s 2 1) < 1. Hence buyer 1 gets the object at (s 1,s2 1) for all s 1 S 1 (λ,s2 1), λ (λ 12 (s 2 1 ),1]. As there s a postve measure of nformaton states at whch buyer 1 s value s strctly greater than buyer 2 s value, there s a postve measure of nformaton states at whch buyer 1 s allocated the object. A smlar argument establshes that buyer 2 s allocated the object at a postve measure of nformaton states. Hence, the mechansm s non-trval. Nether buyer s allocated the object when ther valuatons are close to each other. Ths occurs at nformaton states s = (s,s j ) such that λ (s ) < λ j (s j ) and λ j (s ) < λ j (s ). Such nformaton states have postve measure unless ether () buyers ndfference curves n S space for each fxed s j, = 1, 2, j are dentcal or () Assumpton 3 s not satsfed. From Jehel and Moldovanu (2001) we know that any ncentve compatble mechansm, ncludng ths one, s (genercally) neffcent. 16 The source of the neffcency n the current mechansm s the cost of enforcng ncentves when buyer valuatons are close. A mechansm s condtonally effcent f, when the object s allocated, t s gven to the buyer wth the hghest valuaton. Condtonal effcency s a desrable property for a seller who wshes to prevent resale. The mechansm constructed n Theorem 1 s condtonally effcent. To see ths, suppose that buyer s allocated the object at nformaton state s = (s,s j ). Let λ (s,s j ) and λ j (s j ) be defned at ths state n the usual manner. Then, from the proof of Theorem 1 t s clear that λ (s,s j ) > λ j (s j ) and therefore 16 Eső and Maskn (2000) descrbe a mult-dmensonal sgnal envronment n whch an effcent mechansm exsts.

11 Theoretcal Economcs 1 (2006) Ex post mplementaton 379 V (s,s j ) > V m j (λ (s,s j ),s j ) V j (s j,s ). In the next secton we show that ths mechansm allocates the object at more nformaton states, and s therefore more effcent, than any other condtonally effcent mechansm. Recall that any sgnal s n the pvotal ndfference curve S (λ j,s j ) s a pvotal sgnal for buyer at s j. Wth buyer j s sgnal fxed at s j, buyer wns (loses) at sgnals greater (less) than a pvotal sgnal. Thus a pvotal sgnal s an nfmum of wnnng sgnals. The prce pad by a wnnng buyer equals the valuaton of ths buyer at a pvotal sgnal (provded that λ j [0,1)). Ths s smlar to the generalzatons of Vckrey auctons n Ausubel (1999) and n Dasgupta and Maskn (2000), where buyers have one-dmensonal sgnals. 17 However, unlke n these models, n the mechansm of Theorem 1 the valuatons of buyers and j need not be equal at a pvotal sgnal of buyer ; at a pvotal sgnal of buyer, buyer s valuaton equals the most that buyer j s valuaton can be n the pvotal ndfference curve of buyer. The dfference arses because n one-dmensonal models, ndfference curves of buyer sgnals are sngletons and hence for a gven realzaton of s j there can be only one pvotal sgnal for buyer. A second dfference s the role of the sngle-crossng property or Assumpton 2. Wth one-dmensonal sgnals, the sngle-crossng property s suffcent for the exstence of an effcent ex post ncentve compatble mechansm whereas wth mult-dmensonal sgnals Assumpton 2 s suffcent for the exstence of an ex post ncentve compatble mechansm (whch s neffcent). Next, we llustrate the constructon n Theorem 1 wth an example from