REPUTATION WITHOUT COMMITMENT

Transcription

1 REPUTATION WITHOUT COMMITMENT JONATHAN WEINSTEIN AND MUHAMET YILDIZ A. In the reputaton lterature, players have commtment types whch represent the possblty that they do not have standard payoffs but nstead are constraned to follow a partcular plan. In ths paper, we show that arbtrary commtment types can emerge from ncomplete nformaton about the stage payoffs. In partcular, any fntely repeated game wth commtment types s strategcally equvalent to a standard fntely repeated game wth ncomplete nformaton about the stage payoffs, such that the types wth dentcal solutons have almost dentcal pror probablty n two games. Then, classc reputaton results can be acheved wth uncertanty concernng only the stage payoffs. JEL Numbers: C72, C I The reputaton lterature reles on the exstence of commtment types. These types are not strategc but are certan to follow a partcular plan. Snce the semnal work of Kreps, Mlgrom, Roberts, and Wlson 1982) henceforth, Gang of Four), t has been well-establshed that ncluson of commtment types may alter predcted outcomes dramatcally, as ths may entce the orgnal ratonal types to mtate the commtment types, n order to form a reputaton for playng accordng to the commtted plan. Buldng on ths nsght, a large lterature has emerged, wth applcatons n a wde range of areas. 1 Of course, commtment types can be modeled by usng a payoff functon that rewards a player who follows a specfc plan. Nevertheless, such payoff functons often contradct the underlyng structure of the orgnal game. For example, the tt-for-tat types used by Gang Date: Frst Verson: August 2012; Ths Verson: June Wensten: Washngton Unversty n St Lous; j.wensten@wustl.edu. Yldz: MIT Economcs Department; myldz@mt.edu. We thank the referees and the semnar partcpants at Harvard, Koc, Prnceton, and Yale Unverstes, n partcular, Dlp Abreu and Stephen Morrs, for useful comments. 1 We refer to the textbook of Malath and Samuelson 2006) for a revew. 1

2 2 JONATHAN WEINSTEIN AND MUHAMET YILDIZ of Four n the analyss of fntely repeated prsoners dlemma cannot be justfed by a payoff functon that s consstent wth a repeated game,.e. a possbly dscounted) sum of stagegame payoffs. The only commtment types that arse wthn such an addtve structure are those who commt to playng the same acton throughout the game, and these would not have any mpact on behavor n a repeated prsoners dlemma game. The justfcaton for commtment types s mportant for the nterpretaton of the results. If one can derve an outcome say cooperaton n the fnte prsoner s dlemma) usng only types that arse from a natural payoff structure such as a repeated game wth dfferent stage payoffs), one can then nterpret the result as an outcome of ncomplete nformaton wthn a ratonal framework. On the other hand, some payoff functons, such as one that rewards a player only f he follows tt-for-tat, seem rratonal n a repeated-game context. They may nstead reflect psychologcal anomales and super-game concerns motvatons that le outsde the game), such as mantanng reputaton n a broader context. If one needs to ntroduce such payoff functons, then the result s probably best seen as reflectng the ssues relatng to the super-game concerns, psychologcal anomales or rratonalty that have been overlooked n the orgnal model. That may be why the commtment types are often referred to as crazy types and treated wthn the context of modelng rratonalty, as n Kreps 1990).) In ths paper, we show that for any gven plan, a commtment type who s requred to follow ths plan can be mmcked by a twn whose belefs cause hm to follow the same plan. Specfcally, the twn knows t s common knowledge that payoffs follow a repeatedgame strcuture, and hs unque ratonalzable acton s to follow the gven plan. Moreover, by embeddng a collecton of such twns nto a sngle type space, every game wth commtment types can be converted to a standard repeated game wth ncomplete nformaton about the stage-game payoff functon, such that the twns have pror probabltes almost dentcal to the commtment types. Therefore, any model of reputaton formaton n fntely repeated games, where players form a reputaton for commtment, can be converted to a strategcally equvalent model n whch they form a reputaton for certan belefs about the stage-game payoffs. Ths s true provded that one allows suff cent varatons n stage-game payoffs and consders a rch set of nformaton structures. Of course, one may also wsh to restrct the the stage-game payoff functons. For example, n a standard prsoners dlemma game, one mght want to assume that t s common

3 REPUTATION WITHOUT COMMITMENT 3 knowledge that cooperaton s not domnant. Under such restrctons, twns may not exst for some commtment types. Indeed, we also prove an opposng benchmark, showng that one needs some amount of varatons n the stage-game payoffs n order to have any reputatonal effect. We show that f the stage game s domnance-solvable and the stage game payoffs are restrcted to be a suff cently small neghborhood of the orgnal stage-game payoff functon, then the unque sequental equlbrum of the repeated game wth ncomplete nformaton prescrbes all players to repeat the stage-game soluton throughout the game as n the subgame-perfect equlbrum of the complete nformaton verson), regardless of the length of the game. 2. P R In ths secton, we prevew our result more carefully on the example analyzed by Gang of Four: the fntely-repeated prsoner s dlemma game n whch player 1 may be commtted to tt-for-tat, though ths has small ex-ante probablty. Consder the repeated game n whch the followng prsoner s dlemma s repeated t tmes: PD) Cooperate Defect Cooperate 5, 5 0, 6 Defect 6, 0 1, 1 All the prevous moves are publcly observable perfect montorng), and the payoff of a player n the repeated game s the sum of hs payoffs n the stage game above. A smple applcaton of backward nducton n ths game yelds the play of Defect, Defect) at every hstory. Indeed, t s well known that the only Nash equlbrum outcome s playng Defect, Defect) at every perod. Gang of Four consder an ncomplete nformaton game G n whch player 1 may be commtted to playng tt-for tat. Player 1 has two types, a ratonal type τ 1, whose payoffs and avalable moves are as n the repeated-prsoners dlemma game above, and a commtted type τ T 4T 1 whch can only play tt-for-tat. That s, the latter type must play cooperate n the frst round and mtate the last move of player 2 n the subsequent perods. The pror probablty of τ T 4T 1 s some small ε > 0. Player 2 stll has one type τ 2, whch s ratonal as n the orgnal game. Gang of Four shows that n any sequental equlbrum of the new game each ratonal type τ must play Cooperate at all but few perods. Interestngly, by

