Abstract Single Crossing and the Value Dimension

Transcription

1 Abstract Sngle Crossng and the Value Dmenson Davd Rahman September 24, 2007 Abstract When auctonng an ndvsble good wthout consumpton externaltes, abstract sngle crossng s necessary and suffcent to mplement an effcent allocaton n ex post equlbrum. JEL Classfcaton: D21, D23, D82. Keywords: auctons, sngle crossng, mult-dmensonal types. Department of Economcs, Unversty of Mnnesota. I owe many thanks to Sushl Bkchandan and Angel Hernando for useful dscussons. Please send comments to dmr@umn.edu.

2 1 Introducton Ths short note characterzes determnstc, ex post ncentve compatble (EPIC) allocatons as those that are weakly monotone n an envronment where a sngle, ndvsble good must be allocated. Consumpton externaltes are not allowed, but arbtrarly nterdependent values are. Moreover, the characterzaton apples to any type space, not necessarly convex or ordered. An mmedate corollary of ths result s an abstract sngle crossng condton (ASC) on ndvdual utlty functons that s both necessary and suffcent for the EPIC mplementaton of an effcent allocaton. Weak monotoncty and ASC both pont to the dmenson of value as the relevant one. Intutvely, weak monotoncty reads as follows. Fx an ndvdual, say, a profle of types for others, and consder any two types for. If ndvdual s allocated the good at hs frst type and values t more at hs second type then he should also be allocated the good at hs second type. Smlarly, ASC requres that f ndvdual values t more than others at hs frst type and he values t more at hs second type than t hs frst, then he should value t more than others at hs second type, too. Weak monotoncty and ASC solate the determnants of EPIC and effcent EPIC allocatons, respectvely, regardless of whether or not type spaces are convex, ordered, utlty functons are monotone and dfferentable, n contrast wth the lterature. Maskn (1992), to whom the noton of sngle crossng n auctons s attrbutable, noted that multdmensonal sgnals may lmt the effcency propertes of auctons. Snce then, an extensve lterature on the sngle crossng condton n auctons has emerged, mostly wthn the standard framework (ordered, convex type spaces wth monotone, dfferentable utlty functons). As such, most results rely on ths framework, and yeld condtons that to some readers appear techncal, unntutve, or specfc. An excepton to ths trend s a recent paper by Saks and Yu (2005), whose argument s consderably streamlned by Vohra (2007). Buldng on Bkhchandan et al. (2006), they prove that weak monotoncty s necessary and suffcent for an allocaton to be EPIC on convex domans n otherwse arbtrary envronments. Ther result may be vewed as a refnement of one by Rochet (1987) characterzng EPIC allocatons as cyclcally monotone on arbtrary type spaces. A techncal contrbuton of ths note s to show that when auctonng off a sngle ndvsble good wthout consumpton externaltes, convexty may be dropped. Another contrbuton s to steer away from multdmensonalty per se and towards condtons that only nvolve utlty rankngs. 1

3 2 Model Consder a populaton of fntely many ndvduals, collected n the set I = {1,..., n}. Each ndvdual has prvate nformaton represented by a nonempty type space T wth typcal element t. The product space of type profles s gven by T = n T, =1 wth typcal element t = (t 1,..., t n ). A sngle, ndvsble good s to be allocated. For each, let v : T [0, ) denote Mr. s utlty from the good, and let v = (v 1,..., v n ) be the profle of utlty functons. If Mr. does not obtan the good then he draws zero utlty. A determnstc allocaton s any map z : T I {0}, where z(t) stands for the ndvdual to whom the good s allocated when t s profle of types, and z(t) = 0 means that nobody gets t. Indvduals have quas-lnear preferences over money payments. A payment scheme s any functon ξ : I T R that maps profles of reported types to ndvdual money payments. Thus, ξ (t) denotes the money pad by Mr. f t s the profle of types. Defnton 1. An allocaton z s EPIC f there s a payment scheme ξ such that I, t T, s T, v (t, z(s, t )) v (t, z(t)) ξ (s, t ) ξ (t), where, gven s and t, v (t, z(s)) := v (t)1 {z=} (s) and 1 {z=} s the ndcator functon for whether or not gets the good (so 1 {z=} (s) = 1 f z(s) = and zero otherwse). The left-hand sde represents Mr. s expected utlty gan from ms-reportng assumng observaton of t, whereas the rght-hand sde captures Mr. s expected contractual loss assocated therewth. Hence, the nequaltes requre that every ndvdual s utlty gan from ms-reportng must be outweghed by ts assocated contractual loss. Defnton 2. An allocaton z s weakly monotone f I, t T, s T, v (t) < v (s, t ) & z(t) = z(s, t ) =. 2