Jehel et al. (2006). EXAMPLE 1. Two buyers compete for an ndvsble object. Each gets a par of sgnals (p,c ), = 1, 2. Buyer s valuaton for the object s V (p,c,p j,c j ) = p + c c j, j. Further, each buyer s sgnal les n the unt square: (p,c ) [0,1] 2, = 1, 2. Fx buyer j s sgnals at (p j,c j ). Buyer s ndfference curves n c,p space are straght lnes wth slope c j. (Buyer j s ndfference curves are vertcal lnes, as V j does not depend upon p.) It may be verfed that the pvotal ndfference curve goes through the pont p = c = (p j + c j )/(1 + c j ) and V = V m j = p j + c j on ths ndfference curve. Therefore, usng (6), defne personalzed prces t (p j,c j ) = p j + c j, t j (p,c ) = p + c. Theorem 1 mples (and t may be drectly verfed) that these prces are admssble. Let h be the allocaton rule supported by the prces t1, t 2. In the ex post ncentve compatble mechansm (h,t ), the buyers report ther prvate sgnals. The mechansm desgner allocates the object to buyer for a payment equal to hs personalzed prce t (p j,c j ) = p j + c j f and only f V (p,c,p j,c j ) = p + c c j exceeds t (p j,c j ). As noted n the dscusson after Theorem 1, ths mechansm s effcent condtonal on the object beng allocated to a buyer. Ths s checked drectly as V = p + c c j > p j + c j p j + c c j = V j. 17 There s one dfference n Dasgupta and Maskn (2000). The mechansm desgner (auctoneer) does not know the mappng from buyer sgnals to valuatons. Hence, buyers submt contngent bds rather than report ther prvate sgnals.

12 380 Sushl Bkhchandan Theoretcal Economcs 1 (2006) h 1 (a 1 ) h 1 (a 1 ) h 1 (a 0 ) h 1 (a 2 ) h 1 (a 2 ) a. Two alternatves b. Three alternatves FIGURE 3. Doman of buyers sgnals, S Let (h ) 1 (a k ) be the set of nformaton states that are mapped onto a k by ths allocaton mechansm. We have (h ) 1 (a ) = {(p,c,p j,c j ) [0,1] 4 p p j > c j c c j }, = 1,2 (h ) 1 (a 0 ) = {(p,c,p j,c j ) [0,1] 4 c c j c p p j c j c c j }. The boundary between the sets (h ) 1 (a 1 ) and (h ) 1 (a 2 ) s (h ) 1 (a 1 ) (h ) 1 (a 2 ) = {(p 1,c 1,p 2,c 2 ) [0,1] 4 p 1 = p 2, c 1 = c 2 = 0}, where A s the closure of set A. The boundary between two subsets of the unt cube n 4 can be three-dmensonal; here, the boundary between between (h ) 1 (a 1 ) and (h ) 1 (a 2 ) s one-dmensonal. We refer to ths as the boundary beng of less than full dmenson, and return to ths pont below. Relatonshp wth Jehel et al. (2006) Consder a settng where the object s always allocated,.e., the outcome a 0 s ruled out. Defne µ 1 (s ) V 1 (s ), µ 2 (s ) V 2 (s ), the dfference between the two buyers valuatons for outcomes a 1 and a 2. The doman of sgnals, S, s shown schematcally n Fgure 3a. Any non-trval allocaton rule h parttons S nto two subsets, dependng on whether h(s ) = a 1 or h(s ) = a 2. The boundary between these two sets s the broken lne n Fgure 3a. Jehel et al. show that for any non-trval ex post ncentve compatble allocaton rule ths boundary s full dmensonal and ex post ncentve compatblty mples that the gradents of µ 1 (s ) and µ 2 (s ) on the boundary must be, roughly speakng, co-drectonal. They show that ths condton s mpossble to satsfy for generc reservaton values and hence the generc non-exstence of ex post ncentve compatble mechansms. The Jehel et al. proof depends on the assumpton that each buyer s not ndfferent between the two outcomes a 1 and a 2. Suppose we add back the thrd outcome, a 0. For ther argument to extend t must be case that each buyer s not ndfferent between any two of the three outcomes. Ths assumpton does not hold n the prvate goods model of ths secton as buyer s ndfferent between a j, j, and a 0 at every nformaton