4 4 JONATHAN WEINSTEIN AND MUHAMET YILDIZ varyng the set of commtment types, one can generate a rch set of equlbrum behavor, obtanng a Folk Theorem for long games Fudenberg and Maskn, 1986)). As we mentoned n the ntroducton, one can replcate the above equlbrum behavor wth payoff uncertanty by assgnng the payoff functon of τ T 4T 1 as 1 at hstores at whch player 1 plays accordng to tt-for-tat and 0 at all other hstores. Here, the soluton concept s sequental equlbrum wth the restrcton that player 2 assgns probablty 1 on τ 1 off the path. Such a payoff functon s ncompatble wth the repeated game payoff structure, and one cannot replcate the commtment to tt-for-tat by smply modfyng the stage-game payoff functon for τ T 4T 1. Indeed, such modfcatons can lead to only two commtment types: the type that plays Cooperate throughout and the type that plays Defect throughout. Commtment to cooperaton can be justfed by the stage-game payoff functon CC) Cooperate Defect Cooperate 1 1 Defect 0 0 for example. The ncluson of such smple commtment types cannot affect the behavor of ratonal types n ths game, though n other games such as Cournot duopoly, t could result n a player becomng a Stackelberg leader by convncng the other player he s commtted to a certan acton. Fortunately, the austere nformaton structure above s not the only structure we can consder. For any ε > ε, applyng our Proposton 1 to the game G n Gang of Four generates a game G wth the followng propertes. Ex-ante Proxmty: The pror probablty of the ratonal type profle τ 1, τ 2) s at least 1 ε, and each τ knows that hs stage-game payoffs are as n PD). Repeated-Game Structure: All types can play all strateges and maxmze the sum of stage-game payoffs, whch need not be as n PD). Strategc Equvalence: G and G are strategcally equvalent n the followng sense. 1) G contans types τ 1, τ 2, a twn ˆτ T 4T 1 of the tt-for-tat type τ T 4T 1 n G, and a number of other new types of both players) that we use to encode the belefs of type ˆτ T 4T 1. 2) Though ˆτ T 4T 1 s allowed to play any plan of acton, tt-for-tat s hs unque ratonalzable plan.

5 REPUTATION WITHOUT COMMITMENT 5 3) Ratonal type τ 1 s certan that he faces the ratonal type τ 2, and the ratonal type τ 2 n turn puts probablty 1 ε on τ 1 and probablty ε on the twn ˆτ T 4T 1 of τ T 4T 1. By strategc equvalence property, the strategc stuaton the ratonal types face s the same as n G, except now τ 2 thnks that ˆτ T 4T 1 plays tt-for-tat as a result of some ratonal reasonng under ncomplete nformaton rather than as a result of commtment or an unconventonal payoff functon. Therefore, under the broad set of soluton concepts that are nvarant to such changes, the soluton sets for ratonal types τ 1, τ 2) are dentcal n G and G. The condtonal probabltes specfed above are acheved by a pror dstrbuton n G puttng probablty 1 ε on τ 1, τ 2), ε 1 ε T 4T ) / 1 ε) on ˆτ 1, τ 2) and the remanng small probablty ε ε) / 1 ε) on the newly constructed types. Three ponts are worth emphaszng. Frst, when ε ε s small compared to ε, the pror probabltes of τ 1, τ T 4T 2) and ˆτ 1, τ 2) are approxmately 1 ε and ε, respectvely, wth much smaller probablty on the new types. Hence, the type spaces of G and G are nearly dentcal, and the twn ˆτ T 4T 1 assgns much larger probablty to the standard type τ 2 than to the new types. Despte ths, ˆτ T 4T 1 has a unque ratonalzable plan because ˆτ T 4T 1 beleves that hs own plan has non-neglgble mpact on hs payoff only f he faces one of the newly constructed types. He fnds these types unlkely, but they are lkely enough to be hs man concern. Second, the unque ratonalzable plan emerges under ntrcate belefs that requre a large number of new types for encodng, especally when the game s long. Nonetheless, we are able to encode such belefs by puttng only a neglgble amount of pror probablty on the new types. Thrd, our proof s based on reward and punshment mechansms typcal n repeated games as well as theores of socal learnng that rely on the exstence of payoff functons as n CC). One could object that such stage-games are too dfferent from the orgnal prsoners dlemma game to be concevable. Accordngly, our second benchmark Proposton 5) allows only small varatons n the stage-game payoffs. Snce Nash equlbrum s robust to such perturbatons and Defect, Defect) s the only Nash equlbrum here, one cannot expect to have any dscontnuty n that front. Ths s smlar to the fact that for a gven t, Defect, Defect) becomes the unque equlbrum outcome as ε 0 n the Gang of Four example. Nevertheless, Gang of Four demonstrates that such contnuty s msleadng: for any gven ε, long cooperaton s obtaned when t s large enough. In ths paper, we prove somethng stronger: a unform contnuty

6 6 JONATHAN WEINSTEIN AND MUHAMET YILDIZ result wth respect to the stage game payoffs. The dstncton n these results arses because we mpose common knowledge of the constrant on payoffs. Accordng to our result, when the stage-game payoffs are suff cently close to PD) throughout the type space, Defect, Defect) s the only sequental equlbrum outcome, no matter how large t s. Therefore, one needs to allow some substantal amount of varaton n stage-game payoffs n order to provde an ncomplete-nformaton foundaton for the commtment types. Whle the amount of necessary varaton may depend on the detals of the game and the commtment types at hand, our man result shows that one can always provde such a foundaton as long as there s enough varaton n allowable stage-game payoff functons. Our results here buld on our prevous work on non-robustness n repeated games. In Wensten and Yldz 2013) we showed that, n any nfntely repeated game, any ndvdually ratonal and feasble outcome s the unque ratonalzable outcome of an approprately chosen perturbaton whch mantans common knowledge of the repeated-game structure and dscountng crteron. A key lemma leadng to ths result showed that for any plan whatsoever, there s a type who follows ths plan as a unque ratonalzable acton, although he beleves n common knowledge of the repeated-game structure. An extenson of ths lemma to fntely repeated games plays an mportant role n our constructon. Our contrbuton to our prevous work s twofold. Frst, extendng the above lemma to fntely repeated games nvolves a consderably more nuanced constructon, as provdng effectve future ncentves s harder n fntely-repeated games. Second, and more mportantly, perturbatons consdered n the two work are very dfferent: here we use the ex-ante perturbatons that are used n the tradtonal reputaton lterature and also by Kaj and Morrs 1997) n the context of robustness), whle n Wensten and Yldz 2013) we use perturbatons of nterm belefs n unversal type space. In general, ex-ante perturbatons are sgnfcantly more restrctve, and the two approaches often yeld qute dfferent results. One reason the results here can be acheved wth ex-ante perturbatons s that our constructon centers around perturbng the commtment types, who do not have set belefs. The man dff culty turns out to be embeddng types constructed n the lemma nto a commonpror model wthout affectng the types ratonalzable actons, whle keepng the ex-ante probabltes of the new types arbtrarly small.

