Common Belief Foundations of Global Games

Transcription

1 Common Belef Foundatons of Global Games Stephen Morrs Prnceton Unversty Hyun Song Shn Bank for Internatonal Settlements November 2015 Muhamet Yldz M.I.T. Abstract We study coordnaton games under general type spaces. We characterze ratonalzable actons n terms of the propertes of the belef herarches and show that there s a unque ratonalzable acton played whenever there s approxmate common certanty of rank belefs, defned as the probablty the players assgn to ther payoff parameters beng hgher than ther opponents. We argue that ths s the drvng force behnd selecton results for the specfc type spaces n the global games lterature. 1 Introducton Complete nformaton games often have many equlbra. Even when they have a sngle equlbrum, they often have many actons that are ratonalzable, and are therefore consstent wth common certanty of ratonalty. The nablty of theory to make a predcton s problematc for economc applcatons of game theory. Carlsson and van Damme 1993 suggested a natural perturbaton of complete nformaton that gves rse to a unque ratonalzable acton for each player. They ntroduced the dea of global games where any payoffs of the game are possble and each player observes the true payoffs of the game wth a small amount of nose. They showed for the case of two player two acton games that as the nose about payoffs becomes small, there s a unque equlbrum; moreover, the perturbaton selects a partcular equlbrum the rsk-domnant one of the underlyng game. Ths result has snce been generalzed n a number of Ths paper ncorporates materal from a workng paper of the same ttle crculated n 2007 Morrs and Shn The frst two authors are grateful to the Natonal Scence Foundaton Grant SES for fundng ths research. We are grateful for comments from edtors, anonymous referees, and semnar partcpants at Columba, Iowa State, Northwestern, and UCLA on ths teraton of the project. We thank Anton Tsoy for detaled comments. The vews expressed n ths paper reflect those of the authors and not necessarly those of the Bank for Internatonal Settlements. 1

2 2 drectons and wdely used n applcatons. 1 When the global game approach can be appled to more general games, t can be used to derve unque predctons n settngs where the underlyng complete nformaton game has multple equlbra, makng t possble to carry out comparatve statc and polcy analyss. It has been nformally argued that multplcty partly reles on the unrealstc "complete nformaton" assumpton, and the natural perturbaton underlyng global games captures the more realstc case. However, the global game selecton result uses a partcular form of perturbaton away from "complete nformaton." Complete nformaton entals the assumpton that a player s certan of the payoffs of the game, certan that other players are certan, and so on. Wensten and Yldz 2007 consder more general perturbatons, sayng that a stuaton s close to a complete nformaton game f players are almost certan that payoffs are close to those complete nformaton game payoffs, almost certan that other players are almost certan that payoffs are close to those payoffs, and so on. Formally, they consder closeness n the product topology on the unversal belef space. They show that for any acton whch s ratonalzable for a player n a complete nformaton game, there exsts a nearby type of that player n the product topology for whom ths s the unque ratonalzable acton. Thus by consderng a rcher but also ntutve class of perturbatons, they replcate the global game unqueness result but reverse the selecton result. In ths paper, we dentfy the drvng force behnd global game unqueness and selecton results. In partcular, we do not want to take lterally the mplct assumpton n global games that there s common certanty among the players of a technology whch generates condtonally ndependent nosy sgnals observed by the players. Rather, we want to argue that global game perturbatons are a metaphor, or a convenent modellng devce, for a more general ntutve class of relaxatons of common certanty. We want to characterze and analyze the key property of that more general class, whch must also be more restrctve than the product topology perturbatons of Wensten and Yldz Our baselne analyss s carred out for a two player, two acton game. Each player must decde whether to "nvest" or "not nvest". Payoffs are gven by the followng matrx: nvest not nvest nvest x 1, x 2 x 1 1, 0 not nvest 0, x 2 1 0, 0 1 Each player knows hs own payoff parameter, or return to nvestment, x but may not know the other player s payoff parameter. There are strategc complementartes, because a player has a loss of 1 f the other player does not nvest. If x 1 and x 2 are both n the nterval [0, 1], then there are multple equlbra, 1 Morrs and Shn 1998 analyzed a global game wth a contnuum of players makng bnary choces, and ths case has been studed n a number of later applcatons. See Morrs and Shn 2003b for an early survey of some theory and applcatons of global games. Frankel, Morrs, and Pauzner 2003 study global game selecton n general games wth strategc complementartes.

3 3 both nvest and both not nvest, under complete nformaton. In the symmetrc case, wth x 1 = x 2 = x, the rsk-domnant equlbrum n ths game s the equlbrum that has each player choose a best response to a 50/50 dstrbuton over hs opponent s acton. Thus both nvest s the rsk-domnant equlbrum f x > 1 2. If x 1 and x 2 are both n the nterval [0, 1], both actons reman ratonalzable for player f there s approxmate common certanty of payoffs. 2 Ths s a well known suff cent condton for multple ratonalzable actons, gong back to Monderer and Samet But t s a strong condton. A key concept to understand unqueness n ths settng s a player s "rank belef" that s, hs belef about whether he has the hgher payoff parameter so he has rank 1 or the lower payoff parameter so he has rank 2. A player has the unform rank belef f he assgns probablty 1 2 to each rank. If there s common certanty of unform rank belefs, then a player has a unque ratonalzable acton. In partcular, acton nvest s unquely ratonalzable for a player f t s rsk-domnant. A rough argument for ths s as follows. Let x be the smallest payoff parameter such that nvest s unquely ratonalzable whenever there s common certanty of unform rank belefs for a player wth payoff parameter greater than or equal to x. If x were strctly greater than 1 2, a player wth payoff parameter close to x and common certanty of unform rank belefs would assgn probablty close to 1 2 to hs opponent nvestng and would therefore have a strct ncentve to nvest. Thus we would have a contradcton. Thus x must be less than or equal to 1 2. Ths proves the result n the standard case wth one dmensonal types. Under some addtonal contnuty assumptons, essentally ths argument goes though for general stuatons wthout reference to nosy sgnals or one dmensonal type spaces usng the propertes of the belef herarchy at hand. Ths s how our result provdes a prmtve common belef foundaton for global game selecton. One of our man results wll be a formalzaton of an approprate weakenng of the above suff cent condton: approxmate common certanty of approxmately unform rank belefs. And, more generally, the common belef foundatons results focus attenton on the propertes of hgher-order belefs that matter for global game results rather than the condtonally ndependent nosy sgnal story that generates them. Our characterzaton of ratonalzablty n terms of hgher-order belefs provdes a conceptual foundaton for global games approach, makng explct certan ntutons present n the exstng lterature, as well as extendng the approach to rcher and possbly more nterestng type spaces. The applcatons n global games lterature have been manly confned to undmensonal type spaces wth ether an approxmately unform pror or a monotone supermodular structure e.g. players sgnals have a common shock and an dosyncratc shock both dstrbuted normally. In such type spaces t suff ces to focus on equlbra wth 2 There s approxmate common certanty of an event f both beleve t wth probablty close to 1, both beleve wth probablty close to 1 that both beleve t wth probablty close to 1, and so on.

