Implementation in Mixed Nash Equilibrium

Transcription

1 Department of Economcs Workng Paper Seres Implementaton n Mxed Nash Equlbrum Claudo Mezzett & Ludovc Renou May 2012 Research Paper Number 1146 ISSN: ISBN: Department of Economcs The Unversty of Melbourne Parkvlle VIC

2 Implementaton n Mxed Nash Equlbrum Claudo Mezzett y & Ludovc Renou z 8th May 2012 Abstract A mechansm mplements a socal choce correspondence f n mxed Nash equlbrum f, at any preference pro le, the set of all (pure and mxed) Nash equlbrum outcomes concdes wth the set of f-optmal alternatves for all cardnal representatons of the preference pro le. Unlke Maskn s de nton, our de nton does not requre each optmal alternatve to be the outcome of a pure equlbrum. We show that set-monotoncty, a weakenng of Maskn s monotoncty, s necessary for mxed Nash mplementaton. Wth at least three players, set-monotoncty and no-veto power are su cent. Important correspondences that are not Maskn monotonc can be mplemented n mxed Nash equlbrum. Keywords: Implementaton, Maskn monotoncty, pure and mxed Nash equlbrum, set-monotoncty, socal choce correspondence. JEL Class caton Numbers: C72; D71. We thank Olver Terceux, three referees and the assocate edtor for nsghtful comments on the paper. y Department of Economcs, Unversty of Melbourne, Parkvlle, VIC 3010, Australa. cmez@unmelb.edu.au z Department of Economcs, Unversty of Lecester, Lecester LE1 7RH, Unted Kngdom. lr78@le.ac.uk 1

3 1 Introducton Ths paper studes the problem of mplementaton n mxed Nash equlbrum. Accordng to our de nton, a mechansm mplements an ordnal socal choce correspondence f n mxed Nash equlbrum f, at any preference pro le, the set of all (pure and mxed) equlbrum outcomes corresponds to the set of f-optmal alternatves for all cardnal representatons of the preference pro le. Crucally, and unlke the classcal de nton of mplementaton, ths de nton of mplementaton does not gve a predomnant role to pure equlbra: an f-optmal alternatve does not have to be the outcome of a pure Nash equlbrum. At the same tme, we mantan an entrely ordnal approach. We assume that a socal choce correspondence f maps pro les of preference orderngs over alternatves nto subsets of alternatves (not lotteres) and we requre that a gven mechansm mplements f rrespectve of whch cardnal representaton s chosen. Most of the exstng lterature on Nash mplementaton does not consder equlbra n mxed strateges (see Jackson, 2001, and Maskn and Sjöström, 2002, for excellent surveys). 1 Perhaps, the emphass on pure equlbra expresses a dscomfort wth the classcal vew of mxng as delberate randomzatons on the part of players. However, t s now accepted that even f players do not randomze but choose de nte actons, a mxed strategy may be vewed as a representaton of the other players uncertanty about a player s choce (e.g., see Aumann and Brandenburger, 1995). Moreover, almost all mxed equlbra can be vewed as pure Bayesan equlbra of nearby games of ncomplete nformaton, n whch players are uncertan about the exact pro le of preferences, as rst suggested n the semnal work of Harsany (1973). Ths vew acknowledges that games wth commonly known preferences are an dealzaton, a lmt of near-complete nformaton games. Ths nterpretaton s partcularly mportant for the theory of mplementaton n Nash equlbrum, whereby the assumpton of common knowledge of preferences, especally on large domans, s at best a smplfyng assumpton. Furthermore, recent evdence n the expermental lterature suggests that equlbra n mxed strateges are good predctors of behavor n some classes of games e.g., coordnaton games and chcken games (see chapters 3 and 7 of Camerer, 2003). Snce, for some preference pro les, a mechansm can nduce one of those games, payng attenton to mxed equlbra s mportant f we want to descrbe or predct players behavor. Whle we nd no compellng reasons to 1 Two notable exceptons are Maskn (1999) for Nash mplementaton and Serrano and Vohra (2010) for Bayesan mplementaton. These authors do consder mxed equlbra, but stll requre each f-optmal alternatve to be the outcome of a pure equlbrum; pure equlbra are gven a specal status. Whle Maskn (1999) shows that elmnatng unwanted mxed strategy equlbra mposes no addtonal restrcton to Nash mplementaton, Serrano and Vohra (2010) show that sgn cant addtonal restrctons are requred to mplement socal choce correspondences n Bayesan equlbrum. 2

4 gve pure Nash equlbra a specal status, we follow an ordnal approach because we beleve t mposes a welcome degree of robustness on socety s preferences and the mechansm adopted. Thus, we requre that the set of f-optmal outcomes only depends on players ordnal preferences, and that the mechansm adopted mplements f for all possble cardnal representatons of those ordnal preferences. Our de nton of mxed Nash mplementaton yelds novel nsghts. We demonstrate that the condton of Maskn monotoncty s not necessary for full mplementaton n mxed Nash equlbrum. Intutvely, consder a pro le of preferences and an alternatve, say a, that s f-optmal at that pro le of preferences. Accordng to Maskn s de nton of mplementaton, there must exst a pure Nash equlbrum wth equlbrum outcome a. Thus, any alternatve a player can obtan by unlateral devatons must be less preferred than a. Now, f we move to another pro le of preferences where a does not fall down n the players rankng, then a remans an equlbrum outcome and must be f-optmal at that new pro le of preferences. Ths s the ntuton behnd the necessty of Maskn monotoncty for Nash mplementaton. Unlke Maskn s de nton of mplementaton, our de nton does not requre a to be a pure equlbrum outcome. So, suppose that there exsts a mxed equlbrum wth a as an equlbrum outcome. 2 The key observaton to make s that the mxed equlbrum nduces a lottery over optmal alternatves. Thus, when we move to another pro le of preferences where a does not fall down n the players rankng, the orgnal pro le of strateges does not have to be an equlbrum at the new state. In fact, we show that a much weaker condton, set-monotoncty, s necessary for mplementaton n mxed Nash equlbrum. Set-monotoncty states that the set f() of optmal alternatves at state s ncluded n the set f( 0 ) of optmal alternatves at state 0 whenever, for all players, ether all alternatves n f() are top-ranked at state 0, or the weak and strct lower contour sets at state of all alternatves n f() are ncluded n ther respectve weak and strct lower contour sets at state 0. Moreover, we show that set-monotoncty and noveto power are su cent for mplementaton of socal choce correspondences n envronments wth at least three players. To substantate our clam that set-monotoncty s a substantally weaker requrement than Maskn monotoncty, we show that the strong Pareto and the strong core correspondences are set-monotonc on the doman of sngle-top preferences, whle they are not Maskn monotonc. Smlarly, on the doman of strct preferences, the top-cycle correspondence s set-monotonc, but not Maskn monotonc. The next secton llustrates our approach wth an example. Secton 3 de nes mxed Nash mplementaton. Secton 4 de nes set-monotoncty and shows that t s necessary for mxed 2 More precsely, let be the mxed Nash equlbrum and P ;g the dstrbuton over alternatves nduced by the strategy pro le and the allocaton rule g. Then a belongs to the support of P ;g. 3