7 REPUTATION WITHOUT COMMITMENT 7 We ntroduce the basc defntons and formulatons n Secton 3. In Secton 4, we present our constructon of a new type space n whch the commtment types are replaced by types for whch the commtted acton plan s unquely ratonalzable. In Secton 5, we show that, for the orgnal ratonal types, the constructed game s strategcally equvalent to the model wth commtment types, under a very broad set of soluton concepts. We present the most general verson of our result n Secton 6. After presentng our contnuty result n Secton 7, we conclude n Secton 9. Some of the more complcated proofs are relegated to the Appendx. 3. B D We study, for smplcty, a standard 2-player fntely repeated game wth perfect montorng and normal-form stage games; see Secton 6 for the general case. We wrte N = {1, 2} for the set of players, T = {0, 1,..., t} for the set of dates t, and fx a fnte set A = A 1 A 2 of stage-game acton profles a = a 1, a 2 ). 2 Note that, snce we have perfect montorng, the non-ntal hstores n the repeated game are of the form h t = a 0,..., a t 1 ) where a t denotes the stage-game strategy profle played at date t T. We wrte h 0 for the empty ntal hstory, and wrte H for the set of all non-termnal hstores. An outcome path or a termnal hstory s a lst a 0,..., a t ) ; the set of all termnal hstores s denoted by Z. The payoff vector from an outcome path a 0, a 1,..., a t ) n repeated game s smply the sum 3 of the stage game payoffs: ) 3.1) u a 0, a 1,..., a t g = g a 0) + g a 1) ) + + g a t, where the functon u = u 1, u 2 ) denotes the payoffs from the repeated game and the functon g = g 1, g 2 ) denotes the payoffs from the stage game. Whle the partcular stage payoffs are not necessarly known, ths formula wll be common knowledge throughout the games we study here. That s, t s common knowledge that the stage payoff functon g s fxed throughout the game and that the players smply maxmze the sum of these payoffs. 2 Followng the conventon n game theory, we wrte for the player j and drop the subscrpt to denote the profles, e.g., x = x 1, x 2 ) X = X 1 X 2 and X 1 = X 2. 3 Dscountng would not affect our results; settng the dscount rate to 1 smplfes the expressons.

8 8 JONATHAN WEINSTEIN AND MUHAMET YILDIZ We wrte G = [0, 1] A for the set of all possble stage-game payoff functons g : A [0, 1]. Here, we put a unform bound on the stage game payoffs so that small varatons of the probablty dstrbutons on stage payoffs lead to small varatons n expected payoffs, as n the reputaton lterature. Ths restrcton strengthens our results. We fx a complete-nformaton repeated game n whch t s common knowledge that the stage-game payoffs are a fxed g1, g2). The payoff functon n the repeated game s u g ), gven by the formula n 3.1). Ths could, for example, be the repeated prsoner s dlemma game, wth g defned as n PD). In the complete-nformaton game, a strategy of a player s a mappng s : H A, whch maps each non-termnal hstory to a strategy n the stage game. Because we analyze ncomplete nformaton games, however, we wll avod the word strategy for ths mappng and call t nstead an acton plan, because we reserve the word strategy for mappngs from types to acton plans. We refer to the strateges n the stage game as moves.) The set of all acton plans s denoted by S. The outcome path nduced by a profle s 1, s 2 ) s denoted by z s 1, s 2 ). We also allow behavoral) mxed strateges and wrte Σ for the set of mxed acton plans σ : H A ) for player. We consder two knds of elaboraton, correspondng to two dstnct ways n whch the common-knowledge assumpton n the complete nformaton game may be relaxed. The frst noton of relaxaton s the one consdered n the reputaton lterature through commtment types. Defnton 1. An ε-elaboraton wth one-sded) commtment types s a Bayesan game such that the sets of types for players 1 and 2 are {τ 1} C and {τ 2}, respectvely, where C S 1 ; the probablty of τ 1 s 1 ε and the probablty of each c C s some π c) [0, 1] that add up to ε; the set of moves avalable to τ s as n the repeated game above, whle the only avalable acton plan for type c C s c, the payoffs are as n the complete nformaton game. Such an elaboraton can be denoted by a par C, π) of commtment types C and a vector of probabltes on C.

9 REPUTATION WITHOUT COMMITMENT 9 Here, each acton plan c C corresponds to a type of Player 1 who can only play c. The ncomplete nformaton s only about whether Player 1 can play all acton plans or has commtted to a partcular acton plan. The type τ 1 that can play all plans s called the ratonal type whle the types c C, who can play only accordng to one plan of acton, are called commtment types. The commtment types are also sometmes called crazy types. A commtment type c also be modeled by a payoff functon that gves 1 f Player 1 plays accordng to c throughout and 0 f he ever devates. Such a payoff functon clearly volates the addtve structure n 3.1). Indeed, except for the trval commtment types who play the same stage-game strategy throughout the repeated game, commtment types cannot be justfed by a new payoff functon that has the addtve structure above. We wll next ntroduce type spaces n whch all payoff functons satsfy the addtve structure of 3.1). Whle they enforce ths restrcton on the repeated structure of the game, these spaces are also rcher n that they may have a larger varety of types. Our type spaces wll have two-sded ncomplete nformaton wth a common pror. Formally, by a type space, we mean a lst T, π, g) where T = T 1 T 2 s the set of type profles τ = τ 1, τ 2 ), π T ) s the common pror on type profles, and g = g 1, g 2 ) s the profle of type-dependent stagegame payoff functons g : T G. A Bayesan repeated game wthout commtment 4 types) s a lst N, A, T, π, g)). We should emphasze that ths notaton suppresses many mportant common-knowledge assumptons, such as the fact that the game s repeated, all prevous actons are publcly observable.e. perfect montorng), and the payoffs n the repeated game are gven by the formula 3.1). We also note that players do not observe ether ther own or other players payoffs at each stage of the game. A strategy of a player n a Bayesan repeated game N, A, G, T, π)) s a mappng σ : T Σ. The second noton of elaboraton allows rcher type spaces and two-sded ncomplete nformaton, but does not allow any payoff functon outsde of the addtve structure n 3.1): Defnton 2. An ε-elaboraton wthout commtment types s a Bayesan game N, A, T, π, g)) wth dstngushed types τ 1, τ 2 where 1) g τ ) = g for each N and 2) π τ ) = 1 ε. 4 Here, X) denotes the set of all probablty measures on the fnte set X.