4 4 cutoff strateges. One can study those equlbra drectly wthout needng our machnery here although the rank belefs clearly play an mportant role n those equlbra. Unfortunately, such a drect approach does not extend to hgher-dmensonal types spaces, where monotone equlbra would have hgher-dmensonal boundares, and to the type spaces wthout monotone supermodular structure, so that monotone equlbra may not even exst. One can stll apply our suff cent condtons to those models and fnd out whether the rsk-domnance gves the unque ratonalzable soluton. We llustrate ths pont n two examples modfyng the standard normalty assumptons. In one of them, we smply alter the dstrbuton of varables to Pareto dstrbutons whch often arse when there s a model uncertanty. In ths example, monotoncty propertes fal but our results stll lead to a sharp dchotomy: the rsk-domnant acton s unquely ratonalzable when the dosyncratc shock has a thnner tal, and both actons are ratonalzable when the common shock has a thnner tal. Ths sharply contrasts wth the normal example, where one needs a pecular rate of convergence, on whch many exstng applcatons are bult. In the second example, we smply ntroduce uncertanty about varances, so that a player does not know how much the other player knows, resultng n a two-dmensonal type space. We cannot solve for equlbra n ths game, but we can easly extend the conclusons from the standard exercse to ths case, by applyng our results. We present our results n the context of the smple strategc settng descrbed above and focussng on unform rank belefs. We do ths n order to focus on the common belef foundatons rather than the detals of the strategc settng. We can then descrbe how our results can be mapped back to more general settngs. Our results extend to the case where there s approxmate common certanty of the approxmate rank belef p, where p 1 2. The selected acton wll then depend on the rank belef p. In dong so, we provde common belef foundatons for the results of Izmalkov and Yldz Whle ths case of p 1 2 has typcally been motvated by non-common prors, we wll show that t s possble to have approxmate common certanty of the rank belef p for any p 1 2, even under the common pror assumpton. We state our results for symmetrc two player two acton games, but they extend to symmetrc many player two acton games. The common belef characterzaton of ratonalzablty becomes more complex, because the relevant belef operators depend on belefs about more than one event. However, the unqueness and multplcty results extend cleanly. Common certanty of rank belefs corresponds n the N player case to always assgnng probablty 1 N to exactly n players havng hgher payoff states, for each n = 0, 1,..., N 1. Morrs and Shn 2003b dubbed ths the Laplacan assumpton, and n ths sense we are formalzng a known ntuton about global game selecton. Ths selecton s key n the vast majorty of appled work usng global games. However, the global game selecton results reported n the paper rely on bnary actons and symmetrc payoffs; n partcular, t s requred for common certanty of approxmately unform rank belefs to be the relevant suff cent condton for unque ratonalzable actons. Hgher-order belef foundatons can be

5 5 provded for many acton and asymmetrc payoff global game results, but they are qualtatvely dfferent from those provded for symmetrc two acton games here. We present the basc defntons and prelmnary results n Secton 2. In Secton 3, we present our characterzaton of ratonalzablty n terms of hgher-order belefs. In Secton 4, we report suff cent condtons for multple ratonalzable outcomes based on approxmate common certanty of payoffs and for unque ratonalzable outcomes based on approxmate common certanty of approxmately unform rank belefs. In Secton 5, we report the proof of the man result as well as an example showng that a key techncal assumpton s not superfluous. In Secton 6, we dscuss extensons to more general strategc settngs and the relaton to the global games lterature. 2 Model There are two players, 1 and 2. Let T 1 and T 2 be the sets of types for players 1 and 2, respectvely. A mappng x : T R descrbes a payoff parameter of nterest to player, and a mappng π : T T j descrbes player s belefs about the other player. We assume that T s endowed wth a metrzable topology and Borel sgma-algebra, under whch the mappngs x and π are measurable. We make a couple of mnmal contnuty assumptons: x and π are contnuous, 3 and the pre-mage x 1 [a, b] of every compact nterval [a, b] s sequentally compact. Ths type space can be arbtrarly rch, and n partcular can encode any belefs and hgher-order belefs, and thus our results apply f the type space s the unversal prvate value belef space of Mertens and Zamr We start by descrbng the belef and common belef operators, as n Monderer and Samet The state space s T = T 1 T 2. An event s a subset of T. An event s smple f E = E 1 E 2 where E T. For our game theoretc analyss, we wll be nterested n smple events and we restrct attenton to such events n the analyss that follows. For any such smple event E, we wrte E 1 and E 2 for the projectons of E onto T 1 and T 2, respectvely. Now, for probablty p, wrte B p beleves E wth probablty at least p : B p E = {t 1, t 2 t E and π E j t p }. E for the set of states where player 3 We use the standard defnton for the contnuty of the belefs: as t,m t, π t,m converges to π t under the weak topology on the probablty dstrbutons as n the "convergence n dstrbuton". That s, the expectaton of any contnuous and bounded functon under π t,m converges to ts expectaton under π t. Our results are also vald under an alternatve noton of contnuty: π T j s contnuous for every measurable subset T j T j. 4 Mertens and Zamr 1985 constructed a space that encodes all belefs and hgher-order belefs about a common state space. We mantan the assumpton that the state space s a par of payoff types of the players, where each player knows hs own payoff type. It s a smple adapton to the classc constructon to buld n ths restrcton, see for example Hefetz and Neeman The classcal constructon assumes a compact state space. We need to allow the state space to be R 2 but nstead mpose sequental compactness of types wth payoffs wthn a compact nterval.