13 Theoretcal Economcs 1 (2006) Ex post mplementaton 381 state. Now consder a non-trval ex post ncentve compatble allocaton rule h that yelds each of the three outcomes a 0, a 1, and a 2. Jehel et al. s theorem mples that the boundary between h 1 (a 1 ) and h 1 (a 2 ) has less than full dmenson. Ther theorem does not mpose any restrcton on the dmensonalty of the boundary between a 0 and a, = 1, 2. In partcular, the possblty that h parttons S as shown n Fgure 3b s not ruled out. In fact, ths fgure s a schematc representaton of Example 1 where we demonstrated exstence of an ex post ncentve compatble mechansm n whch the boundary between h 1 (a 1 ) and h 1 (a 2 ) s of less than full dmenson. More generally, suppose there are = 1,2,...,n buyers and L outcomes labeled a l, l = 1,2,..., L. Suppose that there exst two outcomes a l and a k such that all buyers except one are ndfferent between these two outcomes at every nformaton state: a l (s ) a k (s ) s, j. (8) The Jehel et al. theorem does not rule out generc exstence of a non-trval ex post ncentve compatble mechansm wth outcomes a l and a k. Consder the allocaton of a bundle of prvate goods to n buyers. Each outcome s an assgnment of objects among the n buyers, where we allow the possblty that not all objects are allocated to the buyers. Let a l be any assgnment and let a k be another assgnment that dffers from a l only n the allocaton that buyer j receves. 18 Condton (8) s satsfed for each assgnment a l. A full range ex post ncentve compatble mechansm s a possblty. 19 Thus, n order to obtan ther non-exstence result, the noton of genercty that Jehel et al. consder necessarly ncludes perturbatons that add externaltes to the model. Theorem 1 s true for a class of models, namely those satsfyng Assumptons 1 to 3, that s not non-generc n the space of quaslnear models wthout allocatve externaltes. Theorem 1 s easly extended to allow for n buyers (see Secton A of the appendx). Non-trval ex post ncentve compatble mechansms exst also when there are several objects for sale and buyers have subaddtve preferences (see Secton B). 4. CONSTRAINED EFFICIENCY We obtan a necessary condton and a suffcent condton for second-best effcent mechansms wthn the class of ex post ncentve compatble mechansms. Throughout ths secton, strengthened versons of Assumptons 1 and 2 (of Secton 3) are assumed, wth strct nequalty replacng weak nequalty. Thus, V s contnuous, ncreasng, and strct sngle-crossng holds. A mechansm domnates another mechansm f at each nformaton state the sum of reservaton values attaned under the frst mechansm s weakly greater than the sum of reservaton values attaned under the second mechansm, wth strct mprovement at 18 Thus, not all the objects are allocated n at least one of the two assgnments a l, a k. 19 A mechansm has full range f each outcome s mplemented at a postve measure of nformaton states.

