Coalition-Proof Equilibrium

Transcription

1 GAMES AD ECOOMIC BEHAVIOR 7, ARTICLE O Coalton-Proof Equlbrum Dego Moreno Departamento de Economıa, Unersdad Carlos III de Madrd, Getafe ( Madrd ), Span and John Wooders Department of Economcs, Unersty of Arzona, McClelland Hall, Tucson, Arzona 8572 Receved January 9, 995 We characterze the agreements that the players of a noncooperatve game may reach when they can communcate pror to play, but they cannot reach bndng agreements: A coalton-proof equlbrum s a correlated strategy from whch no coalton has an mprovng and self-enforcng devaton. We show that any correlated strategy whose support s contaned n the set of actons that survve the terated elmnaton of strctly domnated strateges and weakly Pareto domnates every other correlated strategy whose support s contaned n that set, s a coalton-proof equlbrum. Consequently, the unque equlbrum of a domnance solvable game s coalton-proof. Journal of Economc Lterature Classfcaton umbers: C72, D Academc Press, Inc. ITRODUCTIO When the players of a noncooperatve game have the opportunty to communcate pror to play, they wll try to reach an agreement to coordnate ther actons n a mutually benefcal way. The am of ths paper s to characterze the set of agreements that the players may reach. Snce we consder stuatons where agreements are nonbndng, only those agreements that are not subject to vable Ž.e., self-enforcng. devatons are of * We are grateful to Mark Walker for many nterestng dscussons. Ths author gratefully acknowledges fnancal support from the Mnstero a Asuntos Socales admnstered through the Catedra Gumersndo Azcarate, and from DGICYT Grants PB and PB Ths author gratefully acknowledges support from the Spansh Mnstry of Educaton $8.00 Copyrght 996 by Academc Press, Inc. All rghts of reproducton n any form reserved. 80

2 COALITIO-PROOF EQUILIBRIUM 8 nterest. As preplay communcaton allows the players to correlate ther play, we take the set of all correlated strateges as the space of feasble agreements. We characterze the set of coalton-proof equlbra as the set of agreements from whch no coalton has a self-enforcng devaton makng all ts members better off. Admttng correlated strateges as feasble agreements alters the set of coalton-proof equlbra of a game n a fundamental way Žvz., no ncluson relatonshp between the noton of coalton-proofness that we propose and others prevously ntroduced s to be found.. In fact, there are games where the only plausble agreements are correlated Ž and not mxed. agreements. We provde examples wth ths feature and we show that the noton of coalton-proof equlbrum that we propose dentfes these agreements. Unfortunately, as wth other notons of coalton proofness prevously ntroduced, the exstence of an equlbrum cannot be guaranteed. We are able to establsh, however, that f there s a correlated strategy whch Ž. has a support contaned n the set of actons that survve the terated elmnaton of strctly domnated strateges, and Ž. weakly Pareto domnates every other correlated strategy whose support s contaned n that set, then ths strategy s a coalton-proof equlbrum. Consequently, the unque equlbrum of a domnance solvable game s coalton-proof. Other authors have explored the mplcatons of preplay communcaton when agreements are mxed strategy profles. Aumann Ž 959. ntroduced the noton of strong ash equlbrum, whch requres that an agreement not be subject to an mprovng devaton by any coalton of players. Ths requrement s too strong, snce agreements must be resstant to devatons whch are not themselves resstant to further devatons. Recognzng ths problem, Bernhem, Peleg, and Whnston Ž 987. Žhenceforth referred to as BPW. ntroduced the noton of coalton-proof ash equlbrum Ž CPE., whch requres only that an agreement be mmune to mprovng devatons whch are self-enforcng. A devaton s self-enforcng f there s no further self-enforcng and mprovng devaton avalable to a proper subcoalton of players. Ths noton of self-enforceablty provdes a useful means of dstngushng coaltonal devatons that are vable from those that are not resstant to further devatons. Only vable devatons can upset potental agreements. A defcency of CPE, however, s that t does not allow players to agree to correlate ther play. Although the possblty that players correlate ther actons when gven the opportunty to communcate was recognzed as early as n Luce and Raffa Ž 957., only recently dd Eny and Peleg Ž 995. Ž E & P. ntroduce a concept of coalton-proof communcaton equlbrum. The dfference between E & P s noton and ours can be better understood f we assume that correlated agreements are carred out wth the assstance of a meda-

3 82 MOREO AD WOODERS tor. The medator selects an acton profle accordng to the agreement and then makes a Ž prvate and nonbndng. recommendaton of an acton to each player. E & P consder stuatons where the players may plan devatons only after recevng recommendatons. In our framework, however, players plan devatons before recevng recommendatons, and no further communcaton s possble after recommendatons are ssued. Ths dfference manfests tself most clearly n two-person games where an agreement s coalton-proof n our sense only f t s Pareto-effcent wthn the set of correlated equlbra, whle an agreement that s coalton-proof n E & P s sense need not be. We provde an example wth ths feature n Secton 4. The second dfference s that n our framework devatons may nvolve the members of a coalton jontly msreportng ther types, whle ths possblty s not consdered by E & P s noton. In Secton 4 these dfferences are dscussed n detal. Ray Ž 996. proposes a noton of coalton-proof correlated equlbrum n whch the players possbltes of correlatng ther play are lmted by an exogenously gven correlaton dece, and he shows that there are coalton-proof equlbra whch cannot be attaned by means of drect deces Ž.e., devces n whch players messages are ther acton spaces.. Ths fndng rases the queston whether allowng nondrect devces mght alter the coalton-proof correlated equlbra Ž CPCE. of a game when, as n our defnton, players possbltes of correlatng ther play are not exogenously gven. We do not have a general answer to ths queston. For games whch satsfy the suffcent condtons we provde for exstence of a unque CPCE, however, the set of equlbra we dentfy as coalton-proof s the same regardless of whether or not nondrect devces are avalable to the players. As the followng example llustrates, correlated play naturally arses when communcaton s possble Žand regardless of whether or not players have access to a correlaton devce.. Therefore one should take the set of correlated strateges as the set of feasble agreements, and one must consder devatons that nvolve correlated play by members of a devatng coalton. Three-Player Matchng Pennes Game TPMPG. Three players each smultaneously choose heads or tals. If all three faces match, then players and 2 each wn a penny whle player 3 loses two pennes. Otherwse, player 3 wns two pennes whle players and 2 each lose a penny. The matrx representaton of ths game s gven n Table I. Ths game has two pure-strategy and one mxed-strategy ash equlbra: one pure-