6 6 For a par of probabltes p 1, p 2, say that event E s p 1, p 2 -beleved f each player beleves event E wth probablty at least p. Wrtng B p 1,p 2 E for the set of states where E s p 1, p 2 -beleved, we have: B p 1,p 2 E = B p 1 1 E Bp 2 2 E. Say that there s common p 1, p 2 -belef of event E f t s p 1, p 2 -beleved, t s p 1, p 2 -beleved that t s p 1, p 2 -beleved, and so on. We wrte C p 1,p 2 E for set of states at whch E s common p 1, p 2 -belef. Thus C p 1,p 2 E = n 1 [B p 1,p 2 ] n E. An event s p 1, p 2 -evdent f t s p 1, p 2 -beleved whenever t s true. Generalzng a characterzaton of common knowledge by Aumann 1976, Monderer and Samet 1989 characterzes common p 1, p 2 - belefs through p 1, p 2 -evdent events. Ths characterzaton s stated along wth other useful facts next see the proof of Lemma 2 below for a proof. Lemma 1 Monderer and Samet 1989 The followng are true for all smple events E: [ ] 1. C p 1,p 2 E B p 1,p 2 n+1 [ ] E B p 1,p 2 n p E B 1,p 2 E E. 2. C p 1,p 2 E s the largest p 1, p 2 -evdent event F wth F E. 3. If E s closed, so are C p 1,p 2 E and B p 1,p 2 E. In our formulaton, we make the belef operators type dependent, and the above propertes generalze mmedately to ths case. For any f : T R, we say that event E s f -beleved by type t of player f he beleves t wth probablty at least f t : B f E = {t 1, t 2 t E and π E j t f t }. Clearly we can make rcher statements about belefs and hgher-order belefs n ths language. We wll contnue to wrte B p E for the orgnal p -belef operator, where p s now understood as the constant functon of types takng the value p. Note that we allow f to take values below 0 and above 1. Ths conventon gves a specal role to the events B f and B f T, snce a player always beleves an event wth probablty at least 0 and never beleves an event wth probablty greater than 1. Thus B f = {t 1, t 2 f t 0} B f T = {t 1, t 2 f t 1}. These operators behave just lke the type-ndependent ones. In partcular, wrtng B f 1,f 2 states where E s f 1, f 2 -beleved, we have: B f 1,f 2 E = B f 1 1 E Bf 2 2 E. E for the set of

7 7 Say that there s common f 1, f 2 -belef of event E f t s f 1, f 2 -beleved, t s f 1, f 2 -beleved that t s f 1, f 2 -beleved, and so on. We wrte C f 1,f 2 E for set of states at whch E s common f 1, f 2 -belef: [ C f 1,f 2 E = n 1 B f 1,f 2 ] n E An event s f 1, f 2 -evdent f t s f 1, f 2 -beleved whenever t s true. Lemma 1 generalzes to our case as follows. Lemma 2 The followng are true for all smple events E. [ 1. C f 1,f 2 E B f 1,f 2 ] n+1 E [ B f 1,f 2 2. C f 1,f 2 E s the largest f 1, f 2 -evdent event F wth F E. ] n f E B 1,f 2 E E. 3. Assume f 1 and f 2 are contnuous. If E s closed, so are C f 1,f 2 E and B f 1,f 2 E. A proof of ths Lemma n ths context and notaton followng standard arguments s reported n the Appendx. In the baselne model, we consder the followng acton space and payoff functon: nvest not nvest nvest x 1, x 2 x 1 1, 0 not nvest 0, x 2 1 0, 0 Note that any two-player two-acton game wth a pure-strategy Nash equlbrum s best-response equvalent to such a game, so that payoffs can be normalzed nto payoffs of ths form wthout changng strategc behavor. Player knows hs own payoff parameter x t but does not necessarly know the other player s payoff parameter x j t j. Moreover, he gets a return x t f he nvests but faces a penalty 1 f the other player does not nvest. Hence, he only wants to nvest f the probablty he assgns to hs opponent nvestng s at least 1 x t. Remark 1 In our general setup and n the above payoff functon, we have a "prvate value" envronment, n whch a player knows hs payoff functon at the nterm stage. 5 envronment s often used n global games lterature by specfyng the payoff functon as nvest not nvest nvest θ, θ θ 1, 0 not nvest 0, θ 1 0, 0 In contrast, a "common-value" 5 See Morrs and Shn 2005 for some theory and Argenzano 2008 for an applcaton of prvate value global games.

8 8 where the payoff parameter θ s unknown. Ths envronment s also ncluded n our model by takng x t = E [θ t ]. In both the common-value formulaton and the prvate-value formulaton wth x t = E [θ t ] a player nvests f and only f the probablty he assgns to hs opponent nvestng s at least 1 x t. Ths bestresponse equvalence leads to strategc equvalence under many soluton concepts, ncludng ratonalzablty and Bayesan Nash equlbrum. We prefer the prvate value formulaton because t leads to a more drect and clearer analyss. Also, n applcatons n whch one studes the lmt propertes of the soluton as n the examples at the end of Secton 4, the prvate value formulaton avods dff cultes related to the convergence of expectatons E [θ t ]. 6 Throughout the paper, we wll use ratonalzablty as the soluton concept. We defne ratonalzablty n the context of ths game as follows t corresponds to standard general defntons. Say that an acton s k + 1th-level ratonalzable f t s a best response to kth-level ratonalzable play of hs opponent; and say that any acton s 0th-level ratonalzable. Wrte R k for the set of types of player for whom acton nvest s kth-level ratonalzable and let R 0 = T. 3 Common-Belef Characterzaton of Ratonalzablty In ths Secton, we provde a useful characterzaton of ratonalzablty n terms of hgher-order belefs about the payoffs. We start wth carefully descrbng the set R 1 of types for whom nvest s 1st-level ratonalzable n terms of our type-dependent belef operators. On the one hand, for any type t of player, acton nvest s 1st-level ratonalzable for t f and only f x t 0; ths s n response to the belef that hs opponent nvests wth probablty 1. On the other hand, snce player always assgns probablty 1 to T, he assgns probablty at least 1 x t to T f and only f x t 0. Thus, R 1 = B 1 x T. That s, the types for whch nvest s frst-level ratonalzable concde wth those n B 1 x T. Now, acton nvest s 2nd-level ratonalzable for type t of player f, n addton, he assgns probablty at least 1 x t to x j t j 0; thus R 2 = B 1 x B 1 x 1,1 x 2 T. 6 An addtonal smplfyng assumpton here s that payoffs are addtvely separable between a component that depends on the opponent s acton and a component that depends on an unknown payoff state. Ths addtve separablty allows for a tghter descrpton of the connecton. Extensons are possble here also, but are messy and nvolve tedous contnuty arguments. Also, lke much of the appled lterature, we focus on a game whch s symmetrc across players. However, global game results go through wth asymmetrc games. One can state analogous hgher-order belef propertes drvng global game results relatng to translaton nvarance but they are not as clean. These ssues are dscussed n Morrs and Shn 2007.