5 Nash mplementaton. Secton 5 proves that set-monotoncty and no-veto power are su cent wth at least three players. Secton 6 bulds a brdge between our approach and the standard approach wth ordnal preferences. Secton 7 concludes by applyng our results to some well known socal choce correspondences. 2 A Smple Example Example 1 There are two players, 1 and 2, two states of the world, and 0, and four alternatves, a, b, c, and d. Players have state-dependent preferences represented n the table below. For nstance, player 1 ranks b rst and a second n state, whle a s ranked rst and b last n state 0. Preferences are strct b c a c a a d d c b c a d d b b The desgner ams to mplement the socal choce correspondence f, wth f() = fag and f( 0 ) = fa; b; c; dg. We say that alternatve x s f-optmal at state f x 2 f(). We rst argue that the socal choce correspondence f s not mplementable n the sense of Maskn (1999). Maskn s de nton of Nash mplementaton requres that for each f-optmal alternatve at a gven state, there exsts a pure Nash equlbrum (of the game nduced by the mechansm) correspondng to that alternatve. So, for nstance, at state 0, there must exst a pure Nash equlbrum wth b as equlbrum outcome. Maskn requres, furthermore, that no such equlbrum must exst at state. However, f there exsts a pure equlbrum wth b as equlbrum outcome at state 0, then b wll also be an equlbrum outcome at state, snce b moves up n every players rankng when gong from state 0 to state. Thus, the correspondence f s not mplementable n the sense of Maskn. In other words, the socal choce correspondence f volates Maskn monotoncty, a necessary condton for mplementaton n the sense of Maskn. In contrast wth Maskn, we do not requre that for each f-optmal alternatve at a gven state, there exsts a pure Nash equlbrum correspondng to that alternatve. We requre nstead that the set of f-optmal alternatves concdes wth the set of mxed Nash equlbrum outcomes. So, at state 0, there must exst a mxed Nash equlbrum wth b correspondng to an acton pro le n the support of the equlbrum. Wth our de nton of mplementaton, the correspondence f s mplementable. To see ths, consder the mechansm where each player has 4

6 two messages m 1 and m 2, and the allocaton rule s represented n the table below. (Player 1 s the row player.) For example, f both players announce m 1, the chosen alternatve s a. m 1 m 2 m 1 a b m 2 d c At state, (m 1 ; m 1 ) s the unque Nash equlbrum, wth outcome a. At state 0, both (m 1 ; m 1 ) and (m 2 ; m 2 ) are pure Nash equlbra, wth outcomes a and c. Moreover, there exsts a mxed Nash equlbrum that puts strctly postve probablty on each acton pro le (snce preferences are strct), hence on each outcome. 3 Therefore, f s mplementable n mxed Nash equlbrum regardless of the cardnal representaton chosen for the two players, although t s not mplementable n the sense of Maskn. We conclude ths secton wth an mportant observaton. Alternatve d s f-optmal at state 0, and t moves down n player 1 s rankng when movng from 0 to. Ths preference reversal guarantees the set-monotoncty of the correspondence f, whch, as we shall see, s a necessary condton for mplementaton n mxed Nash equlbrum. 3 Prelmnares An envronment s a trplet hn; X; where N := f1; : : : ; ng s a set of n players, X a nte set of alternatves, and a nte set of states of the world. 4 Assocated wth each state s a preference pro le < := (< 1; : : : ; < n), where < s player s preference relaton over X at state. The asymmetrc and symmetrc parts of < are denoted and, respectvely. We denote wth L (x; ) := fy 2 X : x < yg player s lower contour set of x at state, and SL (x; ) := fy 2 X : x yg the strct lower contour set. For any (; ) n N and Y X, de ne max Y as fx 2 Y : x < y for all y 2 Y g. We assume that any preference relaton < s representable by a utlty functon u (; ) : X! R, and that each player s an expected utlty maxmzer. We denote wth U all possble cardnal representatons u (; ) of < at state, and let U := 2N U. the set of A socal choce correspondence f :! 2 X n f;g assocates wth each state of the world, a non-empty subset of alternatves f() X. Two classc condtons for Nash mplementaton are 3 Note that at state, a s the unque ratonalzable outcome, whle all outcomes are ratonalzable at state 0, so that f s mplementable n ratonalzable outcomes. See Bergemann, Morrs and Terceux (2011) for the study of socal choce functons mplementable n ratonalzable outcomes. 4 In some applcatons (e.g., exchange economes) t s natural to work wth more general outcome spaces. As n Abreu and Sen (1991), our results extend to the case n whch X s a separable metrc space and the socal choce correspondence maps nto a countable, dense subset of X. 5

7 Maskn monotoncty and no-veto power. A socal choce correspondence f s Maskn monotonc f for all (x; ; 0 ) n X wth x 2 f(), we have x 2 f( 0 ) whenever L (x; ) L (x; 0 ) for all 2 N. Maskn monotoncty s a necessary condton for Nash mplementaton (à la Maskn). A socal choce correspondence f sats es no-veto power f for all 2, we have x 2 f() whenever x 2 max X for all 2 N N, wth N havng at least n 1 elements. Maskn monotoncty and no-veto power are su cent condtons for Nash mplementaton (à la Maskn) when there are at least three players. For any subset Y X, let (Y ) be the set of all probablty measures over Y. We vew (Y ) as a subset of (X) wth the property that P (x) = 0 for all x 2 X n Y f P 2 (Y ). A mechansm (or game form) s a par h(m ) 2N ; g wth M the set of messages of player, and g : 2N M! (X) the allocaton rule. Let M := j2n M j and M := j2nnfg M j, wth m and m generc elements. A mechansm h(m ) 2N ; g, a state and a pro le of cardnal representatons (u (; )) 2N of (< ) 2N nduce a strategc-form game as follows. There s a set N of n players. The set of pure actons of player s M, and player s expected payo when he plays m and hs opponents play m s U (g(m ; m ); ) := X g(m ; m )(x)u (x; ); x2x where g(m ; m )(x) s the probablty that x s chosen by the mechansm when the pro le of messages (m ; m ) s announced. The nduced strategc-form game s thus G(; u) := hn; (M ; U (g(); )) 2N. Let be a pro le of mxed strateges. We denote wth P ;g the probablty dstrbuton over alternatves n X nduced by the allocaton rule g and the pro le of mxed strateges. 5 De nton 1 The mechansm h(m ) 2N ; g mplements the socal choce correspondence f n mxed Nash equlbrum f for all 2, for all cardnal representatons u(; ) 2 U of <, the followng two condtons hold: () For each x 2 f(), there exsts a Nash equlbrum of G(; u) such that x s n the support of P ;g, and () f s a Nash equlbrum of G(; u), then the support of P ;g s ncluded n f(). It s mportant to contrast our de nton of mplementaton n mxed Nash equlbrum wth Maskn (1999) de nton of Nash mplementaton. Frst, part () of Maskn s de nton requres that for each x 2 f(), there exsts a pure Nash equlbrum m of G(; u) wth equlbrum outcome x, whle part () of hs de nton s dentcal to ours. In contrast wth Maskn, we allow 5 Formally, the probablty P ;g (x) of x 2 X s P m2m (m)g(m)(x) f M s countable. If M s uncountable, a smlar expresson apples. 6