10 10 JONATHAN WEINSTEIN AND MUHAMET YILDIZ The frst and second condtons state that the orgnal complete nformaton game s embedded n the elaboraton and has a hgh ex-ante probablty of 1 ε. The last condton states that the ratonal types τ 1, τ 2) know ther payoffs, and ther payoffs are as n the orgnal complete nformaton game. The novelty n ths defnton s that the payoffs under all possble specfcatons n the Bayesan game satsfy the addtve repeated game structure of 3.1). That s, the formula 3.1) remans common knowledge. In that sense, all the types n an elaboraton wthout commtment types are ratonal, although we reserve the term ratonal for types τ 1, τ 2) as n the elaboratons wth commtment types. Both elaboratons above fall under the category of ε-elaboratons as defned by Kaj and Morrs 1997). An ε-elaboraton wthout commtment types s a Kaj-Morrs elaboraton wth the addtonal restrcton that the formula 3.1) s common knowledge. Whle ε-elaboratons wth commtment types were presented above n terms of uncertanty about the strateges, they could also be represented as Kaj-Morrs elaboratons wth specfc smple type space n whch the formula 3.1) fals. Fnally, we revew two standard concepts n game theory. Frst, for any Bayesan game, nterm correlated ratonalzablty henceforth ICR) s the outcome of terated elmnaton of acton plans for types that are never a weak best response, as defned by Dekel, Fudenberg, and Morrs 2007). We wrte S [τ G] for the set of all nterm correlated ratonalzable acton plans for type τ T n game G = N, A, G, T, π)). We wll gve a more detaled defnton of ICR later n the constructon. We just note here that ICR s the weakest known ratonalzablty concept for Bayesan games, and all the acton plans that are played by a type wth postve probablty n any equlbrum are ICR for that type. Second, we say that acton plans s and s are equvalent f z s, s ) = z s, s ) for all acton plans s S,.e., they lead to the same outcome no matter what strategy the other player plays. Note that s and s are equvalent ff s h t ) = s h t ) for every hstory h t n whch played accordng to s throughout; they may dffer only n ther prescrptons for hstores that they preclude. Hence, n reduced form, acton plans can be represented as mappngs that maps the hstory of other players play nto own stage game actons. We wrte S for the set of reduced-form acton plans s ; these map each ) a l to some acton a A n the stage game. 0 l<t

11 REPUTATION WITHOUT COMMITMENT I C T In ths secton, we show that one can replace each commtment type c wth a regular type for whch c s unquely ratonalzable wth a slght perturbaton of the pror dstrbuton. Consequently, one can transform any elaboraton wth commtment types to an elaboraton wthout commtment types, such that, from the pont of vew of ratonal types who beleve n the ICR concept, the two elaboratons are dentcal. Under ICR as well as a broader set of soluton concepts, ths wll lead to the same set of solutons for the ratonal types n the two elaboratons. Proposton 1. For any ε, ε 0, 1) wth ε > ε and for any ε-elaboraton G wth commtment types C, π) there exsts a ε -elaboraton G = N, A, G, T, π )) wthout commtment types n whch the commtment types are replaced by types wth unque ratonalzable acton plans: 1) π g, τ 2 τ 1) = 1 and π g, τ 1 τ 2) = π g, τ 1 τ 2) = 1 ε, and 2) for every c C there exsts τ c 1 T 1 such that all ICR acton plans of τ c 1 are equvalent to c, and π τ c 1 τ 2) = π c τ 2) = π c). Here, the frst condton establshes that the nterm belefs of ratonal types regardng ther own payoffs and ratonalty of ther opponents are dentcal n the two elaboratons. The second condton establshes that each commtment type c s replaced by a type τ c 1 for whch followng c s unquely ratonalzable, and that the ratonal type of player 2 n G assgns the same probablty to the type τ c 1 as the ratonal type n G assgns to the commtment type c. Note that π τ c 1 τ 2) = g 1 π g 1, g2), τ c 1 τ 2), meanng that type τ 2 knows hs own payoff functon. The proposton thereby establshes that any ε-elaboraton G wth commtment types can be converted nto an ε -elaboraton by replacng commtment types c wth types τ c 1 for whom c s the only ratonalzable acton plan n reduced form. These types follow c not because they are commtted or have payoffs that are nconsstent wth playng a repeated game but because ther reasonng under ther nformaton leads them to do so. Moreover, from the pont of vew of the ratonal types these are the only types wth postve probablty, mrrorng the elaboraton wth commtment types. The equvalence s establshed despte the followng constrants:

12 12 JONATHAN WEINSTEIN AND MUHAMET YILDIZ 1) The repeated-game payoff structure s mantaned throughout G. That s, t s common knowledge throughout that the payoff n the repeated game s the sum of the payoffs n the stage game, and that the stage game s fxed throughout the game. Type τ c 1 knows all ths and yet follows c as ts unque ratonalzable plan. 2) The ex-ante dstrbuton π n G can be arbtrarly close to the dstrbuton π n G, n that ε can be arbtrarly close to ε. Note that the commtment plans c can be arbtrarly complex. Hence, the types τ c 1 that replace the commtment types c may hold hghly complcated belefs, and the elaboraton wthout commtment types may contan a large number of new types that are used to encode these belefs wthn the standard common pror type space. Despte ths, the second condton requres that the ex-ante probablty of these types be made arbtrarly small. Proof of Proposton 1. The frst step n our constructon s the followng lemma, whch establshes that any gven acton plan s s the only ratonalzable acton for a type τ s from some common pror model. The proof of Lemma 1 s the lengthest step of the proposton and s relegated to the appendx.) Lemma 1. For any s S, there exsts a Bayesan repeated game G s = N, A, G s, T s, π s )) wth a type τ s T s such that 1) π s g, τ) > 0 for every g, τ) G s T s and 2) for every acton plan s, s S [τ s Gs ] f and only f s s equvalent to s. By relabelng f necessary, we take all of the types above to be dstnct from each other and from τ, fxng also a unque type τ s for each s. We construct G = N, A, G, T, π )) by settng G = {g, 0, g 2)} c C G c T = {τ } T c N) c C 1 ε f g, τ) = g, τ ), π g, τ) = 1 ε 1 ε f g, τ) = 0, g 2), τ c 1, τ 2)), ε ε 1 ε) C πc g, τ) f g, τ) G c T c, 0 otherwse,