4 COALITIO-PROOF EQUILIBRIUM 83 TABLE I The Three-Player Matchng Pennes Game H3 T3 H2 T2 T2 H2 H,, 2,, 2,, 2,, 2 T,, 2,, 2,, 2,, 2 strategy equlbrum conssts of players and 2 each choosng heads Ž tals. and player 3 choosng tals Ž heads.. In the mxed strategy equlbrum each player chooses heads wth probablty 2. The game does not have a CPE, as each of the ash equlbra s upset by a devaton of the coalton of players and 2; n the pure-strategy ash equlbrum where players and 2 both choose heads, they each obtan a payoff of. By jontly devatng Ž both choosng tals nstead. players and 2 each obtan a payoff of. Ths devaton s self-enforcng as players and 2 each obtan ther hghest possble payoffs and therefore nether player can mprove by a further unlateral devaton. ŽA symmetrc argument shows that the other pure-strategy ash equlbrum s not a CPE ether.. In the mxed-strategy ash equlbrum, players and 2 each obtan an expected payoff of 2. Ths equlbrum s not a CPE as players and 2 can jontly devate Ž both choosng heads nstead. and obtan a payoff of zero. Ths devaton s self-enforcng, snce gven that player 3 chooses heads or tals wth equal probablty, nether player can obtan more than zero by a further devaton. Snce a CPE must be a ash equlbrum, ths game has no CPE. evertheless, the game does have an agreement that s resstant to mprovng devatons. Ths agreement s the correlated strategy where wth probablty 2 players and 2 both choose heads and wth probablty 2 both choose tals, and player 3 chooses heads or tals wth equal probablty. Under ths agreement each player has an expected payoff of zero. o sngle player can devate and mprove upon ths agreement: nether player nor player 2 can beneft by unlaterally devatng, as they both lose a penny whenever ther faces do not match. ether does player 3 beneft from devatng: gven the probablty dstrbuton over the moves of players and 2, he s ndfferent between heads and tals. Moreover, snce the nterests of players and 2 are completely opposed to those of player 3, no coalton nvolvng player 3 can mprove upon the gven agreement. Fnally, gven player 3 s strategy, players and 2 obtan at most a payoff of zero, and therefore they cannot beneft by devatng. Hence, no coalton can gan by devatng from the agreement.

5 84 MOREO AD WOODERS TABLE II An Entry Game Enter ot enter Enter 2, 2, ot enter, 0, 0 otce that the agreement descrbed above s not a mxed strategy profle and so t cannot possbly be a CPE. As we shall see, however, when we expand the space of agreements to nclude all the correlated strateges, ths agreement s the unque coalton-proof equlbrum of the game. ŽSee Moreno and Wooders, 995, for an expermental study of ths game.. The possblty of players correlatng ther play arses even when communcaton s lmted. Consder, for nstance, the game descrbed n Table II whch s related to a class of games dscussed n Farrell Ž 987.; n ths game two dentcal frms must smultaneously decde whether to enter a market whch s a natural monopoly. Ths game has three ash equlbra: Ž Enter, ot enter., Ž ot enter, Enter., and a mxed-strategy ash equlb- rum where each frm enters the market wth probablty 2. Each of these ash equlbra s also a CPE. Although the mxed ash equlbrum s a CPE, t s not resstant to mprovng devatons gven the possblty of preplay communcaton. The frms can mprove by augmentng the game wth a round of cheap talk. In the game wth cheap talk each frm smultaneously and publcly announces whether t ntends to Enter or ot enter the market. Followng the announcements each frm makes ts choce. Suppose the frms agree to play the followng ash Žand subgame perfect. equlbrum of the game wth cheap talk. Each frm announces 3 Enter wth probablty 4. If the profle of announcements s ether Ž Enter, ot enter. or Ž ot enter, Enter., then each frm plays ts an- nouncement. Otherwse, each frm plays Enter wth probablty 2. Ths 5 equlbrum yelds an expected payoff for each frm of 6 whle n the mxed ash equlbrum of the orgnal game each frm has an expected payoff of only 2. Preplay communcaton has enabled the frms to correlate ther play. In ths ash equlbrum of the cheap talk game the frms effectvely play the correlated strategy Ž of the orgnal game. gven n Table III. Ths jont probablty dstrbuton s not the product of ts margnal dstrbutons and therefore cannot be obtaned from a mxed strategy profle of the game

6 COALITIO-PROOF EQUILIBRIUM 85 TABLE III The Correlated Strategy Induced by Cheap Talk Enter ot enter 5 Enter ot enter wthout communcaton. Ths correlated devaton from the mxed strategy equlbrum makes both frms better off. Moreover, t s a self-enforcng devaton snce t s a correlated equlbrum of the orgnal game. Expandng the set of feasble agreements from the mxed strateges Žas n CPE. to the set of correlated strateges does not lead smply to an expanson of the set of coalton-proof agreements. In the Three-Player Matchng Pennes game we found a coalton-proof agreement where no CPE exsted. In the entry game we found a CPE that was not coalton-proof. Thus, there s no ncluson between the set of CPE and the set of equlbra that are coalton-proof n our sense. In our framework the prmtves are a set of feasble agreements and the concepts of feasble devaton and of self-enforcng devaton by a coalton from a gven agreement. The set of feasble devatons by a coalton from a gven agreement s the set of all correlated strateges that the coalton can nduce when the complementary coalton behaves accordng to the gven agreement and when the members of the coalton correlate ther play. The defnton of a self-enforcng devaton s recursve. For a coalton of a sngle player any feasble devaton s self-enforcng. For coaltons of more than one player, a devaton s self-enforcng f t s feasble and f there s no further self-enforcng and mprovng devaton by one of ts proper subcoaltons. Wth these concepts, our noton of coalton-proofness s easly formulated; an agreement s coalton-proof f no coalton Ž not even the grand coalton. has a self-enforcng devaton that makes all ts members better off. Our noton of a self-enforcng devaton concdes wth that mplct n the concept of CPE. The dfference between our noton of coaltonproofness and CPE s only that we take the set of correlated strateges as the space of feasble agreements. For games of complete nformaton, f feasble agreements are mxed strateges then our defnton of coaltonproofness concdes wth CPE. Ž Ths s establshed n Appendx B.. In some stuatons t may be natural to restrct the space of feasble agreements Ž e.g., f communcaton s lmted. or to lmt the possbltes of players to form devatons. The framework we propose easly accommodates these knds of changes.