14 382 Sushl Bkhchandan Theoretcal Economcs 1 (2006) some state. An ex post ncentve compatble mechansm s (ex post) constraned effcent f t s undomnated by another ex post ncentve compatble mechansm. The defnton of constraned effcency s due to Holmström and Myerson (1983). In Secton 2 we saw that any mplementable allocaton rule can be mplemented through admssble prces. 20 The allocaton rules supported by a par of admssble prces dffer along pvotal ndfference curves. We modfy (3), the defnton of an allocaton rule supported by admssble prces, so as to allocate the object to buyers on pvotal ndfference curves whenever possble. Defne a 1 f V 1 (s 1,s 2 ) > t 1 (s 2 ) a 1 f V 1 (s 1,s 2 ) = t 1 (s 2 ) and V 2 (s 2,s 1 ) t 2 (s 1 ) h(s 1,s 2 ) a 2 f V 2 (s 2,s 1 ) > t 2 (s 1 ) a 2 f V 2 (s 2,s 1 ) = t 2 (s 1 ) and V 1 (s 1,s 2 ) < t 1 (s 2 ) otherwse. a 0 Ths allocaton rule h s supported by admssble prces t,.e., (h,t ) s ex post ncentve compatble; moreover, t domnates the allocaton rule defned n (3) and s undomnated by any other allocaton rule supported by t. Let (t,t j ) be admssble prces. Fx buyer j s sgnal at s j. At any buyer pvotal sgnal s, we have V (s,s j ) = t (s j ). Admssblty, together wth contnuty of valuatons n sgnals, mples that for any s that s pvotal at s j, we have V j (s j,s ) t j (s ),.e., s j s (weakly) less than pvotal. Wth ths background, consder the followng defntons. A par of sgnals (s,s j ) s mutually pvotal under admssble prces (t,t j ) f s s pvotal for buyer at s j and s j s pvotal for buyer j at s. If at every s j, j = 1, 2, for whch there s a buyer pvotal sgnal, there exst mutually pvotal sgnals (s,s j ), then admssble prces (t,t j ) are mutually pvotal. Ths s a necessary condton for constraned effcency. THEOREM 2. Let (h,t ) be an ex post ncentve compatble mechansm n whch the admssble prces are contnuous functons. If (h,t ) s constraned effcent then the admssble prces must be mutually pvotal. PROOF. Suppose that contnuous admssble prces (t,t j ) are not mutually pvotal. That s, there exsts s j such that V (0,s j ) t (s j ) V (1,s j ) (10) for any s, V (s,s j ) = t (s j ) = V j (s j,s ) < t j (s ). (11) Inequaltes (10) mply that there exsts at least one buyer pvotal sgnal at s j, and (11) states that s j s not buyer j pvotal at any of these buyer pvotal sgnals. Defne t j t j, and t to be dentcal to t except at s j where t (s j ) = t (s j ) ε. The contnuty of V,V j, 20 For ths reason, we sometmes wrte that admssble prces t domnate admssble prces t or that admssble prces t are constraned effcent. (9)

15 Theoretcal Economcs 1 (2006) Ex post mplementaton 383 and t j mples that f ε > 0 s small then t s admssble. (Of course t s not contnuous.) When buyer j s sgnal s not s j, (the allocaton rules mplemented by) t and t lead to dentcal outcomes. However, at s j the subset of buyer sgnals at whch buyer gets the object s strctly larger under t than under t whereas the subset of buyer sgnals at whch buyer j gets the object s dentcal under t and t. Thus, t s domnated by t. Consder the followng defnton. Admssble personalzed prces (t,t j ) are strongly mutually pvotal f for each s j () f V (0,s j ) t (s j ) V (1,s j ) then there exsts s such that t (s j ) = V (s,s j ) = V j (s j,s ) = t j (s ) () f, nstead, V (0,s j ) > t (s j ) then V (0,s j ) V j (s j,0) () f, nstead, V (1,s j ) < t (s j ) then V (1,s j ) V j (s j,1), and s, V j (s j,s ) > t j (s ). Condton () strengthens the requrement of mutually pvotal prces to requre that there exst mutually pvotal sgnals at whch the two buyers valuatons are the same. Condtons () and () mpose restrctons when there do not exst buyer pvotal sgnals and ether buyer always wns or buyer always loses at s j. As valuatons are contnuous, strongly mutually pvotal prces are contnuous for the range of sgnals n whch () holds. For one-dmensonal sgnals, a generalzed Vckrey aucton s frst-best effcent (provded sngle crossng holds); t s also strongly mutually pvotal. Ths condton s suffcent for constraned effcency when sgnals are mult-dmensonal. THEOREM 3. An ex post ncentve compatble mechansm (h,t ) s constraned effcent f (t 1,t 2 ) are strongly mutually pvotal. PROOF. Suppose that (t,t j ) are strongly mutually pvotal. Fx buyer j s sgnal at s j and suppose that a pvotal sgnal exsts for buyer (case () of the defnton of strongly mutually pvotal). Then there exsts s such that t (s j ) = V (s,s j ) = V j (s j,s ) = t j (s ). Suppose that s 0 or 1. For small enough δ, ˆδ d +, δ, ˆδ 0, we have s δ, s + ˆδ [0,1] d. The strct sngle-crossng property mples that V (s δ,s j ) < V j (s j,s δ) V (s + ˆδ,s j ) > V j (s j,s + ˆδ). Let (t,t j ) be another par of admssble personalzed prces. Suppose that t (s j ) < t (s j ) at s j. For small enough δ the object s allocated to buyer j under t and to buyer under t at (s δ,s j ). The frst nequalty above mples that t does not domnate t. Suppose nstead that t (s j ) > t (s j ). For small enough ˆδ the object s allocated to buyer under t and to buyer j under t at (s + ˆδ,s j ). The second nequalty above mples that t does not domnate t. The proof for s = 0 or 1 s smlar, wth only one of the above nequaltes beng used n each case.