8 for mxed strategy Nash equlbra n part () and, thus, restore a natural symmetry between parts () and (). Yet, our de nton respects the sprt of full mplementaton n that only optmal outcomes can be observed by the desgner as equlbrum outcomes. Second, as n Maskn, our concept of mplementaton s ordnal as all equlbrum outcomes have to be optmal, regardless of the cardnal representaton chosen. Also, our approach parallels the approach of Gbbard (1977). Gbbard characterzes the set of strategy-proof probablstc socal choce functons (.e., mappngs from pro les of preferences to lotteres over outcomes). Importantly to us, Gbbard requres each player to have an ncentve to truthfully reveal hs preference, regardless of the cardnal representaton chosen to evaluate lotteres (and announcements of others). 6 Thrd, we allow the desgner to use mechansms that randomze among the optmal alternatves n equlbrum. Ths s a natural assumpton gven that players can use mxed strateges. Although a random mechansm ntroduces uncertanty about the alternatve to be chosen, the concept of mxed Nash equlbrum already encapsulates the dea that players are uncertan about the messages sent to the desgner and, consequently, about the alternatve to be chosen. 7 Fnally, from our de nton of mxed Nash mplementaton, t s mmedate to see that f a socal choce correspondence s Nash mplementable (.e., à la Maskn), then t s mplementable n mxed Nash equlbrum. The converse s false, as shown by Example 1. The goal of ths paper s to characterze the socal choce correspondences mplementable n mxed Nash equlbrum. The next secton provdes a necessary condton. 4 A Necessary Condton In ths secton, we ntroduce a new condton, called set-monotoncty, whch we show to be necessary for the mplementaton of socal choce correspondences n mxed Nash equlbrum. De nton 2 A socal choce correspondence f s set-monotonc f for all pars (; 0 ) 2, we have f() f( 0 ) whenever for all 2 N, one of the followng two condtons holds: ether (1) f() max 0 X or (2) for all x 2 f(), () L (x; ) L (x; 0 ) and () SL (x; ) 6 See also Barberà et al. (1998) and Abreu and Sen (1991) for further dscussons of the ordnal approach. 7 In the lterature on (exact) Nash mplementaton, Benoît and Ok (2008) and Bochet (2007) have studed mechansms n whch randomzaton by the desgner can only occur out of equlbrum; unlke us, they do not attempt to rule out mxed strategy equlbra wth undesrable outcomes. (See also Vartanen, 2007, for the use of random mechansms n the mplementaton of correspondences n (pure) subgame perfect equlbrum.) Our approach also d ers from the use of random mechansms n the lterature on vrtual mplementaton (e.g., see Matsushma, 1998, and Abreu and Sen, 1991), whch heavly explots the possblty of selectng undesrable alternatves wth postve probablty n equlbrum. 7

9 SL (x; 0 ). 8 Set-monotoncty s a weakenng of Maskn monotoncty. As the state changes from to 0, t restrcts the set f ( 0 ) to contan the set f () only when, for all players, all alternatves n f () ether are top ranked at 0 or they do not move down n the weak and strct rankngs. Maskn monotoncty, on the contrary, restrcts f ( 0 ) to nclude each sngle alternatve x 2 f () whch does not move down n any player s weak rankng when movng from to 0. Theorem 1 If the socal choce correspondence f s mplementable n mxed Nash equlbrum, then t sats es set-monotoncty. Proof We begn by provng the followng clam. Clam C. Suppose L (x; ) L (x; 0 ) and SL (x; ) SL (x; 0 ) for all x 2 f(): Then, gven any cardnal representaton u (; ) of <, there exsts a cardnal representaton u (; 0 ) of < 0 such that u (x; 0 ) u (x; ) for all x 2 X and u (x; 0 ) = u (x; ) for all x 2 f(). Proof of Clam C. To prove our clam, consder any par (x; x 0 ) 2 f() f() wth x < x 0. Snce L (^x; ) L (^x; 0 ) for all ^x 2 f(), we have that x x 0 mples x 0 x 0 and x x 0 mples x 0 x 0. Hence, we can assocate wth each alternatve n f() the same utlty at 0 as at. Now, x an alternatve x 2 f() and consder y 2 L (x; ). Snce L (x; ) L (x; 0 ), we must have u (y; 0 ) u (x; 0 ) = u (x; ). If x y, then we can choose u (y; 0 ) u (y; ) = u (x; ). If x choose u (y; 0 ) n the open set ( y, then we must have x 0 y snce SL (x; ) SL (x; 0 ); we can therefore 1; u (y; )) and stll represent < 0 by u (; 0 ). Fnally, f y =2 [ x2f() L (x; ), we have that u (y; ) > u (x; ) for all x 2 f(). If y 2 L (x; 0 ) for some x 2 f(), then we can set u (y; 0 ) u (x; 0 ) = u (x; ) max x 0 2f() u (x 0 ; ) < u (y; ). If y =2 [ x2f() L (x; 0 ), then we can choose u (y; 0 ) n the open set (max x 0 2f() u (x 0 ; ); u (y; )). Ths concludes the proof of our clam. The proof proceeds by contradcton on the contrapostve. Assume that the socal choce correspondence f does not satsfy set-monotoncty and yet s mplementable n mxed Nash equlbrum by the mechansm hm; g. Snce f does not satsfy set-monotoncty, there exst x ; ; and 0 such that x 2 f()nf( 0 ), whle L (x; ) L (x; 0 ) and ether SL (x; ) SL (x; 0 ) for all x 2 f() or f() max 0 X, for all 2 N. Let N := f 2 N : f() 6 max 0 Xg. Snce f s mplementable and x 2 f(), for any cardnal representaton u(; ) of <, there exsts an equlbrum of the game G(; u) wth x n the support of P ;g. Furthermore, snce 8 Alternatvely, a socal choce correspondence f s set-monotonc f x 2 f() n f( 0 ) mples that there exsts a trple (x; x 0 ; y) n f() f() X and a player 2 N such that x 0 y 0 x, or (2) x y and y <0 x. =2 max 0 X and ether (1) x < y and 8