13 REPUTATION WITHOUT COMMITMENT 13 where 0 a) = 0 a A). We now observe that G satsfes the propertes n the proposton. Indeed, ratonal type τ 1 of Player 1 assgns probablty 1 on g, τ 2). Lkewse, we have π G {τ 2}) = 1 ε + 1 ε 1 ε c C πc) = 1 ε + 1 ε 1 ε ε = 1 ε ) / 1 ε) and therefore, n the nterm, τ 2 assgns probablty 1 ε to g, τ 1) and probablty π c) to τ c 1 for each c. On the other hand, snce the belefs of type τ c 1 altered substantally when G c, T c, π c ) was ncorporated n G, t s not clear that τ c 1 follows c as the unque ICR acton. The next lemma states that ths s ndeed the case. Lemma 2. For any c C, N, and any τ T c, S [τ G ] = S [τ G c ]; n partcular, S 1 [τ c 1 G ] = c. Ths lemma completes the proof of the proposton; ts proof s n the appendx. Our proof has two man steps. The frst, found n Lemma 1, s to construct a type space n whch a gven acton plan s unquely ratonalzable for a type. We constructed such a type space n Wensten and Yldz 2013) for nfnte-horzon repeated games, but wthout requrng that the constructed type space has a common pror, a property that s essental for our proposton here. In ths paper, usng the deas n that constructon, we frst construct such a type space for fnte-horzon games wthout common pror and then convert t to a common-pror type space, usng ths tme the deas and the results developed by Lpman 2003) and Wensten and Yldz 2007). The man economc deas nvolved n these constructons come from socal learnng and reward and punshment mechansms n repeated games. Frst, for the class of acton plans that s consstent wth learnng n a sngle decson problem, we construct games n whch players whose payoffs depend only on ther own acton update ther optmal stage game strateges as they observe the past behavor of the other players and learn about ther own payoff from the other players moves, as n socal learnng. We show that such processes can generate any acton plan that s consstent wth ndvdual learnng as the unque ICR acton plan. The generated acton plans can be arbtrarly complex because ndvdual learnng only

14 14 JONATHAN WEINSTEIN AND MUHAMET YILDIZ puts a relatvely mld restrcton on behavor, smlar to the sure-thng prncple. Accordngly, the type space generatng these behavor can be hghly complex, leadng players to update ther belefs about own payoffs, other players belefs, and other players belefs about belefs and so on, as they observe the other players moves. These deas apply to the fnte and nfnte repeated games, and the proof s nearly dentcal for both cases. For nfntely repeated games, n Wensten and Yldz 2013) we extend the result to all acton plans ncludng plans that contradcts the condton for ndvdual learnng), usng a reward and punshment mechansm. Unfortunately, t s harder to come up wth effectve reward and punshment mechansms for fnte horzon games. After all, one cannot provde any future ncentve n the last perod. Hence, here, we use a more nuanced constructon that combnes socal learnng wth a reward and punshment mechansm to extend the result to all acton plans n fntely repeated games. The second man step s to ncorporate the above type spaces n one common pror model, replacng each commtment type wth one of these type spaces. One must do ths n such a way that ) the orgnal complete-nformaton game stll has hgh pror probablty 1 ε ), ) the nterm belefs of the ratonal types are as n the orgnal elaboraton wth commtment types, and ) the types ratonalzable behavor n the constructed type space reman the same after ncorporatng them nto common pror model. The condtons ) and ) oppose each other, makng the constructon more dff cult. To see ths, note that ) and ) requre that the common pror π puts a hgh probablty on τ c 1, requrng that probablty to be 1 ε 1 ε π c) as n our proof. When ε and ε are close, ths probablty s approxmately π c). When ε and ε are close, ths also requres that π puts a very small probablty on T c, the orgnal type profles n the constructed type space n the frst step. That probablty can be at most ε ε) / 1 ε), whch s neglgble wth respect to 1 ε 1 ε π c) when ε and ε are close. These constrants make the belef of type τ c 1 n game G substantally dfferent from the belef of the type τ c 1 n game G c. In our constructon, type τ c 1 n game G assgns probablty 4.1) p c = C 1 ε ) π c) C 1 ε ) π c) + ε ε) π c τ) on type τ 2. Note that, for fxed π c), when ε ε approaches 0, p c approaches 1. 5 In contrast, τ c 1 n game G c assgns zero probablty on τ 2. Consequently, the belef herarches 5 Note also that the technque we use n transformng the model wthout common-pror to the one wth common pror also renders π c τ c 1) small, brngng p c near 1 even when ε and ε are far apart.

15 REPUTATION WITHOUT COMMITMENT 15 of the types n G can be qute dfferent from the belef herarches of the types n G c wth the same label, whch could lead to dstnct set of ICR actons. We crcumvent ths problem wth the followng trck. We set the belefs such that, whenever player 2 has type τ 2, the payoff of type τ c 1 s 0 for every move n the stage game, makng hm ndfferent among all outcomes. Snce p c < 1, hs best responses are dentcal to hs best responses condtonal on the type of player 2 beng other than τ 2, thereby replcatng the best responses of hs twn n G c. Snce ths was the only dfference between the two type spaces, the ratonalzable actons turn out to be dentcal n games G c and G, as shown formally by Lemma 2. Roughly speakng, from the pont of vew of ratonal types, Proposton 1 replaces commtment types by types who follow the same plans as ther unque ratonalzable plan. Hence, under any ratonalzable soluton concept, the ratonal types face the same strategc uncertanty n both games leadng the same set of possble behavor. We wll next establsh such strategc equvalence formally. 5. S E In ths secton, we show that, n Proposton 1, the elaboratons G wth commtment types and G wthout commtment types are strategcally equvalent for ratonal types. By ths we mean that, for a broad set of soluton concepts to be delneated, the set of solutons for each ratonal type are dentcal n games G and G. Therefore, the same set of behavor can be supported by reputatonal models regardless of whether one allows commtment types. In other words, the same set of behavor s supported whether one allows payoff functons that are nconsstent wth the repeated-game structure or mposes ths structure throughout. Ths wll further mply, as we detal n Secton 5.3, that the predctons of models wth commtment types are nearly ndstngushable from that of those wthout commtment types. Clearly, our result here apples to any soluton concept that s nvarant to replacng commtment types wth types that have unque ratonalzable acton plans n reduced form). In general Bayesan games, ths nvarance condton s somewhat stronger than elmnaton of non-ratonalzable strateges, because the new game contans some new types, encodng the belefs of the types wth unque ratonalzable plans. We frst establsh our result for a general class of such nvarant soluton concepts. We also establsh the same strategc