9 9 More generally, acton nvest s k + 1th-level ratonalzable for a type t f he 1 x t -beleves that T s kth-order 1 x 1, 1 x 2 -beleved: R k+1 = B 1 x [B 1 x 1,1 x 2 ] k T. Acton nvest s ratonalzable f t s kth level ratonalzable for all k. Thus, acton nvest s ratonalzable for both players exactly f T s common 1 x 1, 1 x 2 -beleved: R = C 1 x 1,1 x 2 T. 2 By a symmetrc argument, acton not nvest s ratonalzable exactly f T s common x 1, x 2 -belef. The next result states ths characterzaton; recall that C 1 x 1,1 x 2 T = C 1 x 1,1 x 2 1 T C 1 x 1,1 x 2 2 T. Proposton 1 Acton nvest s ratonalzable for type t f and only f t C 1 x 1,1 x 2 T ; acton not nvest s ratonalzable for type t f and only f t C x 1,x 2 T. It s useful to note that, snce R 1 = B 1 x T, C 1 x 1,1 x 2 T corresponds to a hgh common belef n the event that acton nvest s ratonal. Hence, each part of the proposton states that an acton a s ratonalzable for a type t f and only f t assgns suff cently hgh probablty on a suff cently hgh common belef n the event that a s ratonal. That s, he fnds t suff cently lkely that the acton s ratonal, fnds t suff cently lkely that the other player fnds t suff cently lkely that the acton s ratonal,..., ad nfntum. The key nnovaton that yelds such a smple characterzaton s allowng the threshold for the suff cency to depend on the payoffs of the types throughout. 4 Rsk-Domnant Selecton and Multplcty Our focus n ths paper s on when both actons are ratonalzable for both players and when one acton s unquely ratonalzable for both players wthout loss of generalty, we focus on unqueness of acton nvest. Buldng on the characterzaton n the prevous secton, we present ntutve suff cent condtons for each case. Our frst result characterzes the cases wth multplcty and unqueness, as an mmedate corollary to the characterzaton n the prevous secton. Corollary 1 Both actons are ratonalzable for a type t f and only f t C x 1,x 2 T C 1 x 1,1 x 2 T.

10 10 Invest s the unquely ratonalzable acton for a type t f and only f t C 1 x 1,1 x 2 T \C x 1,x 2 T. The frst part characterzes the cases wth multple ratonalzable solutons. It states that both actons are ratonalzable f and only f there s suff cently hgh common belef that both actons are ratonal for both players. The second part characterzes the cases n whch nvest s the only ratonalzable soluton. It states that nvest s unquely ratonalzable f and only f there s suff cently hgh common belef n ratonalty of nvest.e., t C 1 x 1,1 x 2 T but there s not suff cently hgh common belef n ratonalty of not nvest.e., t C x 1,x 2 T. Once agan, we obtan such smple and straghtforward characterzatons by makng the threshold for suff cency type-dependent. Whle such characterzatons are useful conceptually, they may not be of great practcal use. In the rest of ths Secton, we provde smple tractable suff cent condtons for multplcty and unqueness. We start wth a result for multplcty, whch states that both actons are ratonalzable whenever there s approxmate common certanty that payoffs support multple strct equlbra. Proposton 2 For any ε [0, 1/2], both actons are ratonalzable on C 1 ε,1 ε M ε where M ε = {t 1, t 2 ε x t 1 ε for both } ; ndeed, there exst Bayesan Nash equlbra 7 s and s such that s t = nvest, nvest and s t = not nvest, not nvest t C 1 ε,1 ε M ε. Proof. We wll construct an equlbrum s as n the Proposton; constructon of s s dentcal. We construct an auxlary game by alterng the payoffs of some types as follows. We set x t = 2 whenever x t > 1 and set x t = 1 whenever x t < 0, so that domnant actons reman domnant. For types n C 1 ε,1 ε M ε, we assgn payoff 1 for nvest and 0 for not nvest, makng nvest strctly domnant on C 1 ε,1 ε M ε. The auxlary game wth bounded supermodular payoffs satsfes the suff cent condtons of van Zandt 2010 and therefore has a Bayesan Nash equlbrum s, n whch the types n C 1 ε,1 ε M ε must play nvest. But s s also a Bayesan Nash equlbrum n the orgnal game, as we show next. Indeed, for each type t C 1 ε,1 ε M ε, equlbrum acton s t = nvest s a best response to s because x t ε t M ε by Lemma 1 and type t assgns at least probablty 1 ε on C 1 ε,1 ε j M ε by Lemma 1 where s j takes the value of nvest throughout. All the remanng types play a best response because ther best responses are dentcal n the orgnal and the auxlary games. 7 By a Bayesan Nash equlbrum, we mean a strategy profle s : T {nvest, not nvest} 2 such that s t s a best response to s for type t for each t.