10 x =2 f( 0 ), for all cardnal representatons u(; 0 ) of < 0, for all equlbra of G( 0 ; u), x does not belong to the support of P ;g. In partcular, ths mples that s not an equlbrum at 0 for all cardnal representatons u(; 0 ). Thus, assumng that M s countable, there exst a player, a message m n the support of, and a message m 0 such that: 9 X m [U (g(m ; m ); ) U (g(m 0 ; m ); )] (m ) 0, and X m [U (g(m ; m ); 0 ) U (g(m 0 ; m ); 0 )] (m ) < 0: It follows that X m [U (g(m ; m ); ) U (g(m ; m ); 0 )] (m ) > (1) X m [U (g(m 0 ; m ); ) U (g(m 0 ; m ); 0 )] (m ) Frst, assume that 2 N. By Clam C, we can construct cardnal representatons such that u (x; 0 ) u (x; ) for all x 2 X and u (x; ) = u (x; 0 ) for all x 2 f(). Snce f s mplementable, we have that the support of P ;g s ncluded n f(). Therefore, U (g(m ; m ); ) = U (g(m ; m ); 0 ) for all m n the support of. Hence, the left-hand sde of nequalty (1) s zero. Furthermore, we have that U (g(m 0 ; m ); ) U (g(m 0 ; m ); 0 ) for all m. Hence, the rght-hand sde of (1) s non-negatve, a contradcton. Second, assume 2 N n N. Snce f() max 0 X, t follows that for all x 2 f(), u (x; 0 ) u (x 0 ; 0 ) for all x 0 2 X, regardless of the cardnalzaton u (; 0 ) of < 0. Consequently, player has no pro table devaton at state 0, a contradcton. Ths completes the proof. Note rst that Theorem 1 remans vald f we restrct ourselves to determnstc mechansms, so that set-monotoncty s a necessary condton for mplementaton n mxed Nash equlbrum, regardless of whether we consder determnstc or random mechansms. Second, t s easy to verfy that set-monotoncty s also a necessary condton for mplementaton f we requre the Nash equlbra to be n pure strateges, but allow random mechansms. Thrd, whle we have restrcted attenton to von Neumann-Morgenstern preferences, the condton of setmonotoncty remans necessary f we consder larger classes of preferences that nclude the von Neumann-Morgenstern preferences. Ths s because we follow an ordnal approach and requre that f be mplemented by all admssble preference representatons. Fourth, set-monotoncty s related to the concepts of almost monotoncty (Sanver, 2006) and quasmonotoncty (Cabrales and Serrano, 2011). Quasmonotoncty and almost monotoncty restrct f when a sngle 9 If the mechansm s uncountable, a smlar argument holds wth approprate measurablty condtons. 9

11 alternatve n f () moves up n the rankngs of all players. The socal choce correspondence f s quasmonotonc f for all pars (; 0 ) 2 and x 2 f(), we have x 2 f( 0 ) whenever for all 2 N, SL (x; ) SL (x; 0 ). The socal choce correspondence f s almost monotonc f for all pars (; 0 ) 2 and x 2 f(), we have x 2 f( 0 ) whenever for all 2 N, the followng two condtons hold: () L (x; ) L (x; 0 ) and () SL (x; ) SL (x; 0 ). On the unrestrcted doman of preferences, set-monotoncty s nether weaker nor stronger than quasmonotoncty or almost monotoncty. Note, however, that for socal choce functons, set monotoncty s weaker than quasmonotoncty and almost monotoncty. For an example of a set-monotonc socal choce correspondence that s nether quasmonotonc nor almost monotonc, see Example 1. Conversely, for an example of a quasmonotonc (and almost monotonc) socal choce correspondence that s not set-monotonc, see Example 2. Example 2 There are three players, 1, 2 and 3, two states of the world, and 0, and three alternatves a, b and c. Preferences are represented n the table below b a c b a b c c b a c a a c b a c b The socal choce correspondence s f() = fag and f( 0 ) = fbg. It s not set-monotonc snce L (a; ) L (a; 0 ) for all 2 f1; 2; 3g, SL (a; ) SL (a; 0 ) for all 2 f1; 3g and a 2 max 0 2 fa; b; cg and yet a =2 f( 0 ). However, t s quasmonotonc and almost monotonc. Yet, f max X s a sngleton for each 2 N, for each 2, then set-monotoncty s weaker than quasmonotoncty and almost monotoncty. Indeed, whenever max X s a sngleton for each 2 N, for each 2, the requrement that f() be nested n f( 0 ) n the de nton of set-monotoncty s equvalent to: For all 2 N, for all x 2 f(), () L (x; ) L (x; 0 ) and () SL (x; ) SL (x; 0 ). We refer to ths doman of preferences as the sngle-top preferences. 10 As we shall see n Secton 7, mportant correspondences, lke the strong Pareto correspondence, the strong core correspondence and the top-cycle correspondence are set-monotonc on the doman of sngle-top preferences, whle they fal to be Maskn monotonc. Lastly, the condton of set-monotoncty s related to the condton of extended monotoncty of Bochet and Manquet (2010). They consder the problem of vrtual mplementaton n pure-strategy Nash equlbrum, wth the addtonal restrcton that the approxmate lottery correspondence to be exactly mplemented has a restrcted support h. Roughly, t means that 10 Note that the doman of strct preferences s a subset of the doman of sngle-top preferences. 10

12 for all 2, for all x 2 f(), the lottery that s "-close to 1 x must have support h(; x). In partcular, f we mpose that h(; x) = f() for all x 2 f(), then the approxmate lottery correspondence can assgn postve probablty only to f-optmal alternatves, and vrtual mplementaton comes to resemble, to some extent, mxed strategy mplementaton as we de ne t. Under the restrcton of strct preferences, they show that extended monotoncty wth respect to the admssble support h s necessary and su cent for vrtual mplementaton wth support h. On the doman of strct preferences, the condton of set-monotoncty mples the condton of extended monotoncty wth respect to the support f, and thus vrtual mplementaton wth support f. In general, however, the two condtons are qute d erent, because unlke set-monotoncty, extended monotoncty does not requre nestedness of the strct lower contour sets. For nstance, when f s a functon (.e., sngle valued) extended monotoncty wth respect to f s equvalent to Maskn monotoncty, whle set-monotoncty s less restrctve. 5 A Su cent Condton We now show that n any envronment wth at least three players, set-monotoncty and no veto-power are su cent for mplementaton n mxed Nash equlbrum. Theorem 2 Let hn; X; be an envronment wth n 3. If the socal choce correspondence f s set-monotonc and sats es no-veto power, then t s mplementable n mxed Nash equlbrum. Proof Let U = [ 2 U and de ne the set U as f(; u) 2 U : u 2 U g. Consder the followng mechansm hm; g. For each player 2 N, the message space M s U f : : X 2! Xg X Z ++. In words, each player announces a state of the world and a pro le of cardnal representatons consstent wth that state of the world, a functon from alternatves and pars of states nto alternatves, an alternatve, and a strctly postve nteger. A typcal message m for player s (( ; u ); ; x ; z ). (Note that we denote any nteger z n bold.) Let M := 2N M wth typcal element m. Let ff 1 (); : : : ; f K ()g = f() be the set of f-optmal alternatves at state ; note that K = jf()j. For any 2, for any u 2 U, let 1 > " u > 0 be such that for all 2 N, for all pars (x; y) 2 X X wth x y, we have u (x; ) (1 " u )u (y; ) + " u max w2x u (w; ) + " u (jxj 1)(max w2x u (w; ) mn w2x u (w; )). Snce X and N are nte, such an " u exsts. Let 1[x] 2 (X) be the lottery that puts probablty one on outcome x 2 X. The allocaton rule g s de ned as follows: Rule 1: If m = ((; u); ; x; 1) for all 2 N (.e., all agents make the same announcement m ) and (f k (); ; ) = f k () for all f k () 2 f (), then g(m) s the unform lottery over 11