4 Proposton 1. An allocaton z s EPIC f and only f t s weakly monotone. Proof. Accordng to Rochet (1987), an allocaton s EPIC f and only f t s cyclcally monotone. But cyclc monotoncty mples weak monotoncty, yeldng suffcency. For necessty, fx any ndvdual, profle of others types t, and any fnte cycle (t 0,..., t m ) of types for. Let (z 0,..., z m ) be the nduced cycle of socal choces,.e., z k = z(t k, t ), and (v 0,..., v m ) the nduced cycle of values,.e., v k = v (t k, t ). Let x k = 1 f player gets the good,.e., z k =, and 0 otherwse. Defne the ndex sets M = {k : x k = 0 & x k+1 = 1} and N = {l : x l = 1 & x l+1 = 0}, where m + 1 = 0. Clearly, M = N. For cyclc monotoncty, we requre that 0 v (t k, t, z k ) v (t k+1, t, z k ) = (v k v k+1 )x k = = k M v k+1 (x k+1 x k ) v k+1 (x k+1 x k ) + l N v l+1 (x l+1 x l ) = k M v k+1 x k+1 l N v l+1 x l = k M v k+1 l N v l+1. By weak monotoncty, v l+1 v k+1 for every k M and l N. Snce M has the same cardnalty as N, t follows that k M v k+1 v l+1. l N Therefore, z s cyclcally monotone, hence EPIC, too. Ths result holds for mult-dmensonal types. It follows that any weakly monotone allocaton s EPIC. In fact, f we focus on effcent allocatons, t redefnes snglecrossng n a way that s ndependent of the type space s dmenson. Defnton 3. The profle v = (v 1,..., v n ) exhbts abstract sngle crossng (ASC) f (, t, s ), v (t) < v (s, t ) & j, v (t) v j (t) j, v (s, t ) v j (s, t ). An allocaton z s effcent f t T, z (t) arg max v (t). I By Proposton 1, weak monotoncty mples the followng. 3

5 Corollary 1. An effcent allocaton s EPIC f and only f the profle v = (v 1,..., v n ) of utlty functons exhbts abstract sngle crossng. ASC says that f t s effcent to gve the good to Mr. then t should also be effcent to gve t to hm when hs type yelds a hgher payoff and everyone else s type s fxed. Agan, abstract sngle crossng s ndependent of the dmensonalty of the type space. 3 Examples Consder the followng obvous counterexample. In a sngle good envronment wth two ndvduals, suppose that v (t, t j ) = t + 2t j, where t [0, 1]. Then the effcent allocaton fals to satsfy weak monotoncty. However, t s easy to check that lots of others do. For nstance, the allocaton gven by z(t) = j(t) where j(t) arg max t s EPIC but neffcent. Also, t s easy to see that abstract sngle crossng fals n Example 4.2 (wth multdmensonal type spaces) of Jehel and Moldavanu (2001). To see how abstract sngle crossng apples to envronments wth mult-dmensonal type spaces, suppose that there are two ndvduals, wth respectve type spaces gven by T 1 = [0, 1] and T 2 = [0, 1] [0, 1], wth typcal elements t 1 T 1 and (t 2, s 2 ) T 2. Utlty functons are gven by v 1 (t 1, t 2, s 2 ) = t 1 +s 2 and v 2 (t 1, t 2, s 2 ) = t 2 +s 2. Mr. 1 s type descrbes hs prvate value, t 1, for the good, whereas Mr. 2 s type descrbes both hs prvate value, t 2, and a common value component, s 2. Ideally, Mr. 1 should not get the good f and only f t 1 < t 2. However, such allocaton s not weakly monotone. In fact, by Proposton 1 every EPIC allocaton must be adapted to each ndvdual s so-value lnes, whch may be thought of as bds n ts ndrect mplementaton. s 2 t 1 s 2 t 1 s 2 + t 2 = const. B s 2 + t 2 = const. B z * = 2 z * = 1 z * = 2 z * = 1 A t 2 A t 2 Fgure 1: Falure of abstract sngle crossng (left) and weak monotoncty (rght). 4