11 11 Note that, f the payoffs were known and as n M ε, then both nvest, nvest and not nvest, not nvest would have been 1 ε-domnant Nash equlbra. Proposton 2 then states that both actons are played n an equlbrum and hence ratonalzable whenever there s common 1 ε, 1 ε-belef that both actons would have been 1 ε-domnant Nash equlbrum under complete nformaton. Ths follows a key observaton n the robustness lterature, gong back to Monderer and Samet 1989 whch states that any p-domnant equlbrum of a game can be extended to a larger type space n whch the orgnal game s p-evdent. In an earler workng-paper verson, we have used the standard technques n the robustness lterature to prove multplcty of ratonalzable actons. 8 Such technques can also be used to prove exstence of multple equlbra under addtonal techncal condtons on the type space. Here, we prove exstence of multple equlbra wthout addtonal assumptons by usng an equlbrum exstence result for supermodular games. Note that, n our proof, we specfy the equlbrum actons only on C 1 ε,1 ε M ε and the domnance regons. The equlbrum strateges can be hghly complex outsde of these regons n the general setup consdered here. In contrast, smple cutoff strateges suff ce n the type spaces consdered n global games lterature. We next turn to establshng suff cent condtons under whch there s a unque ratonalzable acton. Our key concept wll be approxmate unformty of "rank belefs", whch we now defne. We wll wrte r t for the probablty that a player assgns to hs payoff parameter beng greater than or equal to that of the other player, and r t for the probablty that t s strctly greater. We wll refer to these expressons as "rank belefs" as they reflect the player s belef about hs rank f the players are ordered by the payoff parameter. Formally, we defne r t = π {t j x j t j x t } t, r t = π {t j x j t j < x t } t, and r t = r t + r t /2. We refer to r t, r t, and r t as the upper rank belef, the lower rank belef and the rank belef of type t, respectvely. When the dstrbuton of x j s atomless accordng to type t, all these belefs concde: r t = r t = r t. We defne upper and lower rank belefs separately n order to deal wth pont masses. Such pont masses may arse, for example, under complete nformaton. Rank belef of a type t s unform f he fnds t equally lkely that ether player s value s hgher. Formally, we say that rank belef of a type t s ε-unform f 1 2 ε r t r t ε. 8 The technque nvolves settng the actons on C 1 ε,1 ε M ε as desred and allowng the remanng types play a best response.

12 12 We wrte URB ε for the set of type profles t 1, t 2 where both players have ε-unform rank belefs: { URB ε = t 1, t ε r t r t 1 } + ε for each. 2 We sometmes nformally say that rank belefs are approxmately unform to mean that they are ε-unform for some suff cently small ε. Our second concept s a strct verson of rsk-domnance. We say that acton nvest s ε-strctly rskdomnant for t f x t > ε. We wrte SRD ε for the set of type profles for whch nvest s ε-strctly rsk-domnant for each player, so { SRD ε = t 1, t 2 x t > 1 } 2 + ε for each. We say that acton nvest s strctly rsk-domnant f nvest s ε-strctly rsk-domnant for ε = 0. Proposton 3 For any ε 0, assume that C 1 ε,1 ε URB ε s closed. Then, nvest s the unquely ratonalzable acton for both players f t s 2ε-strctly rsk-domnant for both players and there s common 1 ε-belef of ε-unform rank belefs,.e., f t 1, t 2 SRD 2ε C 1 ε,1 ε URB ε. Proposton 3 provdes a useful suff cent condton for unqueness, dentfyng common features of the unqueness results n the global games lterature. It states that nvest s unquely ratonalzable f t s strctly rsk-domnant and there s approxmate common certanty of approxmately unform rank belefs. Observe that our result establshes ths result wthout explctly assumng some of the crtcal features of global games, such as exstence of domnance regons. Moreover, t allows arbtrary type spaces wth mnmal contnuty and compactness propertes. In the next secton, we wll present the proof of our result and further dscuss the assumpton that the set C 1 ε,1 ε URB ε s closed. In the rest of ths secton, we wll present some applcatons of our result. We start wth presentng a weaker suff cent condton: Corollary 2 For any ε > 0, let { DURB ε = t 1, t 2 x t [0, 1] 1 2 ε r t r t 1 } 2 + ε be the event that each player has ether ε-unform belefs or a domnant acton. Assume that C 1 ε,1 ε DURB ε s closed. Then, nvest s unquely ratonalzable for both players on SRD 2ε C 1 ε,1 ε DURB ε.

13 13 That s, nvest s unquely ratonalzable whenever t s strctly rsk-domnant for both and there s approxmate common certanty that each player has ether approxmately unform rank belefs or a domnant acton. Note that ths suff cent condton merges the unform rank belefs and strct rsk-domnance propertes nto a sngle condton. We smply drop the restrcton on rank belefs when a player has a domnant strategy. Ths wll clearly not matter for strategc results. The corollary mmedately follows from applyng Proposton 3 to an augmented game n whch the belefs of the types wth a domnant acton s modfed so that they have unform rank belefs by addng new types as necessary. Our results have mmedate applcatons n low dmensonal type spaces, whch are often used n applcatons. Here, we wll llustrate a couple of them. In the frst two examples, the type spaces wll be undmensonal,.e., T 1 = T 2 = R, and n all of them the payoff functons wll be lnear,.e., x = t + y, where y R s the ex-ante mean of the payoffs. Undmensonal, Lnear-Normal Model In the prevous framework, we specfy t = θ + σe θ = τη where σ and τ postve real numbers and η, e 1, and e 2 are ndependent standard normal random varables. Here, each player s payoff has a common component θ and an dosyncratc component e. Usng the standard formulas, one can compute that the rank belef of a player wth type t s σ r σ,τ t = Φ 2 σ 2 + 2τ 2 σ 2 + τ 2 t, where Φ s the cumulatve dstrbuton functon of standard normals; recall that player assgns probablty r σ,τ t to the event that the other player s type s lower than hs own. 9 What can we say about σ hgher-order belefs n ths case? Observe that, as σ 0 and τ 0, 2 σ 2 +2τ 2 σ 2 +τ 2 converges σ σ σ to f and converges to 0 f 0. Thus f, we obtan approxmate common τ 2 τ 2 τ 2 σ certanty of payoffs and we have lmt multplcty. If 0, we do have pontwse convergence τ 2 to unform rank belefs, so that r σ,τ t 1 2 for each t. But ths convergence s not unform n the tals, so that for any σ, τ, r σ,τ t 1 as t and r σ,τ t 0 as t. Thus there s never 9 Ths known formula for rank-belefs has been extensvely used n the exstng lterature, e.g., Morrs and Shn 2001 and Morrs and Shn 2003a.