16 384 Sushl Bkhchandan Theoretcal Economcs 1 (2006) Next, suppose that there s no buyer pvotal sgnal at s j. Frst, consder case (). By sngle-crossng V (s,s j ) V j (s j,s ) for all s. Under t buyer gets the object at (s,s j ) for all s and hence the allocaton cannot be mproved. Suppose, nstead, we are n case (). By sngle-crossng, for all s the best use of the object at (s,s j ) s to allocate t to buyer j, whch s the allocaton under t. The noton of constraned effcency s dfferent from that of condtonal effcency. There usually does not exst a mechansm that satsfes both. As already noted, the mechansm constructed n Theorem 1 s condtonally effcent. However, t need not satsfy the mutually pvotal prces condton of Theorem 2 and therefore s, n general, not constraned effcent. (Recall that the admssble prces n ths mechansm are contnuous.) Ths s llustrated next, n a contnuaton of Example 1. EXAMPLE 2. Consder the same settng and valuaton functons as n Example 1 of the prevous secton. The allocaton rule supported by the personalzed prces t (p,c ) = p + c s the mechansm of Theorem 1. We drectly verfed that ths mechansm s condtonally effcent. Theorem 4 below mples that t domnates any other condtonally effcent mechansm. However, as shown below, ths mechansm s not constraned effcent. Fx buyer 2 s sgnal at (p 2,c 2 ), c 2 > 0. Buyer 1 s pvotal ndfference curve s p 1 = p 2 + c 2 c 1 c 2. For any (p 1,c 1 ) on ths ndfference curve, t 2 (p 1,c 1 ) = p 1 + c 1 = p 2 + c 2 c 1 c 2 + c 1 > p 2 + c 1 c 2 = V 2 (p 2,c 2,p 1,c 1 ). Thus, (11) mples that t1 and t 2 are not mutually pvotal. As t 1 and t 2 are contnuous, Theorem 2 mples that ths mechansm s not constraned effcent. A constraned effcent mechansm t j (p,c ) p + c 2. Suppose that Defne personalzed prces t (p j,c j ) p j + c 2 j and V (p,c,p j,c j ) = p + c c j > p j + c 2 j = t (p j,c j ). Then p p j > c 2 j c c j c c j c 2 mples that V j (p j,c j,p,c ) = p j + c c j < p + c 2 = t j (p,c ). Consequently, personalzed prces p 2 + c2 2 for buyer 1 and p 1 + c1 2 for buyer 2 are admssble. Let h be the allocaton rule supported by the prces t 1,t 2 (as defned n (9)). The mechansm (h,t ) s ex post ncentve compatble. From the fact t (p j,c j ) t (p j,c j ), wth strct nequalty whenever c j < 1, we conclude that () (h,t ) domnates (h,t ), and () (h,t ) s non-trval. 21 We show that (t 1,t 2 ) are strongly mutually pvotal. Fx buyer 2 s sgnal at (p 2,c 2 ) = (p,c) [0,1] 2. As V 1 ((0,0),(p,c)) t 1 (p 2,c 2 ) = p +c 2 V 1 ((1,1),(p,c)), condton () of the 21 In fact, f (p j,c j ) (0,0) or (1,1) then each of the outcomes a 0, a 1, and a 2 s mplemented for a postve Lebesgue measure of buyer sgnals.