13 alternatves n f(); that s, g(m) = 1 X K 1 [f K k ()] : k=1 Rule 2: If there exsts j 2 N such that m = ((; u); ; x; 1) for all 2 N n fjg, wth (f k (); ; ) = f k () for all f k () 2 f () ; and m j = (( j ; u j ); j ; x j ; z j ) 6= m, then g(m) s the lottery: 1 K XK k=1 k (m)(1 " k (m))1 j (f k (); ; j ) + k (m)" k (m)1 x j + (1 k (m)) 1 [f k ()] ; wth 8 < f j (f k (); ; j ) 2 L j (f k (); ) k (m) = : 0 f j (f k (); ; j ) 62 L j (f k (); ) for 1 > > 0, and 8 < " u f j (f k 0(); ; j ) 2 SL j (f k 0(); ) for some k 0 2 1; :::; K " k (m) = : 0 f j (f k 0(); ; j ) 62 SL j (f k 0(); ) for all k 0 2 : 1; :::; K That s, suppose all players but player j send the same message ((; u); ; x; 1) wth (f k (); ; ) = f k () for all k 2 f1; : : : ; K g. Let m j = (( j ; u j ); j ; x j ; z j ) be the message sent by player j. Consder the set F Lj () of all outcomes f k () such that j (f k (); ; j ) 2 L j (f k (); ). Frst, suppose that for some k 0 2 1; :::; K, j (f k 0(); ; j ) selects an alternatve x n player j strct lower-contour set SL j (f k 0(); ) of f k 0() at state. Then, the desgner mod es the unform lottery by replacng each outcome n the set F Lj () wth the lottery that attaches probablty (1 " u ) to j (f k (); ; j ), probablty " u to x j, and probablty (1 ) to f k (). Second, suppose that for all k 0 2 1; :::; K, j (f k (); ; j ) selects an alternatve x that s not n SL j (f k (); ). Then, the desgner replaces the outcomes n the set f Lj () wth wth the lottery that attaches probablty to x and probablty (1 ) to f k (). Rule 3: If nether rule 1 nor rule 2 apples, then g ( ; u ); ; x ; z = x ; wth a 2N player announcng the hghest nteger z. (If more than one player selects the hghest nteger, then g randomzes unformly among ther selected x.) Fx a state and a cardnal representaton u 2 U of < for each player. Let u be the vector of cardnal representatons. We dvde the rest of the proof n several steps. Step 1. We rst show that for any x 2 f( ), there exsts a Nash equlbrum of G( ; u ) such that x belongs to the support of P ;g. Consder a pro le of strateges such that = (( ; u ); ; x; 1) for all 2 N, so that rule 1 apples. The (pure strategy) pro le s a 12

14 Nash equlbrum at state. By devatng, each player can trgger rule 2, but none of these possble devatons are pro table. Clearly, f player s devaton s such that (f k 0( ); ; ) =2 SL (f k 0( ); ) for all k 0 2 f1; : : : ; K g, player s devaton cannot be pro table. So, let us assume that player s devaton s such that (f k 0( ); ; ) 2 SL (f k 0( ); ) for some k 0. It follows that " k = " u > 0 for all k 2 f1; : : : ; K g. There are three cases to consder. Frst, for all k 2 f1; : : : ; K g such that (f k ( ); ; ) =2 L (f k ( ); ), we have k = 0 and, thus, there s no shft n probablty from f k ( ). Second, for all k 2 f1; : : : ; K g such that (f k ( ); ; ) f k ( ), we have k = and, thus, there s a probablty shft from f k ( ) to a lottery wth mass (1 " u ) on the alternatve (f k ( ); ; ) nd erent to f k ( ) and mass " u on x. In partcular, f x f k ( ), the shft n probabltes leads to a lottery strctly preferred over f k ( ). Thrd, for all k 2 f1; : : : ; K g such that f k ( ) (f k ( ); ; ), we agan have k = and, thus, there s a probablty shft from f k ( ) to a lottery wth mass (1 " u ) on the alternatve (f k ( ); ; ) strctly less preferred than f k ( ) and mass " u on x. By de nton of " u, ths shft of probabltes leads to a lottery worse than f k ( ). It s mportant to note that ths last case exsts snce we have assumed that (f k 0( ); ; ) 2 SL (f k 0( ); ) for some k 0. Thus, the best devaton for player conssts n choosng such that (f k ( ); ; ) 2 SL (f k ( ); ) for a unque k, (f k ( ); ; ) f k ( ) for all other k, and x 2 max X. The maxmal d erence n payo s between playng and devatng s therefore (up to the multplcatve term =K ): u (f k ( ); ) (1 " u )u ( (f k ( ); ; ); ) " u u (x ; )+ X u (f k ( ); ) u (x ; ) ; k6=k " u whch s postve by constructon of " u. Therefore, player has no pro table devaton. Lastly, under, the support of P ;g s f( ). Hence, for any x 2 f( ), there exsts an equlbrum that mplements x. Step 2. Conversely, we need to show that f s a mxed Nash equlbrum of G( ; u ), then the support of P ;g s ncluded n f( ). Let m be a message pro le and denote wth g O (m) the set of alternatves that occur wth strctly postve probablty when m s played: g O (m) = fx 2 X : g(m)(x) > 0g. Let us partton the set of messages M nto three subsets correspondng to the three allocaton rules. Frst, let R 1 be the set of message pro les such that rule 1 apples,.e., R 1 = fm : m j = ((; u); ; x; 1) for all j 2 N, wth (f k (); ; ) = f k () for all f k () 2 f ()g. Second, f all agents j 6= send some message m j = ((; u); ; x; 1) wth (f k (); ; ) = f k () for all f k () 2 f (), whle agent sends a d erent message m = ( ; u ); ; x ; z, then rule 2 apples and agent s the only agent d erentatng hs message. Let R 2 be the set of these message pro les and de ne R 2 = [ 2N R 2. Thrd, let R 3 be the set of message pro les such that rule 3 apples (.e., R 3 s the complement of R 1 [ R 2 n M). 13 (2)