7 86 MOREO AD WOODERS In fact, our exstence results are easly modfed to provde condtons for the exstence of a CPE; namely, any mxed strategy profle whose support s contaned n the set of acton profles that survve terated elmnaton of strctly domnated strateges and whch weakly Paretodomnates any other mxed strategy profle whose support s contaned n ths set s a CPE. For games wth strategc complementartes, Mlgrom and Roberts Ž 994. have ndependently obtaned analogous results. The paper s organzed as follows: n Secton we dscuss our framework and defne our noton of equlbrum for games of complete nformaton. In Secton 2 we establsh condtons for exstence of these equlbra and show by means of an example that an equlbrum does not always exst. In Secton 3 we extend the concept of coalton-proofness to games of ncomplete nformaton. Of course, the noton of coalton-proof equlbrum for games of ncomplete nformaton reduces to that formulated for games of complete nformaton when every player has a sngle type. We present separately the noton of coalton-proofness for games of complete nformaton, as the noton s smplcty n ths context facltates the dscusson and because we want to stress the fact that our noton of coaltonproofness can be formulated wthout resortng to games of ncomplete nformaton. In Secton 4 we compare our noton of coalton-proof equlbrum and E & P s noton of coalton-proof communcaton equlbrum, and we present some concludng remarks.. GAMES OF COMPLETE IFORMATIO A game n strategc form s defned as, Ž A., Ž u., where s the set of players, and for each, A s player s set of actons Ž or pure strateges. and u s player s utlty Ž payoff. functon, a real-valued functon on A A. Assume that and A are nonempty and fnte. For any fnte set Z, denote by Z the set of probablty dstrbutons over Z. In partcular, denote by A the set of probablty dstrbutons over A, and refer to ts members as correlated strateges. Gven a correlated strategy, player s expected utlty when players actons are selected accordng to s Ý U Ž a. u Ž a.. aa A coalton of players S s a member of 2. When S conssts of a sngle 4 player, we wrte t as rather than the more cumbersome. For

8 COALITIO-PROOF EQUILIBRIUM 87 each S 2, S, denote by AS the set Ł S A. Gven a A, we wrte a Ž a, a. S S, where as AS and as A S. If S, then Ž a, a. a a. S S S Coalton-Proof Correlated Equlbrum We conceve of communcaton and play as proceedng n two stages. In the frst stage players communcate, reachng an agreement and possbly plannng devatons from the agreement. Gven an agreement A, the players mplement t wth the assstance of a medator who recommends the acton profle a A wth probablty Ž a.. In the second stage, each player prvately receves hs component of the recommendaton and then chooses an acton. Ž o further communcaton occurs n ths stage.. A devaton by a coalton s a plan for ts members to correlate ther play n a way dfferent from that prescrbed by the agreement. We take a broad vew of the ablty of coaltons to plan devatons; for every dfferent profle of recommendatons receved by ts members, a devatng coalton may plan a dfferent correlated strategy. Therefore, a devaton for a coalton S s a mappng from the set AS of profles of recommendatons for ts members to the set AS of probablty dstrbutons over the set of the coalton s acton profles. Gven an agreement, f a coalton S plans to devate accordng to S: AS A S, and f the members of the complement of S play ther part of the agreement,.e., they obey ther recommendatons, then the nduced probablty dstrbuton over acton profles for the grand coalton s gven for each a A by Ý S S S S S Ž a. Ž, a. Ž a.. SAS It wll be convenent to defne the feasble devatons for coalton S as those correlated strateges A whch the coalton can nduce, rather than as mappngs from AS to A S. Thus, a correlated strategy s a feasble devaton by coalton S from a gven agreement f the members of S, usng some plan to correlate ther play, can nduce the correlated strategy when each member of the complementary coalton obeys hs recommendaton. DEFIITIO.. Let A and S 2, S. We say that A s a feasble deaton by coalton S from f there s an S: AS A S, such that for all a A we have Ž a. Ý,a Ž a.. SAS S S S S S We llustrate our defnton of a feasble devaton by descrbng a procedure that can be thought of as mmckng the process by whch players select agreements and plan devatons. Gven an agreement,

9 88 MOREO AD WOODERS suppose that the medator mplementng mals to each player a sealed envelope contanng the player s recommendaton. A coalton S devates from by employng a new medator to whch each member of S sends the Ž unopened. envelop t receved from the medator mplementng. The new medator opens the envelops, reads the recommendatons S, and then selects a new profle of recommendatons accordng to the correlated strategy Ž. S S. The medator then mals to each player S a sealed envelope contanng hs recommended acton. When each player opens hs envelope and obeys the recommendaton t contans, the nduced correlated strategy s gven by the equaton n Defnton.. Gven a coalton S 2, S, and an agreement A, let DŽ,S. denote the set of feasble devatons by coalton S from ; note that DŽ, S., snce a coalton always has the trval devaton consstng of each member of the coalton obeyng hs own recommendaton. Also note that for every A, we have DŽ,. A. A correlated equlbrum s a correlated strategy from whch no ndvdual has a feasble mprovng devaton. Correlated Equlbrum. A correlated strategy s a correlated equlbrum f no ndvdual, has a feasble devaton DŽ,., such that UŽ. UŽ.. The defnton of strong ash equlbrum suggests the followng defnton of strong correlated equlbrum : a strong correlated equlbrum s a correlated strategy from whch no coalton has a devaton whch makes every member of the coalton better off. DEFIITIO.2. A correlated strategy A s a strong correlated equlbrum f no coalton S 2, S, has a feasble devaton DŽ,S., such that for each S, we have U Ž. U Ž.. The agreement descrbed n the ntroducton for the Three-Player Matchng Pennes game s, for example, the unque strong correlated equlbrum of that game. Lke strong ash equlbrum, the noton of strong correlated equlbrum s too strong. A strong correlated equlbrum must be resstant to any feasble devaton by any coalton. In partcular, t must be resstant to devatons whch are not themselves resstant to further devatons. Consder, for example, the Prsoners Dlemma game descrbed n Table IV. Ths game has a unque correlated equlbrum where Ž D, D. s played wth probablty one. Ths correlated equlbrum s not a strong correlated equlbrum snce the correlated strategy consstng of playng Ž C,C. wth probablty one s a feasble devaton whch makes both players better off. Snce a strong correlated equlbrum must be a corre- A noton of strong correlated equlbrum was nformally proposed n Mouln 98.