17 Theoretcal Economcs 1 (2006) Ex post mplementaton 385 defnton apples. Observe that (p 1,c 1 ) = (p,c) and (p 2,c 2 ) = (p,c) are mutually pvotal sgnals that satsfy the restrcton n condton (): V 1 ((p,c),(p,c)) = p + c 2 = V 2 ((p,c),(p,c)). Theorem 3 mples that (h, t ) s constraned effcent. Next observe that at (p 1,c 1 ) = (0.5,0.2), (p 2,c 2 ) = (0.4,0.8), V 1 = 0.66 > 0.56 = V 2. However, as t 1 = 1.04 and t 2 = 0.54, buyer 2 s allocated the object at the nformaton state (p 1,c 1,p 2,c 2 ) = (0.5,0.2,0.4,0.8). Thus, the mechansm s not condtonally effcent. The next result shows that any mechansm that domnates the mechansm of Theorem 1 must be condtonally neffcent n that a buyer wth a lower valuaton s sometmes allocated the object. Thus, constraned effcency and condtonal effcency are conflctng objectves. THEOREM 4. The allocaton rule supported by the admssble prces defned n (6) domnates any other condtonally effcent mplementable allocaton rule. PROOF. Let t be defned n (6), and let h be a mechansm supported by t. Let (h,t ) be any other ex post ncentve compatble mechansm. Suppose that t (s j ) > t (s j ) for some s j. Let λ and λ be the ndexes of buyer ndfference curves that are pvotal at s j under t and t, respectvely. Clearly, λ > λ. Let s m S(λ,s j ) be the pont on the λ -ndfference curve at whch V j s maxmzed. That s, V j (s j,s m ) = V m j (λ,s j ). Further, as S(λ,s j ) s pvotal at s j under t, the constructon n Theorem 1 mples that V (s m,s j ) = V j (s j,s m ). Take any λ (λ,λ ) and s S(λ,s j ) such that s < s m. As V (s m,s j ) = V j (s j,s m ), sngle crossng mples that V (s,s j ) < V j (s j,s ). However, at (s,s j ) buyer gets the object under (h,t ). Thus, (h,t ) s not condtonally effcent. Hence, for any other condtonally effcent mechansm (h,t ), we have t ( ) t ( ), = 1, 2. Therefore, whenever the object s allocated (to a buyer) under (h,t ), t s also allocated under (h,t ), wth both mechansms allocatng to the buyer wth the hghest valuaton. As (h,t ) (h,t ), at some nformaton state (h,t ) allocates the object whle (h,t ) does not. Consequently, among all condtonally effcent mechansms, the set of nformaton states at whch the object s allocated s the largest n the mechansm of Theorem 1. However, ths mechansm s usually not constraned effcent as t s unlkely to satsfy the necessary condton of Theorem 2. To see ths, we wrte the mutually pvotal condton for ths mechansm. Take any s j for whch (10) holds. Let λ be the ndex of the buyer pvotal ndfference curve at s j. That s, λ s such that there exsts s m S (λ,s j ) satsfyng V (s m,s j ) = V j (s j,s m ) V j (s j,s ) s S (λ,s j ). Further, there exsts s S (λ,s j ) at whch s j s buyer j pvotal. That s, lettng λ j be the ndex of the buyer j pvotal ndfference curve at s, we have s j S j (λ j,s ) where s m j S j (λ j,s ) s.t. V j (s j,s ) = V j (s m j,s ) = V (s,s m j ) V (s,s j ) s j S j (λ j,s ).