15 Consder an equlbrum of G( ; u ) and let M be the set of message pro les that occur wth postve probablty under. (M g O (m ) f( ) for all m 2 M := 2N M. s the support of.) We need to show that Step 3. For any player 2 N, for all m = (( ; u ); ; x ; z ) 2 M ; de ne the (devaton) message m D (m ) = (( ; u ); D ; x D ; z D ), where: 1) D d ers from n at most the alternatves assocated wth elements (f k (); ; ) for all 2, for all k 2 f1; : : : ; K g; that s, we can only have D (f k (); ; ) 6= (f k (); ; ) for some k 2 f1; : : : ; K g and some 2, whle we have D (x; 0 ; 00 ) = (x; 0 ; 00 ), otherwse, 2) x D 2 max X, and 3) z D > z and for 1 > 0; the nteger z D s chosen strctly larger than the ntegers z j selected by all the other players j 6= n all messages m 2 M, except possbly a set of message pro les M M havng probablty of beng sent less than. (Note that can be chosen arbtrarly small, but not necessarly zero because other players may randomze over an n nte number of messages.) Consder the followng devaton D for player from the equlbrum strategy : 8 < D (m ) f m = m D (m ) for some m 2 M (m ) = : : 0 otherwse Step 4. Frst, note that under ( D ; ), the set of messages sent s a subset of R 2[R 3 : ether rule 2 apples and all players but player send the same message or rule 3 apples. Second, whenever rule 3 apples, player gets hs preferred alternatve at state wth arbtrarly hgh probablty (1 ). Thrd, suppose that under, there exsts m 2 R j 2 wth j 6=. Under ( D ; ), wth the same probablty that m s played, (m D (m ); m ) 2 R 3 s played (rule 3 apples) and wth probablty at most, the lottery g((m D (m ); m )) under (m D (m ); m ) mght be less preferred by player than the lottery g(m ). (Wth probablty 1, g((m D (m ); m )) = max X.) Yet, snce can be made arbtrarly small and utltes are bounded, the loss can be made arbtrarly small. Consequently, by settng D (f k (); ; ) < (f k (); ; ) for all and all k 2 f1; : : : ; K g, player can guarantee hmself an arbtrarly small, worst-case loss of u, n the event that m 2 [ j6= R j 2 under. Step 5. Let us now suppose that there exsts (m ; m ) 2 R 1 ; that s, for all j 6= ; m j = m = ((; u); ; x; 1). Assume that there exsts k 0 2 f1; : : : ; K g such that f k 0() =2 max X. In the event the message sent by all others s m j, player strctly gans from the devaton f D (f k 0(); ; ) 2 L (f k 0(); ) and ether (1) D (f k 0(); ; ) f k 0() or (2) D (f k 0(); ; ) 2 SL (f k 0(); ), D (f k 0(); ; ) < f k 0() (snce " u > 0 and f k 0() =2 max X). Now assume that, n addton, there exsts k 00 2 f1; : : : ; K g such that f k 00() 2 max X. Player strctly gans from the devaton f (3) D (f k 00(); ; ) 2 SL (f k 00(); ) and D (f k 00(); ; ) snce the devaton shfts probablty mass from f k 0() =2 max player would not gan from the devaton f f() max 14 X.) X to x D 2 max f k 00(), X. (Note that

16 Snce the expected gan n ths event can be made greater than u by approprately choosng, (1) and (2) cannot hold for any player and any k such that f k 0() =2 max X. Smlarly, (3) cannot hold for any player and any k 00 such that f k 00() 2 max X unless f() max X. It follows that for to be an equlbrum, we must have (1) L (f k (); ) L (f k (); ) for all k and (2) ether SL (f k (); ) SL (f k (); ) for all k or f() max X. Therefore, by set-monotoncty of f, we must have f() f( ). Ths shows that g(m ; m ) f( ) for all (m ; m ) 2 R 1 : Step 6. Let us now suppose that there exsts (m ; m ) 2 R 2; that s, for all j 6= ; m j = ((; u); ; x; 1) 6= m : In ths case, any player j 6= strctly gans from the devaton D j whenever z D s the largest nteger, whch occurs wth a probablty of at least 1, unless g(m ; m ) max j X. Snce can be made arbtrarly small, t must be g(m ; m ) max j X for all j 6=. Therefore, by no-veto power, t must be g(m ; m ) f( ) for all (m ; m ) 2 R2: Step 7. It only remans to consder messages (m ; m ) 2 R 3 : For such messages the argument s analogous to messages n R 2: For no player to be able to pro t from the devaton D, t must be g(m ; m ) max X for all 2 N. Therefore, the condton of no-veto power mples g(m ; m ) f( ) for all (m ; m ) 2 R 3. Some remarks are n order. Frst, the mechansm constructed n the proof s nspred by the mechansm n the appendx of Maskn (1999), but ours s a random mechansm. As we have already explaned, we beleve ths s natural gven that we consder the problem of mplementaton n mxed Nash equlbrum. Second, the assumpton of von Neumann-Morgenstern utlty functons s not easly dspensed wth. For nstance, we mght just assume that preferences over lotteres are monotone n probabltes, so that shfts n probablty mass to strctly preferred alternatves yeld preferred lotteres. However, as n Abreu and Sen (1991, Secton 5), ths s not su cent, snce our mechansm nvolves addng and subtractng non-degenerate lotteres, thus creatng compound lotteres that are not comparable f we just assume that preferences are monotone n probablty. For a concrete example, suppose that there are three alternatves x, y, and z, that player at state prefers x to y to z and f() = fyg. Accordng to rule 1, f all players report (; ; y; 1) wth (y; ; ) = y, player faces the lottery (0; 1; 0). Now, accordng to rule 2, f player devates to (; ; x; 1) wth (y; ; ) = z, he faces the lottery ("; 1 ; (1 ")). The two lotteres are not comparable f we only mpose the axom of monotoncty n probabltes. Whether we can desgn an mplementng mechansm that only requres the axom of monotoncty n probabltes s an open ssue. Lastly, Theorem 2 strongly reles on the condton of set-monotoncty, a weakenng of Maskn monotoncty, whch s relatvely easy to check n applcatons. We have not tred 15