10 COALITIO-PROOF EQUILIBRIUM 89 TABLE IV A Prsoners Dlemma Game C D C,, 2 D 2, 0,0 lated equlbrum, ths game has no strong correlated equlbrum. otce, however, that ether player can unlaterally devate from and ncrease hs payoff. Hence should not undermne an agreement to play Ž D, D. wth probablty one. In order to be able to dstngush those devatons that are vable from those that are not Žand whch therefore should not upset an agreement as coalton-proof. we ntroduce the noton of self-enforcng devaton; a correlated strategy s a self-enforcng devaton by coalton S from correlated strategy f s a feasble devaton and f no proper subcoalton of S has a further self-enforcng and mprovng devaton. Ths noton of self-enforceablty s dentcal to the one mplct n the concept of CPE. DEFIITIO.3. Let A and S 2, S. The set of self-en- forcng deatons by coalton S from, SEDŽ, S., s defned, recursvely, as follows: Ž. If S, then SEDŽ, S. DŽ, S.; S If S, then SED, S D, S R 2 S, R, SED Ž, R. such that R: U Ž. U Ž.4. Snce a coalton consstng of a sngle player has no proper nonempty subcoaltons, any feasble devaton by a one-player coalton s self-enforcng. Wth ths noton of a self-enforcng devaton, a coalton-proof correlated equlbrum s defned to be a correlated strategy from whch no coalton has a self-enforcng and mprovng devaton. DEFIITIO.4. A correlated strategy s a coalton-proof correlated equlbrum Ž CPCE. f no coalton S 2, S, has a devaton SEDŽ, S., such that for each S, we have U Ž. U Ž.. It s clear that a strong correlated equlbrum s a coalton-proof correlated equlbrum, whch n turn s a correlated equlbrum. For two-player games the set of coalton-proof correlated equlbra s the set of correlated equlbra whch are not strongly Pareto-domnated by other correlated equlbra Ž.e., s a CPCE f t s a correlated equlbrum, and

11 90 MOREO AD WOODERS there s no other correlated equlbrum such that U Ž. U for each.. Thus, for two-player games, the set of coalton-proof correlated equlbra s nonempty. Although exstence of a CPCE cannot be guaranteed n general games, n the next secton we dentfy condtons under whch a CPCE exsts. 2. EXISTECE OF CPCE AD CPE In ths secton we show that a CPCE Ž CPE. exsts whenever there s a correlated Ž mxed. strategy whose support s contaned n the set of acton profles that survve terated elmnaton of strctly domnated strateges and whch weakly Pareto-domnates every other correlated Ž mxed. strategy whose support s contaned n ths set. Frst, we defne formally the noton of strct domnance. DEFIITIO 2.. Let B Ł B A. An acton a B s sad to be j j strctly domnated n B, f there s B such that for each a B, j j j j Ý j Ž a j. u j Ž a j, a j. u j Ž a j, a j.. ajbj ote that f aj s strctly domnated n B, then t s also strctly domnated n A j B j. The set of acton profles that survve terated elmnaton of strctly domnated strateges, whch we wrte as A, s now easly defned. DEFIITIO 2.2. The set A of acton profles that sure the terated elmnaton of strctly domnated strateges s defned by A Ł A, where each A n0 A n, A n s the set of actons that are not strctly domnated n A n Ł A n, and A 0 A. The followng proposton establshes that f s a correlated strategy whose support s contaned n A, then the support of every self-enforcng devaton from by a coalton of more than one player s also contaned n A. For each A and S 2, S, wrte A S for the set a A Ž a, a. 0 for some a A 4, and wrte A S S S S S S for the set A Ž.. PROPOSITIO. Let A be such that A A, and let S 2 be a coalton of more than one player. If SED, S, then A A. Proof. Let S and A be as n the proposton, and let SEDŽ, S.. By the defnton of feasble devaton Ž Defnton.. A Ž. AS A S. We show that n fact A A. Suppose by way of contradcton that A s not contaned n A. Let n be the largest n S S

12 COALITIO-PROOF EQUILIBRIUM 9 Ž. Ž. n such that AS A S. Hence there s j S and aj A j such that n aj s strctly domnated n A. Thus aj s also strctly domnated n A A n ;.e., there s A such that for each a A n, j j j j j j Ý j j j j j j j j ajaj Ž a. u Ž a, a. u a, a. Consder the devaton by player j Ž a proper subcoalton of S. from, where player j chooses an acton accordng to j when recommended aj and takes the recommended acton otherwse. Formally, the devaton j s defned as follows: for each a A such that a a, let Ž a. j j j j j j j f a, and Ž a. 0f a ; for a, let Ž a. Ž a. j j j j j j j j j j j j j j. Ž. n Agan by the defnton of feasble devaton A A A ; then Ý Ý ž Ý / n U Ž. Ž a. u Ž a. Ž, a. a u Ž a.. j j j j j j j j aa aajaj jaj Substtutng as defned above we have j Ý U Ž. Ž a. u Ž a. j j j n Aj aža a 4. Ý j a, a Ž. u Ž, a.. j j Ž j j. ž Ý j j j j j / n aa 4A ja j j j n j j j j Snce a, a 0 for some a A, Eq. mples Ý Ý j j j j j j j Ž 4. n 4 n a Aj aj Aj a aj Aj U a u a a, a u a, a ;.e., U j Ý Ž a. ujž a. UjŽ.. aa Hence s an mprovng and self-enforcng devaton from by player j Ž recall that every feasble devaton by a sngle player s self-enforcng.. Thus, s not a self-enforcng devaton by S from ;.e., SEDŽ, S.. Ths contradcton establshes that A A. The followng corollary establshes that f a correlated strategy whose support s contaned n A weakly Pareto-domnates every other correlated strategy whose support s contaned n A Ž.e., U U Ž., for each., then s a CPCE.