18 386 Sushl Bkhchandan Theoretcal Economcs 1 (2006) Ths condton appears to be dffcult to satsfy, as we saw n Example 2. Whenever that s the case, Theorem 2 mples that the mechansm of Theorem 1 s not constraned effcent. 5. CONCLUDING REMARKS Jehel et al. (2006) queston the exstence of ex post equlbrum n models wth multdmensonal sgnals and nterdependent values. Our paper shows that ex post equlbrum exsts n such models wth prvate goods. Exstence s proved under the assumpton that buyers nformaton satsfes a generalzaton of the sngle-crossng property. The mechansm shares the feature wth the generalzed Vckrey aucton of onedmensonal nformaton models that the prce pad by the wnnng buyer s equal to ths buyer s value at the lowest possble sgnal (ndfference curve) at whch ths buyer wns. Thus, ex post equlbra n aucton models wth one-dmensonal models are robust n that non-trval ex post equlbra exst even when buyers have mult-dmensonal sgnals. To reconcle our postve result wth the negatve result of Jehel et al., observe that no consumpton externalty s a natural assumpton n many economc models. But n the space of all preferences (wth and wthout externaltes), the absence of externaltes s non-generc. At a tny perturbaton away from the no externaltes assumpton, (8) s not satsfed by any par of outcomes. Jehel et al. s theorem would then mply generc non-exstence of ex post ncentve compatble mechansms. However, non-trval mechansms that are approxmately ex post ncentve compatble stll exst at these tny perturbatons away from selfsh preferences. Thus, ex post ncentve equlbrum s a robust equlbrum concept for models wth prvate goods. Under a small departure from the usual assumpton of selfsh preferences n prvate goods models, many results n economcs would be only approxmately true. Three possble notons of effcency are () frst-best effcency wthout regard to ncentve constrants, () effcency subject to ncentve constrants, and () condtonal effcency subject to ncentve constrants. In one-dmensonal sgnal models, the ncentve constrants do not bnd and all three types of effcency are attanable n one mechansm. Wth mult-dmensonal sgnals, ncentve constrants preclude frst-best effcency (Jehel and Moldovanu 2001) and we llustrate that, n the class of mplementable allocaton rules, there s a tenson between constraned effcency and condtonal effcency. The mechansm of Theorem 1 domnates any other condtonally effcent mechansm but s not usually constraned effcent. Concevably, as the dfferences n how the buyers aggregate the multple dmensons of the sgnals decreases, the mechansm of Theorem 1 approaches both constraned effcency and frst-best effcency. APPENDIX A. EXTENSION TO MANY BUYERS We outlne the mnor changes n notaton, assumptons, and analyss requred to extend Theorem 1 to many buyers.

19 Theoretcal Economcs 1 (2006) Ex post mplementaton 387 Each buyer s valuaton depends on the (possbly mult-dmensonal) sgnals of all n buyers. The nformaton states are denoted s = (s 1,s 2,...,s n ) = (s,s ). Change s j to s n Assumpton 2, and requre the assumpton to hold for every V and V j. Assumpton 3 s requred to hold for two dstnct buyers,.e., there exst two sets of nformaton states, A and A j, each set wth postve measure, such that buyer s [j s] value s strctly greater than all other buyers values on the set A [A j ]. We wrte V (s,s ), V m j (λ,s ), g j (λ;s ) nstead of V (s,s j ), V m j (λ,s j ), g j (λ;s j ), etc. The defnton of λ j (s ) s 1 f g j (1;s ) < 0 λ j (s ) max{λ [0,1] g j (λ;s ) = 0} f g j (1;s ) 0 g j (0;s ) 0 f g j (0;s ) > 0, where g j (λ;s ) V (λ1,s ) V m j (λ,s ). Buyer s personalzed prce s t (s ) maxv m j (λ j (s ),s ). j Once agan, buyer