17 to look for necessary and su cent condtons for mxed Nash mplementaton. We suspect that such a characterzaton wll nvolve condtons that are hard to check n practce, as t s the case for Nash mplementaton à la Maskn (e.g., condton of Moore and Repullo, 1990, condton M of Sjöström, 1991, condton of Dutta and Sen, 1991, or essental monotoncty of Danlov, 1992). 6 Cardnal vs Ordnal: a Brdge Followng the suggeston of a referee, ths secton bulds a brdge between the ordnal approach adopted n ths paper and a cardnal approach. To do so, we adopt an alternatve de nton of mxed Nash mplementaton, proposed by the referee. Ths de nton, whch we call mxed* Nash mplementaton, replaces part () n De nton 1 wth: ( ) For each x 2 f(), there exsts a Nash equlbrum of G(; u) such that the support of P ;g s x. Ths alternatve de nton s ntermedate between our de nton of mxed Nash mplementaton and Maskn s de nton of Nash mplementaton. Wth mxed* Nash mplementaton, a necessary and almost su cent condton for mplementaton s cardnal monotoncty. De nton 3 A socal choce correspondence f s cardnally monotonc f for all trples (x; ; 0 ) 2 X wth x 2 f(), we have that x 2 f( 0 ) whenever for each player 2 N, there exsts a cardnal representaton u (; ) of < for all P 2 (X), u (x; ) X x2x and a cardnal representaton u (; 0 ) of < 0 such that P (y)u (y; ) =) u (x; 0 ) X y2x P (y)u (y; 0 ): (3) The necessty of cardnal monotoncty s clear. We now provde a sketch of the proof that cardnal monotoncty together wth no-veto power s su cent (wth at least three players). For smplcty, we consder socal choce functons. Fx a socal choce functon f :! X. De ne the extended state space U = [ 2 U, and let f ~ : U! X be the socal choce functon such that for any 2, for any u 2 U, f(u) ~ = f(). De ne the lower contour set L ~ (x; u) of x at u as fp 2 (X) : u (x) P y2x u (y)p (y)g. 11 Frst, we show that f f s cardnally monotonc and sats es no-veto power, then f ~ s Maskn monotonc and sats es no-veto power. Assume that f s cardnally monotonc and consder any two states of the world u and u 0 such that L ( f(u); ~ u) L ( f(u); ~ u 0 ) for each 2 N. We want to show that f(u) ~ = f(u ~ 0 ). Ths s clearly true f there exsts 2 such that u 2 U and u 0 2 U. So, assume that u 2 U and u 0 2 U 0 11 It s mportant that we de ne lower-contour sets n the space of lotteres, as opposed to the space of alternatves. 16

18 wth 6= 0. We thus have two cardnal representatons u(; ) = u 2 U and u(; 0 ) = u 0 2 U 0 such that u (f(u); ) X P (y)u (y; ) =) u (f(u); 0 ) X P (y)u (y; 0 ): y2x y2x Cardnal monotoncty then mples that ~ f(u(; )) = f() = f( 0 ) = ~ f(u(; 0 )) and, consequently, ~ f(u) = ~ f(u 0 ) (snce ~ f s ordnal). Thus, ~ f s Maskn monotonc. Second, snce ~ f clearly sats es no-veto power, we can apply Maskn s theorem and conclude that t s Nash mplementable. The nal step s to observe that f can be mplemented n mxed* Nash equlbrum by the mechansm used to mplement ~ f n Nash equlbrum. Ths s because, for each 2 and for each u 2 U, the set of equlbrum outcomes of the game nduced at u by the mechansm mplementng ~ f n Nash equlbrum (e.g., the mechansm n Sjöström and Maskn, 2002) concdes wth the sngleton f () : Several addtonal remarks are worth makng. Frst, mxed* Nash mplementaton s stronger than mxed Nash mplementaton, as the smple example n Secton 2 shows. In Example 1, the socal choce correspondence s not cardnally monotonc, and yet t s mxed Nash mplementable. Second, for socal choce functons, set-monotoncty and mxed Nash mplementaton are equvalent to cardnal monotoncty and mxed* Nash mplementaton. Thrd, and most mportant, an appealng feature of our concept of mxed Nash mplementaton s that t smply stpulates that the set of equlbrum outcomes s f() at each state of the world, and does not mpose any addtonal restrctons. Ths contrasts sharply wth Maskn s concept that requres the exstence of a pure equlbrum for each f-optmal alternatve at or wth mxed* Nash mplementaton that requres the exstence of equlbra that put probablty one on each f-optmal alternatve. To clarfy further the connecton between the two concepts, we present an alternatve characterzaton of set-monotoncty, whch explctly consders cardnal representatons and lotteres over alternatves, and s clearly a weaker condton than cardnal monotoncty. De nton 4 A socal choce correspondence f s cardnally set-monotonc f for all pars (; 0 ) 2, we have f() f( 0 ) whenever for each player 2 N, there exsts a cardnal representaton u (; ) of < and a cardnal representaton u (; 0 ) of < 0 P f() 2 (f()), for all P 2 (X), X x2x P f() (x)u (x; ) X x2x P (x)u (x; ) =) X x2x P f() (x)u (x; 0 ) X x2x such that for all P (x)u (x; 0 ): (4) Accordng to De nton 4, the set of optmal alternatves f() at state must be a subset of the set of optmal alternatves f( 0 ) at state 0, whenever there exst a cardnal representaton of preferences at and a cardnal representaton at 0 such that, for any lottery P f() wth 17

19 support n f(), the lower contour set of P f() for the cardnalzaton at state s nested n the lower contour set of P f() for the cardnalzaton at state 0, wth the lower contour sets de ned n the lottery space. In spte of the apparent d erences between cardnal set-monotoncty and set-monotoncty, the next proposton shows that the two condtons are equvalent. Proposton 1 A socal choce correspondence s set-monotonc f and only f t s cardnally set-monotonc. Proof (Only f ) Suppose that f s set-monotonc. Consder two states and 0 and for each player 2 N, two cardnal representatons u (; ) and u (; 0 ) such that Equaton (4) holds. We want to show that f() f( 0 ). To prove ths, we show that Equaton (4) mples that for each player 2 N, ether f() max 0 X or () L (x; ) L (x; 0 ) and () SL (x; ) SL (x; 0 ) for all x 2 f(). The concluson then follows from the set-monotoncty of f. Step 1: Equaton (4) clearly mples that () holds for all x 2 f(). To see ths, consder x k 2 f() and any y 2 L (x k ; ), so that u (x k ; ) u (y; ). Lettng P f() (x k ) = 1 and P (y) = 1, Equaton (4) mples that u (x k ; 0 ) u (y; 0 ),.e., y 2 L (x k ; 0 ), as requred. Step 2: Assume that f() 6 max 0 wth x k 0 X and consder any f k () =2 max 0 X. There exsts x k 2 X f k (). From step 1, t must be x k f k (). Contrary to (), suppose that there exsts y 2 X such that f k () y, but y < 0 f k (). Then there exsts p 2 (0; 1) such that u (f k (); ) pu (x k ; ) + (1 p)u (y; ): Equaton (4) mples that u (f k (); 0 ) pu (x k ; 0 ) + (1 p)u (y; 0 ), a contradcton. Therefore, t must be SL (x; ) SL (x; 0 ) for all x 2 f() such that x =2 max 0 X. Step 3: Contnue to assume that f() 6 max 0 X and consder any f k () 2 max 0 X. Then there exsts f k 0() 62 max 0 X. Snce f k () 0 f k 0(), by step 1 t s also f k () f k 0(). Contrary to (), suppose that there exsts y 2 X such that f k () y, but y 0 f k (). Then there exsts p 2 (0; 1) such that pu (f k 0(); ) + (1 p)u (f k (); ) pu (f k (); ) + (1 p)u (y; ). Equaton (4) then mples that u (f k 0(); 0 ) u (f k (); 0 ), a contradcton. Therefore, t must be SL (x; ) SL (x; 0 ) for all x 2 f() such that x 2 max 0 X. Step 4: Equaton (4) trvally holds whenever f() max 0 X. Ths completes the rst part of the proof. (If ) Suppose that f s cardnally set-monotonc. Consder two states and 0 such that for each player 2 N, ether f() max 0 X or () L (x; ) L (x; 0 ) and () SL (x; ) SL (x; 0 ) for all x 2 f(). We want to show that ths mples the exstence of two cardnal representatons u (; ) and u (; 0 ) such that Equaton (4) holds. The concluson then follows from the cardnal set-monotoncty of f. Frst, consder any player 2 N such that f() max 0 representatons u (; ) and u (; 0 ) satsfy Equaton (4). X. It s mmedate that any cardnal 18