13 92 MOREO AD WOODERS COROLLARY. Let A be such that A A and such that t weakly Pareto-domnates eery other A for whch A A. Then s a CPCE. Proof. Let be as n the corollary. We show that s a CPCE. It s easy to show that no sngle player has a feasble and mprovng devaton from. If a player j has an mprovng devaton from, then he also Ž. has an mprovng devaton from such that A A. Snce weakly Pareto-domnates every correlated strategy whose support s contaned n A, such a devaton cannot exst. Moreover, nether does a coalton of more than one player have a self-enforcng and mprovng devaton, snce the support of every self-enforcng devaton from by such a coalton s contaned n A by the proposton above. Hence s a CPCE. In Appendx B we show that the set of CPE of a game can be characterzed as the set of mxed strateges from whch no coalton has a self-enforcng devaton whch makes all ts members better off. The proposton above s easly modfed to show that f s a mxed strategy profle whose support s contaned n A, then any self-enforcng mxed devaton from by a coalton of more than one player also has ts support contaned n A. Thus, the corollary above establshes condtons under whch a CPE exsts; whenever there s a mxed strategy profle whose support s contaned n A and whch weakly Pareto-domnates every other mxed strategy whose support s contaned n A, then ths strategy s a CPE. In fact, the exstence of a correlated strategy whose support s contaned n A and whch weakly Pareto-domnates every other correlated strategy whose support s also contaned n A mples the exstence of an acton profle a A whch weakly Pareto-domnates every acton Ž profle n A.e., such that for each and each a A, u Ž a. u Ža... Ths acton profle s therefore a Ž pure strategy. coalton-proof correlated Ž and ash. equlbrum. Thus, the condtons that guarantee the exstence of a CPCE also mply exstence of a CPE. An obvous mplcaton of our corollary s that the unque equlbrum of Ž a domnance solable game.e., a game for whch the set A s a sngleton. s the unque CPCE Ž and CPE. of the game. Also f a correlated strategy whose support s contaned n A strongly Paretodomnates every other strategy whose support s contaned n A, then s the unque CPCE of the game Žas any other correlated equlbrum wll be upset by the devaton of the grand coalton n whch players gnore ther recommendatons and play accordng to.. It s worth notcng that the equlbra characterzed by our corollary are strong n the sense that any mprovng devaton by a coalton of players s

14 COALITIO-PROOF EQUILIBRIUM 93 upset by a further devaton by a sngle player. Mlgrom and Roberts Ž 994. refer to such equlbra as strongly coalton-proof. They provde condtons that guarantee the exstence of these equlbra n games wth strategc complementartes. Specfcally, they show that f a game wth strategc complementartes has a unque ash equlbrum, then ths equlbrum s the unque strongly coalton-proof equlbrum of the game Ž Theorem.; f each player s payoff functon s ncreasng Žrespectvely, decreasng. n the other players strateges, then the maxmal Žrespectvely, mnmal. ash equlbrum s the unque strongly coalton-proof equlbrum Ž Theorem 2.. Mlgrom and Roberts establsh ther results usng Tarsk s fxed pont theorem, and they do not rely on domnance arguments. For games wth fnte strategy spaces, Mlgrom and Roberts s results are mpled by our corollary: f a game wth strategc complementares has a unque ash equlbrum, then t s domnance solvable 2 ; hence, ths equlbrum s the unque Ž strongly. coalton-proof ash equlbrum. Moreover, f each player s payoff functon s ncreasng Žrespectvely, decreasng. n the other players strateges, then the maxmal Žrespectvely, mnmal. ash equlbrum weakly Pareto-domnates every other strategy whose support s contaned n A, and therefore our Corollary mples that ths equlbrum s a Ž strongly. coalton-proof ash equlbrum. Of course, the equlbra dentfed by these condtons are also Ž strongly. coaltonproof correlated equlbra. A Game Where a CPCE Does ot Exst Unfortunately, as the followng example shows, there are games wth more than two players wth no coalton-proof correlated equlbra. Consder the three-player game gven n Table V, taken from Eny and Peleg, where player chooses the row, player 2 chooses the column, and player 3 chooses the matrx. TABLE V A Game Where a CPCE Does ot Exst c c2 b b2 b b2 a 3, 2, 0 0, 0, 0 3, 2, 0 0, 3, 2 a 2, 0, 3 2, 0, 3 0, 0, 0 0, 3, See Mlgrom and Shannon s 994 Theorem 2.

15 94 MOREO AD WOODERS We show that there does not exst a coalton-proof correlated equlbrum of ths game. Let be an arbtrary correlated equlbrum and suppose that player has the lowest payoff of the three players. Then 3 U. Ž 3 5 Ths can be proven by maxmzng player 3 s utlty over the set of correlated equlbra satsfyng U maxu Ž., U Ž Moreover, U 5 snce player has the lowest payoff. ow consder the followng devaton from by players and 3: player chooses the bottom row and player 3 chooses the left matrx. Ths devaton s mprovng as players and 3 now receve payoffs of 2 and 3, respectvely. To demonstrate that s not a coalton-proof correlated equlbrum we need only show that ths devaton s self-enforcng. Clearly player 3 does not devate further as he now obtans 3, hs hghest possble payoff. It can 5 be shown that player obtans at most 3 by devatng further and choosng the top row. 3 Ž The detals of ths calculaton are n Appendx A.. Thus, s not a coalton-proof correlated equlbrum as players and 3 have a self-enforcng and mprovng devaton. There was no loss of generalty n assumng that player has the lowest payoff. If player 2 has the lowest payoff, then there s a self-enforcng and mprovng devaton by players 2 and. If player 3 has the lowest payoff, then there s a self-enforcng and mprovng devaton by players 3 and 2. Snce any correlated equlbrum has a self-enforcng devaton by two players whch makes both players better off, ths game has no coaltonproof correlated equlbrum. Ž Ths game does not have a CPE ether.. 3. GAMES OF ICOMPLETE IFORMATIO In ths secton we extend our noton of coalton proofness to games of ncomplete nformaton. A Ž fnte. game of ncomplete nformaton Žor Bayesan game. G s defned by G, Ž T., Ž A., Ž p., Ž u., where s the set of players, and for each, T s the set of possble types for players, A s player s acton set, p : T T s player s pror probablty dstrbuton over the set of type profles for the other players n the game Ž T Ł T., and u : T A s player s j 4 j 3 Followng the devaton by players and 3, player s choosng the bottom row wth probablty one. Hence, when consderng a further devaton by player there s no loss of generalty n restrctng attenton to the devaton where he chooses the top row wth probablty one. If ths devaton does not make hm better off, then no devaton does.