s personalzed prce equals the maxmum valuaton of all other buyers on the pvotal ndfference curve, whch equals s valuaton at a pvotal sgnal whenever max j λ j (s ) [0,1). B. MANY BUYERS AND MANY OBJECTS There are n buyers ndexed by or j, and K objects ndexed by k or l. Each buyer receves a d -dmensonal sgnal s [0,1] d. Buyer s valuaton for object k alone s V k (s,s ); hs valuaton for a subset L {1,2,..., K } s denoted V L (s,s ). Each buyer s preferences over subsets of objects are subaddtve (defned n (14) below). We wrte S k (λ,s ), V m,k j (λ,s ), g k j (λ;s ), λ k j (s ) nstead of S (λ,s j ), V m j (λ,s j ), g j (λ;s j ), λ j (s ), etc. Note that only V k (s,s ) = V k (λ1,s ) for s S k (λ,s ); n general, V l (s,s ) V l (λ1,s ) when s S k (λ,s ), l k. The followng generalzatons of Assumptons 1 and 2 of Secton 3 are suffcent for exstence of admssble prces. ASSUMPTION 1. For all and k, V k ASSUMPTION 2. For all, j, and k, for any s we have V k (s,s ) V k s (a) non-decreasng and (b) contnuous. (s,s ) V k j (s,s ) V k j (s,s ) s > s. Personalzed prces (whch we show to be admssble for subaddtve preferences over subsets of objects) for each object are defned by t k (s ) maxv m,k j (λ k j (s ),s ) s, k,. (12) j

20 388 Sushl Bkhchandan Theoretcal Economcs 1 (2006) Note that t k (s ) 0. Wth Assumptons 1 and 2, the results of Secton 3 generalze so that for any s, f V k (s ) > t k (s ) then V k j (s ) t k j (s j ) for all j. (13) Consder the followng mechansm, whch gves each buyer a surplus maxmzng bundle at personalzed prces t k that satsfy (13) (for nstance, the prces defned n (12)). Buyers report ther sgnals. Personalzed prces t k are computed for each buyer and each object. Every buyer gets a mnmal element n hs demand set at the reported sgnals at these personalzed prces. Thus, f buyers report s = (s 1,s 2,...,s n ) then buyer gets L {1,2,..., K } such that V L (s ) t k (s ) V L (s ) t k (s ) L {1,2,..., K }, k L and f V L k L (s ) k L t k (s ) = V L (s ) k L t k (s ) for some other L then L L. Call ths mechansm a demand mechansm, because each buyer gets a (mnmal) element of hs demand set. No matter what preferences buyers have over subsets of objects, because at each nformaton state each buyer gets an element of hs demand set, the demand mechansm satsfes the ex post ncentve compatblty constrants. However, n general ths mechansm may not be feasble as demand for an object may exceed ts supply (of one unt). We show that for subaddtve preferences (defned below), the demand mechansm s feasble: each object s allocated to at most one buyer. Moreover, we exhbt examples of subaddtve preferences n whch ths mechansm s non-trval. Subaddtve preferences The value of the unon of two dsjont subsets s no greater than the sum of the values of the two subsets. That s, for all s V L L (s ) V L (s ) + V L (s ) L, L {1,2,..., K }, L L =. (14) We menton two specal cases of subaddtve preferences. Clearly, addtve preferences, where the valuaton of a subset s the sum of the valuatons of objects n the subset, satsfy (14) wth equalty. A second example s that of unt demand preferences,.e., the preferences of the assgnment model. Each buyer has utlty for at most one object. If a buyer s gven a subset of objects L, he selects an object wth the hghest valuaton and throws away the rest. Thus, hs reservaton value for any subset L {1,2,..., K } s V L (s ) max {V k (s )}. k L Note that the object that attans the maxmum may vary wth s. It s easly verfed that unt demand preferences also satsfy (14). LEMMA 3. If Assumptons 1 and 2 are satsfed and buyers preferences are subaddtve then the demand mechansm s feasble and ex post ncentve compatble.