16 16 JONATHAN WEINSTEIN AND MUHAMET YILDIZ equvalence for sequental equlbrum; ths requres an addtonal off-path belef restrcton commonly mposed n the reputaton lterature Strategc Equvalence under Invarant Solutons. The followng defntons are standard: A soluton concept Σ maps every Bayesan game G to a set Σ G) of mxed strateges n game G. For any type spaces T and T wth T T and any strategy profle σ on T, σ T denotes the restrcton of σ to T. In the followng defntons, we also use the conventon that two probablty dstrbutons that have common support and agree on ths support are dentcal, gnorng any dfference n domans. Defnton 3. A soluton concept Σ s sad to be nvarant to elmnaton of non-ratonalzable strateges f and only f Σ G) = Σ G ) for any two games G and G wth dentcal type spaces such that ) f an acton plan s s avalable for a type τ n game G then s s avalable for τ n G and ) f s s not avalable for τ n G then s S [τ G ]. Defnton 4. A soluton concept Σ s sad to be nvarant to trval enrchments of the type spaces f and only f Σ G) = {σ T σ Σ G )} for any two games G and G wth type spaces T and T such that ) T T, ) every type n T has dentcal set of avalable acton plans n games G and G, and ) any type n T wth multple acton plans has dentcal nterm belefs n games G and G. Note that the transformaton n the frst defnton allows only elmnaton of non-ratonalzable actons and the transformaton n the second defnton allows only ncluson of new types such that the types who put postve probablty to the new types are trval n that they can play only accordng to one plan. Proposton 1 mples that under any soluton concept that s nvarant to the above transformatons, elaboratons wth or wthout commtment types have the same strategc mplcatons for ratonal types. Due to ts mportance, we state ths corollary as a proposton:

17 REPUTATION WITHOUT COMMITMENT 17 Proposton 2. Let Σ be a soluton concept that s nvarant to elmnaton of non-ratonalzable strateges and to trval enrchment of the type spaces. Then, for any ε, ε 0, 1) wth ε > ε and for any ε-elaboraton G wth commtment types, there exsts an ε -elaboraton G = N, A, G, T, π )) wthout commtment types such that {σ τ ) σ Σ G)} = {σ τ ) σ Σ G )},.e., the set of solutons for ratonal types are dentcal n games G and G. Proof. Note that, n Proposton 1, the elaboraton G can be obtaned from G by 1) ntroducng new types such that only commtted types beleve n the new types, and 2) allowng commtment types to play any acton plan n the repeated game. The frst step s a trval enrchment as n Defnton 3 and the second undoes an elmnaton covered by Defnton 4, so the concluson follows Strategc Equvalence under Sequental Equlbrum. We wll next establsh the same strategc equvalence under sequental equlbrum, whch s defned as follows. Gven any Bayesan repeated game wth a type space G, T, π), a belef structure s a lst µ = µ,τ,h) N,τ T,h H of type specfc belefs µ,τ,h G T ) regardng the underlyng payoffs and the other player s types, belefs that vary wth the hstory of play. 6 An assessment s a par σ, µ) of strategy profle σ : T Σ and a belef structure µ. An assessment σ, µ) s sad to be sequentally ratonal f σ τ ) s a sequental best response to µ,τ,h and σ,.e., the restrcton of σ τ ) to the contnuaton game after every hstory h s a best response to σ and the belefs µ,τ n the contnuaton game. An assessment σ, µ) s sad,h to be consstent f there exsts a sequence σ n, µ n ) σ, µ) such that σ n assgns postve probablty to each avalable move at every hstory and µ n s derved from Bayes rule and σ n. An assessment σ, µ) s sad to be a sequental equlbrum f t s sequentally ratonal and consstent. In an ε-elaboraton wthout commtment types, sequental equlbra are defned as above. In an ε-elaboraton wth commtment types, the defnton of course depends on how one formalzes the commtment types. In partcular, the defnton above mples that Player 2 puts probablty 1 on the ratonal type of Player 1 f the hstory s not consstent wth 6 A more general defnton of a belef structure would also specfy the belefs regardng past actons, but those belefs are trval because of perfect montorng.

18 18 JONATHAN WEINSTEIN AND MUHAMET YILDIZ any commtment type even when the hstory s also nconsstent wth the strategy of the ratonal type. Ths s because the commtment types have only one acton, so that only the ratonal types may tremble. Ths s an addtonal assumpton when the commtment types are represented by payoff perturbatons volatng the addtve repeated game structure). In general, the possble off-the-path belefs can vary dependng on the way the commtment types are formulated, but the above assumpton s usually mantaned. We wll keep ths addtonal assumpton n our defnton for sequental equlbrum wthout commtment types: Assumpton 1. For every hstory h = a 0,..., a t 1 ), µ 2,τ 2,h g, τ 1) = 1 whenever h has zero probablty under every type τ 1 τ 1. In our analyss we wll focus on the behavor of the ratonal types under sequental equlbrum, whch s formally defned as follows. Defnton 5. For any elaboraton G wth or wthout commtment types), we wrte SE G) = {σ τ ) σ, µ) s a sequental equlbrum of G that satsfes Assumpton 1} for the set of sequental equlbrum acton plans for the ratonal types n G. We are now ready to state the strategc equvalence result for sequental equlbrum. Proposton 3. For any ε, ε 0, 1) wth ε > ε and for any ε-elaboraton G wth commtment types C, π) there exsts an ε -elaboraton G = N, A, G, T, π )) wthout commtment types such that SE G) = SE G ),.e., under Assumpton 1, the set of sequental equlbrum acton plans for the ratonal types s same n games G and G. Proof. We wll show that both condtons σ τ ) SE G) and σ τ ) SE G ) are characterzed by the followng condtons, SR1) and SR2). Frst, σ, µ) s a sequental equlbrum of G f and only f the followng three condtons are satsfed. The consstency