6 Ths can easly be seen from Fgure 1 below. It depcts Mr. 2 s type space at some realzaton of Mr. 1 a type, say t 1. The effcent allocaton should gve the good to Mr. 2 when hs type s gven by pont A, and to Mr. 1 when t s gven by pont B. However, t s clear that such an allocaton must weak monotoncty, snce gong from A to B nvolves crossng over so-value lnes. For another example, consder the same envronment except for Mr. 2 s utlty functon, whch becomes v 2 (t 1, t 2, s 2 ) = t 2 + s 2.5. Now the orderng of types s nconsstent between both ndvduals: v 1 s monotone n the common-value component, s 2, whereas v 2 s not. Stll, weak monotoncty characterzes EPIC allocatons to be ones adapted to ndvdual so-value lnes. Ths can be seen n Fgure 2 below. s 2 t 1 s 2 t 1 s 2 + t 2 = const. s 2 + t 2 = const. z * = 1 z * = 1 z * = 2 z * = 2 t 2 t 2 Fgure 2: Falure of ASC (left) and weak monotoncty (rght) wthout order. Of course, nsghts from the examples above apply also to ther dscretzed versons. Therefore, the ntuton above apples even when the type space s not convex. 4 Concluson EPIC allocatons of a sngle, ndvsble good have been characterzed n ths note by weak monotoncty n the sprt of Bkhchandan et al. (2006), and as a corollary, effcent EPIC ones, too, n envronments wthout consumpton externaltes but arbtrarly nterdependent values. These results are smply derved wthout reference to any structural propertes of the envronment except for the value dmenson. Ths suggests that mult-dmensonal types and other smlar restrctons are not essental to understandng mechansm desgn. Ths nterpretaton may also be drawn from Saks and Yu (2005) n ther more general envronment save for a convex type space. 5

7 Saks and Yu s assume convexty to apply the separatng hyperplane theorem. Ths note s restrcted to utlty functons that are only one-dmensonal (the value of the good beng the dmenson), so a hyperplane s just a pont and convexty of sets becomes unnecessary to separate them by a pont. Apart from ts techncal contrbuton, ths note makes the followng basc argument: Economc condtons to descrbe effcency, ncentve compatblty, etc., should not be marred by mathematcal complexty. A frutful area for research seems to be to gve center stage to ths doctrne n the context of aucton theory and mechansm desgn. Ths ncludes restrctng attenton away from mult-dmensonal types per se, monotone/dfferentable utlty functons, etc., despte hstorcal precedent. A Cyclc Monotoncty Theorem 1 (Rochet). An allocaton z s EPIC f and only f t s cyclcally monotone,.e., for every I, t T and fnte cycle (t 0, t1,..., tm+1 ) wth t 0 = tm+1, [v (t k, t, z(t k, t )) v (t k+1, t, z(t k, t ))] 0. ( ) Proof. For necessty, suppose z s ex post mplementable wth transfers ξ and for any (, t ), let (t 0, t1,..., tm+1 ) wth t 0 = tm+1 be a fnte cycle. For all k {0, 1,..., m}, v (t k+1, t, z(t k, t )) v (t k+1, t, z(t k+1, t )) ξ (t k, t ) ξ (t k+1, t ). Addng wth respect to k, the rght-hand sde equals 0, and cyclc monotoncty ( ) follows. For suffcency, fx t 0 T and for any t T, let W (t) := sup [v (t k+1, t, z(t k, t )) v (t k, t, z(t k, t ))], where the supremum s taken wth respect to all fnte sequences (t 0, t1,..., tm+1 ) wth t m+1 = t. By ( ), W (t 0, t ) = 0. For any t T, W (t 0, t ) W (t) + v (t 0, t, z(t)) v (t, z(t)), whch mples that W (t) s fnte. By defnton of W (t), for any s T, W (t) W (s, t ) + v (t, z(s, t )) v (s, t, z(s, t )). Fnally, substtutng W (t) = v (t, z(t)) ξ (t) yelds ex post ncentve compatblty. 6

8 References Bkhchandan, S., S. Chatterj, A. Sen, R. Lav, A. Mu alem, and N. Nsan (2006): Weak Monotoncty Characterzes Determnstc Domnant-Strategy Implementaton, Econometrca, 74, , 5 Jehel, P. and B. Moldavanu (2001): Effcent Desgn wth Interdependent Valuatons, Econometrca, 69, Maskn, E. (1992): Auctons and Prvatzaton, Tubngen Mohr (Sebeck), Rochet, J. C. (1987): A Necessary and Suffcent Condton for Ratonalzablty n a Quas-Lnear Context, Journal of Mathematcal Economcs, 16, , 3, 6 Saks, M. and L. Yu (2005): Weak Monotoncty Suffces for Truthfulness on Convex Domans, Proceedngs of the 6th ACM Conference on Electronc Commerce, EC05, , 5, 6 Vohra, R. V. (2007): Paths, Cycles and Mechansm Desgn, Workng Paper. 1 7