16 COALITIO-PROOF EQUILIBRIUM 95 utlty Ž payoff. functon Ž A Ł A, T Ł T.. We assume that the sets, A, and T are nonempty and fnte. For every coalton of players S 2, S, we denote by TS the set Ł ST. A correlated strategy s a functon : T A. We let C denote the set of all correlated strateges. Gven C, f each player reports hs type truthfully and obeys hs recommendaton, then player s expected payoff when he s of type t T s Ý Ý UŽ t. p Ž t t. Ž at. u Ž a,t.. tt aa otce that n order for the players to play accordng to a correlated strategy, nformaton about the players types must be reealed so that an acton profle can be selected accordng to the probablty dstrbuton specfed by the gven correlated strategy. We therefore must allow devatons by a coalton n whch the players reveal a type profle dfferent from ther true one, as well as devatons where the players take actons dfferent from those recommended. In the conceptual framework of medaton, the members of a coalton can devate from a correlated strategy by msreportng ther type profle to the medator or by dsobeyng the medator s recommendatons. Intutvely, a devaton can be conceved of as follows: a coalton S carres out a devaton by employng a new medator who represents the coalton wth the medator mplementng and wth whom the members of S communcate. Each member of S reports hs type to ths medator who then Ž. selects accordng to some f S: TS TS a type profle for the coalton Ž whch he reports to the medator mplementng. and, upon recevng from the medator mplementng the recommendatons for the members of S, Ž. 2 selects accordng to some S: TS TS AS AS an acton profle Ž whch he recommends to the coalton members.. The acton profle recommended by the new medator depends upon the type profle reported to t, the type profle t reported to the medator mplementng, and the actons recommended by the medator mplementng. Ths devaton generates a new correlated strategy whch can be calculated from fs and S accordng to the formula gven n Defnton 3.. DEFIITIO 3.. Let C and S 2, S. A correlated strategy s a feasble deaton by coalton S from f there are f : T T and S S S : T T A A, such that for each t T and each a A, S S S S S Ý Ý S S S S S S S S S S S S Ž at. f Ž t. Ž, a,t. Ž a,t,.. STS SAS The set of feasble devatons by coalton S from a correlated strategy s the set of correlated strateges that the coalton can nduce by means of

17 96 MOREO AD WOODERS some f and. Gven C and S 2, S, denote by DŽ, S. S S the set of all feasble devatons by coalton S from correlated strategy. As n Secton, for every C and S 2, we have DŽ, S. and DŽ,. C. For expostonal ease, we explctly ntroduce a concept of Pareto-domnance: a correlated strategy Pareto-domnates another correlated strat- egy for coalton S f no member of S s worse off under than under for any type profle and f for at least one type profle every member of S s better off under than under. DEFIITIO 3.2. Let S 2, S, and let, C. We say that Pareto-domnates for coalton S Ž or that Pareto S-domnates. f Ž 3.. and Ž 3.2. below are satsfed. For each S and each t T : U t U Ž t.. Ž 3.. For each S and some t T : U t U t. Ž 3.2. In our framework, the noton of Pareto domnance used determnes whether a devaton s an mprovement for a coalton. Consequently, alternatve notons of Pareto domnance wll lead to dfferent notons of coalton-proof communcaton equlbrum. There are two alternatve notons worth consderng. We say that weakly Pareto S-domnates f no member of S s worse off under than under for any of hs types Ž.e., f Ž 3.. s satsfed., and f at least one member of S s better off under than under for one of hs types Ž.e., f Ž 3.2. s satsfed for some S rather than for all S.. The noton of weak Pareto domnance does not seem approprate; an agreement wll be ruled out f a coalton has a self-enforcng devaton whch makes only a proper subset of ts members better off, even though there are not clear ncentves for such a coalton to form. We say that strongly Pareto S-domnates f each member of S s better off under than under for each of hs types Ž.e., f the nequaltes Ž 3.. are satsfed wth strct nequalty.. Strong Pareto domnance s sometmes too strong. For example, f the utlty functon of some player s constant for one of hs types, then there s no devaton whch s mprovng for ths player. Usng strong Pareto domnance rules out the possblty of ths player partcpatng n any devaton. It s easy to see that a correlated strategy s a communcaton equlbrum f no sngle player has a feasble devaton whch Pareto -domnates. 4 In the sprt of the noton of strong ash equlbrum, a 4 See Forges 986 or Myerson 986 for a defnton of communcaton equlbrum.