14 14 approxmate common certanty of approxmately unform rank belefs. 10 However, as establshed n Corollary 2, t s enough that all types wthout a domnant actons have approxmately unform rank belefs. Snce r σ,τ t approaches to 1/2 unformly over [ ȳ, 1 ȳ], when σ τ 2 s satsfed, and rsk domnance arses as the unque ratonalzable outcome. s small, ths condton Undmensonal Lnear Model wth Fat-taled Dstrbutons In the prevous model assume nstead that the random varables η, e 1, and e 2 have regularly-varyng tals where the dosyncratc terms e 1 and e 2 have tal ndex α and the common term η has tal ndex β. That s, the tal of e s approxmately proportonal to e α and the tal of η s approxmately proportonal to η β. Such models arse often when the players face model uncertanty. When α > β, the dosyncratc term has a thnner tal, and the player attrbutes all large devatons from the mean to a large common shock, assgnng nearly probablty 1/2 to the event that the other player s type s below hs own. In that case, r σ,τ t s approxmately 1/2 near the tal. Indeed, n an earler verson of ths paper, we have shown that we have common knowledge of approxmate unform rank belefs whenever α > β + 1 and σ/τ s suff cently small. In that case, Proposton 3 establshes that whenever an acton s strctly rsk-domnant, t s the unque ratonalzable acton. Conversely, f β > α, then the common term has thnner tals, and the players attrbute all large devatons from the mean to dosyncratc shocks n ther sgnals, belevng that the other player s type s near the ex-ante mean. Consequently, we have approxmate common certanty of the payoffs, and Proposton 2 establshes multple equlbrum outcomes whenever the payoffs support multple strct equlbra under complete nformaton. Thus we fnd that the fat-taled model delvers sharper suff cent condtons for unqueness than the well known normal model does. Lnear-Normal Model wth Precson Uncertanty In the undmensonal lnear normal model, assume that e N 0, v 2 where the varance v [v, v] s prvately known by player, and v 1 and v 2 are strctly postve and ndependently dstrbuted wth cumulatve dstrbuton functons F 1 and F 2, respectvely, for some v > v > 0. Hence, T 1 = T 2 = R [v, v]. Then, σ r σ,τ t, v = E Φ 2 v σ 2 σ 2 vj vj 2/v2 τ 2 2 v 2 + τ 2 t t, v. Observe that, as σ/τ 2 approaches 0, the expresson n the square root goes to 0 for each v 1, v 2, as n the case wth known varances. Snce Φ s bounded, ths mples that r σ,τ t, v also goes to Φ 0 = 1/2, yeldng approxmately unform rank belefs for each t, v. Once agan, the convergence 10 For any ε < 1/4 and σ, τ, snce lm t r σ,τ t = 1, f the closed set C 1 ε,1 ε URB ε were not empty, then t would have a maxmum t. But snce t URB ε, type t would assgn at most probablty r σ,τ t 1/2 + ε < 1 ε, contradctng that t C 1 ε,1 ε URB ε.

15 15 s not unform, but ths s not an ssue snce the varances v 1 and v 2 are bounded: for suff cently small σ/τ 2, the condton n Corollary 2 s satsfed, and we have rsk-domnance as the unque ratonalzable outcome. Interestngly, when v s not bounded, the rank belefs can vary arbtrarly regardless of the sze of σ/τ 2. Indeed, lm r 1 σ,τ t, v = E Φ t v σ 2 vj 2 + τ 2 Hence, when v s large and σ and τ are small, r σ,τ t, v s approxmately 1 for t > 0 and 0 for t < 0. Nevertheless, one can stll obtan common belef of DURB ε at any gven type when σ/τ 2 s suff cently small. To see ths, take any ε > 0 and any t 1, v 1, t 2, v 2. Set v ε = max { F 1 } 1 ε, v for each, and note that the smple event E ε = R 0, v 1 ε] R 0, v 2 ε] s 1 ε, 1 ε-evdent. Moreover, as n the case of bounded varances, when σ/τ 2 s suff cently small, we have E ε DURB ε. Thus, E ε C 1 ε,1 ε DURB ε. The exercse n the undmensonal lnear-normal model s a standard exercse n the global games lterature Morrs and Shn 2001 and Morrs and Shn 2003a. In ths case, one can explctly compute rank belefs and see how they evolve as the dstrbutons shrnk to zero at dfferent rates. However, the other two models hghlght the fact that these results are specal. The undmensonal model wth fattaled dstrbutons dentfes dstnct propertes that gve unqueness and multplcty dependng on the tal propertes of the dstrbutons. In the model wth precson uncertanty, one can easly compute the rank belefs, but that does not help n analyzng the equlbra per se, because the type space s two dmensonal and the event that players nvest s a complex event dependng on both payoff t and varance v and we cannot compute t. Nevertheless, we can stll apply our result and extend the conclusons from the standard exercse to ths case. t. 5 Proof and a Counterexample In ths Secton, we wll present the proof of Proposton 3. A key techncal assumpton n ths Proposton s that common belef of unform rank belefs s a closed set. Ths closure assumpton holds for free n three cases: 1. there s common knowledge that belefs are approxmately unform,.e., URB ε = T ; 2. the upper and lower rank belefs are contnuous by Lemma 1;

16 16 3. π E j t s contnuous n t for each E j, and π {t } t = 0 for each t. 11 We frst establsh by example that ths assumpton s not superfluous, llustratng dff cultes one would face even under the common certanty of unform rank belefs. Example 1 Consder the followng symmetrc type space. For some ˆt 0, 1/2 and y = 1/2, set T = R { [ U t + t ˆt /2, t + ˆt t /2 ] f t ˆt, ˆt π t = δ t x t = t + y, otherwse where UX s the unform dstrbuton on X and δ t assgns probablty 1 on t j = t. 12 The range of t j s plotted as a functon of t n Fgure 1. Ths type space satsfes our general assumptons: the belefs and payoffs are contnuous functons of types, and x 1 [a, b] s compact for every nterval [a, b]. Nonetheless, the rank belefs are dscontnuous: { 1/2 f t ˆt, ˆt r t = 1 r t = Clearly, the rank belefs are unform on ˆt, ˆt : 1 otherwse. URB ε = ˆt, ˆt 2 ε [0, 1/2. Moreover, URB ε s an evdent event: for any t 1, t 2 ˆt, ˆt 2, each player assgns probablty 1 on ˆt, ˆt 2, as vvdly demonstrated n Fgure 1. Therefore, C 1,1 URB ε = URB ε = ˆt, ˆt 2. That s, we have common certanty of unform rank belefs throughout ˆt, ˆt 2. Note however that 0 < x t < 1 on ˆt, ˆt. Hence, for ε = 0, Proposton 2 mples that we have multplcty over C 1,1 URB ε, contradctng the concluson of Proposton 3. Example 1 shows that even assumng common certanty of unform rank belefs would not be enough wthout the closure assumpton. Example 1 also shows that we cannot dspense wth our general techncal 11 In that case, the upper and lower rank belefs concde, and they are contnuous. Ths assumpton holds when π t has a bounded densty f t that s contnuous n t. 12 We could obtan a smlar counterxample wth a common pror by puttng unform dstrbuton on the convex hull of { ˆt, ˆt, ˆt, ˆt, t, t, t, t } for some small but postve t and also on the dagonal outsde of that set. We would consder URB ε for ε = t/ 2ˆt.