19 REPUTATION WITHOUT COMMITMENT 19 condton for τ 2 s C) µ 2,τ 2,h c) = µ σ τ 1) h c) πc) Prh σ τ 1))1 ε)+ πc ) c C h f c C h 0 otherwse h, c) where C h s the set of commtment plans c C that s consstent wth hstory h. Of course, µ σ τ 1) h τ 1) = 1 µσ τ 1) c C h c). The consstency condton for player 1 s trval, as player 2 has only one type. Note that µ σ τ 1) h s a functon of σ τ 1), and hence the followng sequental ratonalty condtons are solely on σ τ ). The sequental ratonalty condtons are SR1): σ τ 1) s a sequental best response to σ τ 2) under g 1, and SR2): at each hstory h, σ τ 2) s condtonal best response to the mxed strategy under g 2. σ µ σ τ 1) h τ 1) σ τ 1) + c C µ σ τ 1) h c) c Snce all the other types are commtted to a sngle plan, there are no other condtons. Ths shows that σ τ ) SE G) f and only f SR1) and SR2) are satsfed. To show that σ τ ) SE G ) mples the condtons SR1) and SR2), consder any sequental equlbrum σ, µ ) of G that satsfes Assumpton 1. Frstly, snce type τ 1 puts probablty one on g, τ 2), the sequental ratonalty condton for that type s SR1). Secondly, snce c s the unque ratonalzable acton plan of τ c 1 n G by Lemma 2) on all hstores h consstent wth c, 5.1) σ c h) h, τ c 1) = 1 c C h, h ). Hence, by Assumpton 1 and consstency, 5.2) µ 2,τ 2,h τ c 1) = µ σ τ 1) h c) h, c), whch of course also mples that µ 2,τ 2,h τ 1) = µ σ τ 1) h τ 1). By 5.1) and 5.2), under the belef of type τ 2, player 1 plays accordng to σ above, and the sequental ratonalty condton for type τ 2 s SR2).

20 20 JONATHAN WEINSTEIN AND MUHAMET YILDIZ To show that SR1) and SR2) are suff cent for σ τ ) SE G ), take any σ τ ) that satsfes SR1) and SR2). We wll construct a sequental equlbrum σ, µ ) of G that satsfes Assumpton 1. Set µ 1,τ 1,h g, τ 2) = 1 and µ 2,τ 2,h = µσ τ 1) h. For each c C, consder a sequental equlbrum σ c, µ c ) of the game n whch the acton plan of type τ 2 s fxed as σ τ 2) as moves of nature, and the type space s T c wth the nterm belefs n G. Set σ τ ) = σ c τ ) and µ,τ,h µc,τ,h for every τ T c and c C. We now show that σ, µ ) s a sequental equlbrum of G and satsfes Assumpton 1. Snce Lemma 2 apples to the case g 2 = 0, n whch case σ τ 2) s ratonalzable for type τ 2, 5.3) σ c c h) h, τ c 1) = 1 c C h, h ). Hence, µ 2,τ 2,h s consstent and satsfes Assumpton 1. The sequental ratonalty condtons for ratonal types are SR1) and SR2) by constructon and 5.3). The sequental ratonalty and consstency for types n T c mmedately follows from the constructon and the fact that σ c, µ c ) s a sequental equlbrum n the auxlary game. The strategc equvalence under sequental equlbrum s somewhat subtle, requrng the lengthy proof above. Ths s because of the ssues relatng to the off-the-path belefs, whch play a central role n sequental equlbrum whle not beng relevant for ICR. If a type τ c 1, who plans to follow c, devates from c, then hs subsequent behavor may be dfferent from c as ICR cannot restrct the behavor at the contngences that are precluded by one s own strategy. In that case, off the path belefs of player 2 at the hstores that are not consstent wth any type could be dfferent. Moreover, consstency may result n unforeseen restrctons on those belefs as t s appled for types n T c and τ 2 smultaneously. Assumpton 1 ensures that Player 2 assgns zero probablty to τ c 1 whenever Player 1 devates from c, resultng n belefs that are dentcal to those wth commtment types, as we show n the proof. Of course, at the hstores that are consstent wth commtment types, the ratonal types n the games G and G face the same uncertanty regardng all relevant aspects, such as whether the other player s ratonal and whch c C he s playng f he s not ratonal. Ths leads to the same set of solutons for ratonal types n both games Indstngushablty of Testable Predctons. The strategc equvalence above mples that the testable predctons wth or wthout commtment types are nearly ndstngushable. Imagne that an emprcal or expermental researcher observes outcomes of games

21 REPUTATION WITHOUT COMMITMENT 21 that essentally look lke a fxed repeated game, as n g, but she does not know the players belefs about possble commtments or payoff varatons. Usng the data, she can obtan an emprcal dstrbuton on outcome paths. Because of samplng varaton, there s some nose regardng the actual equlbrum dstrbuton of the outcomes. The above strategc equvalence mples that the equlbrum dstrbutons for elaboratons wth or wthout commtment types can be arbtrarly close, makng t mpossble to rule out one model wthout rulng out the the other gven the samplng nose. Towards statng ths result formally, let Σ be the set of soluton concepts that are 1) nvarant to the elmnaton of non-ratonalzable plans, 2) nvarant to trval enrchments of the type spaces, and 3) nclude all solutons generated by the sequental equlbra that satsfy Assumpton 1. Gven any soluton concept Σ Σ and any Bayesan game G, a soluton σ leads to a probablty dstrbuton z σ) Z) on the set Z of outcome paths, such that z z σ) = σ s τ) π τ) z Z), τ T s S z where T s the sets of type profles n G, S z = {s S z s) = z} s the set of profles of acton plans that lead to z, π s the nduced) common pror on T, and σ s τ) s the probablty of acton plan s n equlbrum σ when the type profle s τ. A soluton concept Σ yelds a set Z Σ, G) = {z σ) σ Σ G)} of probablty dstrbutons on outcome paths. Towards comparng the dstance between such sets, we endow the set 2 Z) of such subsets wth the Hausdorff metrc d, the standard metrc for sets 7. For any X, Y 2 Z), d X, Y ) λ f and only f for each x X, there exst y Y and p Z) wth x = 1 λ) y + λp, and for each y Y, there exst x X and p Z) wth x = 1 λ) y + λp. Our frst corollary states that the set of dstrbutons on the outcome paths are nearly dentcal wth or wthout commtment types. 7 More specfcally, we use the Hausdorff metrc nduced by the total varaton metrc on Z), but snce we only use the metrc on sets, we wll smply defne the Hausdorff metrc drectly.