20 Second, consder any player 2 N such that f() 6 max 0 X, but L (x; ) L (x; 0 ) and SL (x; ) SL (x; 0 ) for all x 2 f(). The clam n the proof of Theorem 1 shows that, for any cardnal representaton u (; ) of <, there exsts a cardnal representaton u (; 0 ) of < 0 such that u (x; ) u (x; 0 ) for all x 2 X and u (x; ) = u (x; 0 ) for all x 2 f(). It drectly follows that Equaton (4) holds. 7 Conclusons We conclude wth a seres of remarks n whch we provde applcatons of our results to some mportant socal choce rules. Remark 1 On the doman of sngle-top preferences, the strong Pareto correspondence f P O s set-monotonc and hence, by Theorem 2, s mplementable n mxed Nash equlbrum; on the contrary, t fals to be Maskn monotonc. The strong Pareto correspondence s de ned as follows: f P O () := fx 2 X : there s no y 2 X such that x 2 L (y; ) for all 2 N; and x 2 SL (y; ) for at least one 2 Ng: To see that f P O s set-monotonc on the doman of sngle-top preferences, consder two states and 0 such that for all 2 N, ether f P O () max 0 X or for all x 2 f P O (), () L (x; ) L (x; 0 ) and () SL (x; ) SL (x; 0 ). Suppose that x 2 f P O (), but x =2 f P O ( 0 ). Frst, f f P O () max 0 X, t follows from the sngle-top condton that fx g = f P O () = max 0 X and, thus, x 2 f P O ( 0 ). Second, assume that f P O () 6 max 0 X, so that () and () above hold. At state 0, there must then exsts y 2 X such that y =2 SL (x ; 0 ) for all 2 N and y =2 L (x ; 0 ) for at least one 2 N. It follows that y =2 SL (x ; ) for all 2 N and y =2 L (x ; ) for at least one 2 N, a contradcton wth x 2 f P O (). Consequently, f P O () f P O ( 0 ) and f P O s set-monotonc. To see that the strong Pareto correspondence s not Maskn monotonc on the doman of sngle-top preferences (and, therefore, on the unrestrcted doman), consder the followng example. There are three players, 1, 2 and 3, and two states of the world and 0. Preferences are gven n the table below d d b d d b b a a b a b a c b c c c a c d a c d 19

21 The strong Pareto correspondence s: f P O () = fa; b; dg and f P O ( 0 ) = fb; dg. Maskn monotoncty does not hold snce L 2 (a; ) L 2 (a; 0 ) and yet a 62 f P O ( 0 ). 12 Remark 2 Usng arguments that parallel the ones used for the strong Pareto correspondence, t can be ver ed that on the doman of sngle-top preferences, the strong core correspondence f SC s set-monotonc, whle t s not Maskn monotonc. A coaltonal game s a quadruple hn; X; ; v; where N s the set of players; X s the nte set of alternatves, s a pro le of preference relatons, and v : 2 N n f;g! 2 X. An alternatve x s weakly blocked by the coalton S N n f;g f there s a y 2 v(s) such that x 2 L (y; ) for all 2 S and x 2 SL (y; ) for at least one 2 S: If there s an alternatve that s not weakly blocked by any coalton n 2 N n f;g, then hn; X; ; v s a game wth a non-empty strong core. A coaltonal envronment wth non-empty strong core s a quadruple hn; X; ; v; where s a set of preference relatons such that hn; X; ; v has a non-empty strong core for all 2. The strong core correspondence f SC s de ned for all coaltonal envronments wth nonempty strong core as follows: f SC () := fx 2 v(n) : x s not weakly blocked by any ; 6= S Ng : Remark 3 On the unrestrcted doman of preferences, a Maskn monotonc socal choce functon must be constant (Sajo, 1988). It s smple to see that ths s also true for a set-monotonc socal choce functon. Suppose, to the contrary, that f() = x 6= y = f ( 0 ) : Let 00 be such that fx; yg max 00 X for all 2 N: Then set-monotoncty mples fx; yg f( 00 ); contrary to the assumpton that f ( 00 ) s a sngleton. Remark 4 On the doman of strct preferences, the top-cycle correspondence, an mportant votng rule, s set-monotonc, whle t s not Maskn monotonc. Snce t also sats es no-veto power, Theorem 2 apples: on the doman of strct preferences, the top-cycle correspondence s mplementable n mxed Nash equlbrum. We say that alternatve x defeats alternatve y at state, wrtten x y; f the number of players who prefer x to y s strctly greater than the number of players who prefer y to x. At each state ; the top-cycle correspondence selects the smallest subset of X such that any alternatve n t defeats all alternatves outsde t. f T C () := \fx 0 X : x 0 2 X 0 ; x 2 X n X 0 mples x 0 xg: To prove that the top-cycle correspondence s set-monotonc, assume to the contrary that there s at least an alternatve x such that x 2 f T C (); x =2 f T C ( 0 ), and L (x; ) L (x; 0 ) 12 On the unrestrcted doman of preferences, f P O s not set-monotonc. To see ths, suppose alternatve d s not avalable n the example. The strong Pareto correspondence s then f() = fa; bg and f( 0 ) = fbg. Set-monotoncty fals, snce t s L 2 (a; ) L 2 (a; 0 ); fa; bg max 0 2 fa; b; cg and yet a 62 f P O ( 0 ). 20