17 Fgure 1: The range of t j as a functon of t for ˆt = 1/4. assumpton that the premage x 1 [a, b] of any compact nterval [a, b] s compact. Indeed, one can take T = ˆt, ˆt 2, gvng common certanty that belefs are unform, wthout alterng the set of multple solutons. Note that on the boundary of the event URB ε = ˆt, ˆt 2, we must have r ˆt = 1 and r ˆt = 0, so that ˆt URB ε, renderng C 1,1 URB ε = URB ε not closed. As we wll show momentarly, ths s what leads to multplcty wthn C 1,1 URB ε. Indeed, as we show below, when C 1 ε,1 ε URB ε s closed, the set of payoffs are unbounded on C 1 ε,1 ε URB ε, volatng the hypothess of Proposton 2, and allowng contagon from the domnance regons ncluded wthn C 1 ε,1 ε URB ε. Note that the suff cent condton for unqueness n Proposton 3 s local, n that t only refers to the herarchy of the belefs of the type at hand, wthout makng global assumptons about the type space. In contrast, the exstng works on global games usually make structural assumptons on the entre type space, and often rely on the extremal equlbra. As llustrated n Example 1, a subset of types, such as ˆt, ˆt 2, can form a belef-closed subspace where the herarches do not have any meanngful connecton to the rest of the type space. Such herarches may not have substantal amount of nformaton about the types outsde the subspace. Therefore, our local condtons do not lead to global structural condtons n general. It turns out that when C 1 ε,1 ε URB ε s a topologcally closed set, we can use rank belefs to deduce enough structure wthn the set C 1 ε,1 ε URB ε to obtan the desred result, as we do next. Fx a closed p-evdent event E and defne two cutoffs: x E = nf {z z = x t r t for some and t E } x E = sup {z z = x t r t for some and t E }.

18 18 Here, x E s the lowest value for whch any type of ether player wthn E has that value and has a lower lower-rank belef; and x E s the hghest value for whch any type of any player wthn E has that value and a hgher upper-rank belef. The next Lemma establshes that nvest cannot be ratonalzable when the value s lower than x E, and not nvest cannot be ratonalzable when the value exceeds x E. Lemma 3 Let E be a closed p-evdent event. Then, nvest s unquely ratonalzable for any t E wth x t > x E + 1 p and not nvest s unquely ratonalzable for any t E wth x t < x E 1 p. Proof. We wll show that nvest s not ratonalzable for any t E wth x t < x E 1 p. Let ˆx nf {x t t R E } and assume wthout loss that x 1 x 2. 3 When R1 E 1 s empty, our concluson s vacuously true because nvest s not ratonalzable anywhere on E. We wll assume that R1 E 1 s not empty and show that x E 1 p ˆx 1. Now, by defnton, there exsts a sequence t 1,m of types t 1,m R1 E 1 such that x 1 t 1,m [ˆx 1, ˆx 1 + 1] for each m and x 1 t 1,m ˆx 1. Snce x 1 1 [ˆx 1, ˆx 1 + 1] s sequentally compact, there then exsts a convergent subsequence wth some lmt ˆt 1. Snce E s closed, ˆt 1 E 1. Snce x 1 s contnuous and x 1 t 1,m ˆx 1, we have ˆx 1 = x 1 ˆt 1. 4 Now, snce T s closed and x s contnuous, by Lemma 2 and Proposton 1, R s closed, and hence ˆt 1 remans n the closed set R 1 E 1. In partcular, ˆt 1 E 1, and x 1 ˆt 1 1 π1 R 2 ˆt 1. 5 Moreover, snce E s p-evdent, π 1 R 2 \E 2 ˆt 1 π1 T2 \E 2 ˆt 1 1 p. Hence, Therefore, ˆx 1 = x 1 ˆt 1, by 4 π 1 R 2 ˆt 1 π1 R 2 E 2 ˆt p. 6 1 π 1 R 2 ˆt 1, by 5 1 π 1 R 2 E 2 ˆt 1 1 p, by 6 1 π 1 {t2 T 2 x 2 t 2 ˆx 2 } ˆt 1 1 p, by defnton of x2 1 π 1 {t2 T 2 x 2 t 2 ˆx 1 } ˆt 1 1 p, by 3 = π 1 { t2 T 2 x 2 t 2 < x 1 ˆt 1 } ˆt 1 1 p = r 1 ˆt 1 1 p,

19 19 showng that x 2 x 1 x E 1 p. A symmetrc argument establshes that not nvest s not ratonalzable f x t > x E + 1 p. When x E = 1 or x E = 0, Lemma 3 s vacuous, statng that nvest s unquely ratonalzable when t s domnant. When x E < p, Lemma 3 establshes that nvest remans unquely ratonalzable throughout the nterval x E + 1 p, 1. Ths s smlar to the man result of Carlsson and van Damme In ther result, f one can connect a type t to a type t at whch nvest s a domnant acton, va a contnuous path along whch nvest s ether rsk-domnant or domnant, then nvest s unquely ratonalzable at t. Here, as n Example 1, such an assumpton may not be useful, and we do not explctly make any such assumpton. Nonetheless, when x E < p, each type t E wth x E < x t < 1 assgns a substantal probablty.e. a probablty greater than 1 x t on a set of types t j whose values are hgher than that of t and assgn a substantal probablty to yet another set of types wth smlar propertes. In general, such connectons through belef herarches do not necessarly lead to a contagon path. Indeed, as llustrated n Example 1, such a path can reman wthn the open set ˆt, ˆt 2, wthout leadng to a domnance regon. However, when E s closed and p > max t E r t, such a chan necessarly leads to types who assgn a substantal probablty on types for whch nvest s a domnant acton. 13 Such a chan forms a contagon path. Note that the types n the chan have type-dependent thresholds as n Proposton 1. Proof of Proposton 3. Proposton 3 mmedately follows from Lemma 3. Snce C 1 ε,1 ε URB ε URB ε, we clearly have x C 1 ε,1 ε URB ε 1/2+ε. Hence, whenever t 1, t 2 SRD 2ε C 1 ε,1 ε URB ε, we have x t > 1/2 + 2ε x C 1 ε,1 ε URB ε + ε. Snce C 1 ε,1 ε URB ε s 1 ε-evdent, Lemma 3 then mples that nvest s unquely ratonalzable for both players at t 1, t 2. 6 Dscusson: Extensons and Relaton to The Lterature 6.1 Non-Unform Rank Belefs We have focussed on the case of unform rank belefs, whch s the leadng case that arses n the lterature. Our results contnue to hold f we have common certanty that players have rank belef p,.e., they assgn probablty p to havng the hgher rank. Our unqueness results extend mmedately, although the nature of the selecton s then dfferent. Specfcally, defne RBε p for the set of type profles t 1, t 2 where both players have ε-rank belef p: RBε p = {t 1, t 2 p ε r t r t p + ε for each }. 13 Indeed, one can show that when C p,p URB ε s closed for some p > 1/2 + ε, the set x C p,p URB ε of payoff parameters for the types wthn C p,p URB ε s unbounded, nducng domnance regons wthn the subspace.