18 COALITIO-PROOF EQUILIBRIUM 97 strong communcaton equlbrum can be defned as follows: A correlated strategy s a strong communcaton equlbrum f no coalton S has a feasble devaton whch Pareto S-domnates. We only want to requre, however, that an agreement not be Pareto-domnated by self-enforceablty devatons. The noton of self-enforceablty we defne s dentcal to that ntroduced n Secton. DEFIITIO 3.3. Let C and S 2, S. The set of self-en- forcng deatons by coalton S from, SEDŽ, S., s defned, recursvely, as follows: Ž. If S, then SEDŽ, S. DŽ, S.; S If S, then SED, S D, S R 2 S, R, SED Ž, R. such that Pareto R-domnates 4. Wth ths noton of self-enforceablty, a coalton-proof communcaton equlbrum s defned to be any correlated strategy from whch no coalton S has a self-enforcng devaton whch Pareto S-domnates. DEFIITIO 3.4. A correlated strategy s a coalton-proof commun- caton equlbrum CPCE f no coalton S 2, S, has a devaton SED Ž, S. such that Pareto S-domnates. When the set of type profles T s a sngleton, the concepts of strong and coalton-proof communcaton equlbrum reduce to, respectvely, strong and coalton-proof correlated equlbrum. ote that a strong communcaton equlbrum s a coalton-proof communcaton equlbrum, whch n turn s a communcaton equlbrum. In two-player Bayesan games, the set of coalton-proof communcaton equlbra conssts of the communcaton equlbra that are not Pareto -domnated by any other communcaton equlbrum Ž.e., the set of nterm effcent communcaton equlbra.. 5 Hence, for two-player Bayesan games a CPCE always exsts. As establshed by the example n Secton, games wth more than two players need not have a CPCE. 4. DISCUSSIO In ths secton we dscuss the relaton of CPCE to Eny and Peleg s noton of coalton-proof communcaton equlbrum Žwhch we denote by CPCE. EP, and we present some concludng remarks. In CPCE devatons are evaluated pror to the players recevng recommendatons; a devaton s mprovng f t makes each member of the 5 See Holmstrom and Myerson 983.

19 98 MOREO AD WOODERS TABLE VI The Game of Chcken L R T 6, 6 2, 7 B 7, 2 0, 0 TABLE VII Chcken: A Correlated Equlbrum L R T 3 3 B 3 0 devatng coalton better off, condtonal on hs type, for at least one of hs types and no worse off for any of hs types. In contrast, n CPCE EP devatons are consdered after players receve recommendatons; a devaton s mprovng f t makes each member of the devatng coalton better off, condtonal on both hs type and hs recommendaton, for each combnaton of types and recommendatons that occur wth postve probablty. Consequently, for two person games, whle a CPCE must be nterm effcent a CPCE EP need not be. Ths s llustrated by the game Chcken gven n Table VI. Table VII descrbes a correlated equlbrum of Chcken whch yelds an expected payoff of 5 for each player. Ths correlated strategy s not a CPCE as the grand coalton has the self-enforcng devaton gven n Table VIII, whch yelds an expected payoff of 5.25 for each player. ŽThs devaton s self-enforcng snce t s a correlated equlbrum and therefore s mmune to further devatons by a sngle player.. TABLE VIII Chcken: A Devaton by the Grand Coalton L R T 2 4 B 4 0

20 COALITIO-PROOF EQUILIBRIUM 99 onetheless, the correlated strategy gven n Table VII s a CPCE EP. In ths game each player has only a sngle type; therefore, for a devaton to be mprovng n Eny and Peleg s sense, t must make each player better off, condtonal on hs recommendaton, for each of hs possble recommendatons. Consder player gven the recommendaton B. Hs expected payoff condtonal on hs recommendaton s 7. Snce 7 s player s hghest possble payoff, no coalton nvolvng player can mprove upon ths strategy. 6 One nterpretaton of E & P s framework s that players have the opportunty to communcate only after each player has receved hs recommendaton. Thus, when determnng whether or not an agreement s a CPCE EP, the agreement s elevated to the poston of a status quo agreement. It s requred to be resstant to devatons followng recommendatons, but t s not confronted wth alternatve agreements whch are mprovng at the stage pror to each player recevng hs recommendaton. If players have the opportunty to dscuss ther play pror to recevng recommendatons, however, they wll exhaust the opportuntes for mprovements at ths stage. For the game Chcken, f the players must decde whether to play the strategy n Table VII or that of Table VIII, they should choose the latter as ths strategy gves a hgher expected payoff to each player, and t s also resstant to further devatons. The second fundamental way n whch the notons of coalton proofness dffer s that Eny and Peleg do not admt the possblty that members of a coalton jontly msreport ther types. A CPCE EP must be a communcaton equlbrum, and so a CPCE EP s mmune to devatons where a sngle player msreports hs type and dsobeys hs recommendaton. However, n Eny and Peleg s framework, at the stage where devatons are consdered, the players are assumed to have already truthfully reported ther types. Thus, devatons may not nvolve the members of a coalton jontly msreportng ther types, or nvolve one member of a coalton msreportng hs type and another member of the coalton dsobeyng hs recommendaton. An example of a CPCE EP whch fals to be mmune to ths latter knd of devaton s llustrated n the game of ncomplete nformaton below. The game s the same as the Three-Player Matchng Pennes game Ž see Table I., except that player s moves have now become hs types. Ths game s gven n Table IX below. Player now has two possble types H, T 4 and no actons, whle players 2 and 3 both have a sngleton 6 It can be shown that there s no mprovng devaton upon n E & P s sense even wth the weaker requrement that a devaton makes each member of the devatng coalton better off for at least one recommendaton and at least as well off for all recommendatons.

21 00 MOREO AD WOODERS TABLE IX An Incomplete Informaton Verson of the TPMPG t H t T H T T H H,, 2,, 2,, 2,, 2 2 T,, 2,, 2,, 2,, 2 2 type set and ther acton sets reman, respectvely, H, T 4 and H, T Assume that the prors of players 2 and 3 over player s types are, respectvely, p Ž H. p Ž H The correlated strategy gven by Ž H, T H. and Ž T, H T , s a communcaton equlbrum of the game whch yelds expected payoffs of U Ž H. U Ž T., U, and U It s also a CPCE EP; n E & P s framework, a devaton by a coalton s a mappng from the set of type and acton Ž recommendaton. profles for the coalton to probablty dstrbutons over the coalton s set of acton profles. The coalton, 24 has no mprovng devaton snce, f player s of type H, then player 3 moves T3 wth probablty one and players and 2 have a payoff of regardless of the acton taken by player 2. By the same argument, the coalton cannot mprove f player s of type T.o coalton nvolvng player 3 has an mprovng devaton as the nterests of players and 2 are completely opposed to the nterests of player 3. That no sngle player has an mprovng devaton follows from the fact that s a communcaton equlbrum. In contrast, s not a CPCE of the game. Consder the devaton by the coalton, 24, where player reports T when hs type s H and he reports H when hs type s T, and where player 2 moves H2 when recommended T2 and moves T2 when recommended H 2. Ths devaton results n the correlated strategy gven by Ž H, H H. Ž 2 3 T 2, T3 T., whch yelds expected payoffs of U Ž H. U Ž T. and U 2. The devaton makes both players better off and s also self-enforcng Ž as both players attan ther maxmum possble payoff.. Hence s not a CPCE. ote that even f players can communcate only followng the recept of recommendatons, CPCE EP assumes a certan myopa on the part of player. Consder agan the CPCE EP of the Three-Player Matchng Pennes game, where Ž H, T H. and Ž T, H T If player s of type H and f he antcpates the opportunty to communcate followng

22 COALITIO-PROOF EQUILIBRIUM 0 player 2 s recept of hs recommendaton, then player should report type T and, at the communcaton stage, suggest to player 2 that he should move H 2. Player 2 should follow player s suggeston gven that hs nterests are concdent wth player s. Ths game has a unque CPCE Ž whch s also a CPCE. EP, where player 3 moves H3 wth probablty 2 regardless of player s type, and player 2 moves H2 when player s type s H and moves T2 when player s type s T. Ths s essentally the same agreement predcted for the complete nformaton verson of the game. In fact, gven that the nterests of players and 2 are concdent and opposed to those of player 3, ths seems the only reasonable outcome. For the game of Chcken and the ncomplete nformaton verson of the Three-Player Matchng Pennes game we have found correlated strateges whch are CPCE EP and whch are not CPCE. For the CoordnatonDefecton game n Appendx A we fnd a correlated strategy whch s a CPCE and whch s not a CPCE EP. Thus, there s no ncluson relaton between these two notons. We conclude by emphaszng our fndngs. Frst, we show that when players can communcate they wll reach correlated agreements. For example, n the Three-Player Matchng Pennes game the only ntutve agreement s a correlated Ž and not mxed. agreement. Second, we offer a natural defnton of coalton-proof equlbrum when correlated agreements are possble, and we show that no ncluson relatonshp between ths new noton and CPE s to be found. ŽConsequently, the noton of coalton proofness s senstve to the possblty of correlated agreements.. And thrd, we obtan condtons under whch a coalton-proof equlbrum exsts. APPEDIX A In ths appendx we present three examples. The frst example s a game that has no coalton-proof correlated equlbrum. The second example s the Three-Player Matchng Pennes game; we show that the correlated strategy descrbed n the Introducton s the unque coalton-proof correlated equlbrum Ž and the unque strong correlated equlbrum. of the game. The thrd example s a game wth a CPCE whch s not a CPCE EP. A Game wth o Coalton Proof Correlated Equlbrum We show that the game descrbed n Table V has no coalton-proof correlated equlbrum. A correlated strategy for ths game s a vector

23 02 MOREO AD WOODERS Ž. jk, j,k, 24, where jk 0 denotes the probablty that players, 2, and 3 are recommended, respectvely, actons a, b j, and c k. If s a correlated equlbrum, then t satsfes the system of nequaltes Ž I. gven by Ž I.a Ž I.a Ž I.b Ž I.b Ž I.c Ž I.c We show that for each correlated equlbrum there s a coalton of two players whch has an mprovng and self-enforcng devaton. Therefore, snce a coalton-proof correlated equlbrum must be a correlated equlbrum, the set of CPCE of ths game s empty. Let be an arbtrary correlated equlbrum and suppose that player has the lowest payoff of the three players. We show that the coalton of players and 3 has a self-enforcng and mprovng devaton. If player has the lowest payoff n a correlated equlbrum, then player 3 s payoff s 3 no larger than 5, whch s the value of the soluton to the lnear programmng problem max U subject to Ž I., U U Ž., U U Ž.. A We also have U 3 snce player has the lowest payoff. Consder the devaton nduced by players and 3 playng Ž a, c. 2 wth probablty one for each profle of recommendatons. ŽThen , , and jk 0 otherwse.. Gven ths devaton, players and 3 obtan payoffs of, respectvely, 2 and 3, regardless of player 2 s acton. Hence, U 2 U and U 3 U 3 3 and so s an mprovng devaton for, 34. We now show that s self-enforcng. Clearly player 3 does not have a further mprovng devaton as he obtans hs hghest possble payoff. Player has an mprovng devaton f the expected payoff of devatng to a, whch s 3, s greater than U 2 Ž 2 hs expected payoff when he follows a recommendaton to play a 2.. However, ths payoff s not larger 5 than, whch s the value of the soluton to the lnear programmng 3

24 COALITIO-PROOF EQUILIBRIUM 03 problem max 3Ž. A subject to Ž I., U U2Ž., U U3Ž.. The value of the soluton to ths problem s the maxmum payoff that player can obtan by a further devaton to a from the correlated strategy gven that the orgnal agreement was a correlated equlb- rum n whch player had the lowest payoff. Hence, player has no further mprovng devaton. There was no loss of generalty n assumng that player has the lowest payoff. Gven the symmetry of ths game, we can construct the followng self-enforcng and mprovng devatons n each case: If player 2 has the lowest payoff, then players and 2 devate to a, b 4. If player 3 has the lowest payoff, then players 2 and 3 devate to b, c Therefore, ths game has no coalton-proof correlated equlbrum. Three-Player Matchng Pennes In the Introducton we demonstrated that the correlated strategy gven n Table X below s a strong correlated equlbrum of the Three- Player Matchng Pennes game. We now establsh that s the unque coalton-proof correlated equlbrum of ths game. ŽA strong correlated equlbrum s also a coalton-proof correlated equlbrum; therefore s also the unque strong correlated equlbrum.. Let be any correlated strategy. We reduce notaton by wrtng xyz for the probablty Ž x, y, z., where Ž x, y, z. H, T 4 H, T 4 H, T ; e.g., we wrte for Ž T, T, H..If s a correlated equlbrum, then t must TTH 2 3 TABLE X The CPCE of the TPMPG H3 T3 H2 T2 T2 H2 H T