20 20 Say that acton nvest s strctly p-domnant for t f x t > 1 p. We wrte D p for the set of type profles for whch nvest s strctly p-domnant for each player, so D p = {t 1, t 2 x t > 1 p for each }. Proposton 4 For any p, q, ε [0, 1] wth p + q 1 2ε, assume that C 1 ε,1 ε RBε q s closed. Then, nvest s the unquely ratonalzable acton for both players f t s strctly p-domnant for both players and there s common 1 ε-belef of ε-rank belef q,.e., f t 1, t 2 D p C 1 ε,1 ε RBε q. Ths observaton, sutably generalzed to many players, s the drvng force behnd the results of Izmalkov and Yldz The followng example llustrates how approxmate common certanty of non-unform rank belefs can arse. Undmensonal Lnear Model wth Exponental Dstrbutons Assume σ = τ = 1, x = t + y t = θ + e where y < 0 and θ, e 1, e 2 d Exp λ. Observe that, condtonal on t, θ s unformly dstrbuted on [0, t ]; so s e. Hence, r t = 1 t 1 e λt θ dθ = 1 e λt t 0 The rank belefs are plotted n Fgure 2. As λ, t t 0 e λθ dθ = 1 1 e λt λt. r t 1 unformly on [ y,. Thus, f t ε = r 1 1 ε and E ε = { } { } t 1 t 1 t ε t2 t 2 t ε, the event Eε s 1 ε, 1 ε-evdent. The example satsfes the common pror assumpton. One could buld on the example to show that nvest could be made the unque ratonalzable acton. It thus establshes that approxmate common certanty of non-unform rank belefs s consstent wth the unque selecton of any ratonalzable acton n a type that s close to complete nformaton n the product topology, llustratng a result of Wensten and Yldz However, approxmate common certanty of non-unform rank belefs cannot arse wth hgh ex ante probablty under the common pror assumpton, llustratng a result of Kaj and Morrs 1997.

21 21 r t Fgure 2: Rank belefs for λ = 1 and λ = Many Players We focussed on the case of two players. The extenson of the results of ths paper to N players mantanng the symmetry and separablty of payoffs assumptons n ths paper s straghtforward, and we descrbe ths extenson n ths subsecton. Suppose that each player has a payoff type x, the payoff to not nvestng s 0, and the payoff to nvestng s x 1 + ψ l where l s the proporton of other players nvestng and ψ : [0, 1] [0, 1] s ncreasng wth ψ 0 = 0 and ψ 1 = 1. In the specal case of two players, these payoffs reduce to those n the body of ths paper. We can gve a common belef characterzaton of ratonalzablty n ths settng, although the relevant belef operator s more complcated than n the two player case. For a fxed smple event E, agent and nteger n, we can consder derved events correspondng to the set of type profles t of players other than where t j E j for exactly n out of those other players. We wll be nterested n the probablty that player assgns to such derved events and, more specfcally, whether a weghted sum of such probabltes s above some type-dependent level. Thus we defne generalzed belef operators: B f E = { t t E and N 1 n=0 } n ψ π {t # {j t j E j } = n} t f t. N 1 Ths s a generalzaton of the type-dependent belef operators that we ntroduced n the two player case. It s qualtatvely more complcated, though, because t depends on the weghted sum of probabltes

22 22 assgned to a set of events rather than probabltes of one event. However, wth these operators, we can generalze the tght characterzaton of ratonalzablty. For a vector of type dependent probablty functons f = f 1,.., f N, we can now defne f-belef and common f-belef operators as before, B f E = N =1B f E. [ ] n C f E = n=1 B f E. and analogous fxed pont characterzatons wll hold. Now Lemma 2 wll contnue to hold as stated for these modfed operators, as wll Proposton 1 gvng a suff cent condton for a unquely ratonalzable acton n symmetrc games. It s also possble to gve generalzed belef operator characterzatons of ratonalzable actons n more general games, see Morrs and Shn Whle the tght characterzaton of ratonalzablty becomes more complcated when we move to many players, the suff cent condtons we gve for multplcty and unqueness generalze straghtforwardly. The approxmate common certanty suff cent condton for multplcty n Proposton 2 s suff cent to show ratonalzablty of all strct equlbrum actons n general games, as orgnally shown by Monderer and Samet For unqueness suff cent condtons, a player s rank belef now gves the probablty that he assgns to hs payoff type beng ranked kth for each k, and we can defne correspondng upper and lower rank belefs. Thus We say that rank belef of a type t s ε-unform f r k t = π # {t j x j t j x t } = k t, r k t = π # {t j x j t j < x t } = k t. 1 N ε r k t r k t 1 N + ε for each k = 1,.., N. We wrte URB ε for the set of type profles t where both players have ε-unform rank belefs: { URB ε = t 1 N ε r n t r n t 1 N } + ε for each. We say that rank belefs are approxmately unform f they are ε-unform for some suff cently small ε 0. Morrs and Shn 2003b noted that the global game selecton n ths case was the "Laplacan acton", correspondng to a unform belef over the proporton of opponents nvestng. Our second concept s a strct verson of the Laplacan property. We say that acton nvest s ε-laplacan for t f x t 1 1 N N 1 j=0 j ψ + ε. N 1