22 22 JONATHAN WEINSTEIN AND MUHAMET YILDIZ Corollary 1. For any Σ Σ, any ε-elaboraton G wth commtment types, and any ε ε, 1), there exsts an ε -elaboraton G wthout commtment types such that d Z Σ, G), Z Σ, G )) ε ε) / 1 ε). Proof. Defne λ = ε ε) / 1 ε). Consder the ε -elaboraton G n the prevous results. Recall that any type profle τ 1, τ 2) n G has dentcal solutons to a type profle f τ 1 ), τ 2) n G where f τ 1) = τ 1 and f c) = τ c 1. Moreover, π f τ 1 ), τ 2) = 1 λ) π τ 1, τ 2). Hence, y Z Σ, G ) f and only f there exsts σ Σ G ) such that y = 1 λ) x + λp for x and p where x z) = σ s f τ 1 ), τ 2) π f τ 1 ), τ 2) τ T {s S zs)=z} and p z) = τ 2 τ 2 {s S zs)=z} σ s τ) π τ). Snce the set of solutons for τ 1, τ 2) and f τ 1 ), τ 2) are dentcal, x Z Σ, G), and the converse s also true n that there exsts a σ Σ G ) as above for every x Z Σ, G). Suppose that one wants to restrct G to be an ε-elaboraton, so that the pror probablty of ratonal types are dentcal. The results n the reputaton lterature are often contnuous wth respect to ε when the set and the relatve probablty of the commtment types are fxed. In that case, such a restrcton would not make a dfference, as establshed n the next corollary. Corollary 2. Consder any ε-elaboraton G wth commtment types C, π) and a soluton concept Σ Σ such that Σ G α ) s contnuous wth respect to α, where G α s an αεelaboraton wth commtment types C, π/α) for α 1. Then, for any λ > 0, there exsts an ε-elaboraton G wthout commtment types such that d Z Σ, G), Z Σ, G )) λ. Proof. Apply the prevous result startng from G α for some α > 1 that s suff cently close to 1, n partcular where αε λ1 ε) + ε, and then apply contnuty. 6. G C In ths secton, we wll present the result for the n-player case, allowng commtment types for all players. The defntons for the n-player case mrror the case of n = 2, and we wll not repeat them here. Snce we wll allow commtment types for all players, an ε-elaboraton wth commtment types s now defned as a Bayesan game such that the set of types for

23 REPUTATION WITHOUT COMMITMENT 23 each player s {τ } C where C S can be empty, type τ can play any acton plan whle a type c C can play only c, and the probablty π τ ) of ratonal type profle s 1 ε. Note that when ε < 1, there some C s non-empty. Note also that the dstrbuton of commtment type s not restrcted; they can be correlated for example. Such a Bayesan game can be denoted by C 1,..., C n, π) where π {τ 1 } C 1 ) {τ n } C n )) s the pror on the type profles. The result s generalzed to ths case as follows. Proposton 4. For any ε, ε 0, 1) wth ε > ε and for any ε-elaboraton G wth commtment types C 1,..., C n, π) there exsts a strategcally-equvalent ε -elaboraton G = N, A, G, T, π )) wthout commtment types n whch the commtment types are replaced by types wth unque ratonalzable acton plans: 1) for every N, π g, τ τ ) = π τ τ ) ; 2) for every N and c C, there exsts τ c τ c T such that all ICR acton plans of are equvalent to c, and π τ c τ j) = π c τ j) for every j ; 3) for every Σ Σ, {σ τ ) σ Σ G)} = {σ τ ) σ Σ G )}. The frst two condtons all together state that each commtment type s replaced by a type that follows the commtted acton profle as hs unquely ratonalzable plan, and the nterm belefs of the ratonal types reman ntact under ratonalzablty. The last condton states that the two games are strategcally equvalent for ratonal types under any nvarant soluton concept, ncludng sequental equlbra that puts probablty one on ratonal types off the path. An outlne of the proof for ths result can be found n the appendx. 7. N C CK A P In the prevous sectons, whle we mposed the constrant that t s always common knowledge that the payoffs are the sum of dentcal stage-game payoffs, we allowed those payoffs to le anywhere n the nterval [0, 1]. In ths secton, by contrast, we make the strcter requrement that t s common knowledge that payoffs le wthn ε of those n the completenformaton game. Under ths strcter requrement, we show that commtment types are not dspensable n reputaton models. When the stage game s domnance solvable, there s a unque sequental Nash equlbrum outcome, n whch the unque ratonalzable strategy

24 24 JONATHAN WEINSTEIN AND MUHAMET YILDIZ profle of the stage game s played throughout. Here, ε s unform over all type spaces and the number of repettons. For example, n the repeated prsoners dlemma, one cannot have any cooperaton wthout commtment types when t s common knowledge that the payoffs are approxmately those n the prsoner s dlemma. Defne the dstance between two stage-game payoff functons va the sup norm: dg, g) = max g a) ga) a Proposton 5. Fx a complete nformaton stage game g whch has unque ratonalzable profle a. Then, there exsts ε > 0 such that for any ε > 0 and any t, every ε - elaboraton N, A, G, T, π)) wthout commtment types, satsfyng the addtonal requrement that dg, g ) < ε for all g G, has a unque sequental equlbrum n whch a s played by all types at all hstores. Proof. The elmnaton process for the fnte stage game g s fnte. Each tme an acton s elmnated agan by fnteness) t must be that for some δ t s never wthn δ of beng a best reply. Choose ε so that 2ε s smaller than the mnmum of these δ. Now suppose there s a sequental equlbrum strategy profle s whch contradcts the result. Consder one of the latest hstores at whch any volaton of the profle a occurs, and of the volatons at ths hstory, consder an acton a whch s elmnated frst n the elmnaton process for g, say at stage k. When player takes ths acton, he must beleve that a) the profle a s played at all future dates regardless of hs acton and b) no acton elmnated at stage k 1 or earler s played at the current hstory. But then by b), the fact that a s elmnated at stage k, and the choce of ε, hs acton s suboptmal n the stage game; and by a) hs acton cannot affect future play. Ths contradcts the concept of sequental equlbrum. For example, n a repeated prsoners dlemma game, f t s common knowledge that payoffs are close to the prsoners dlemma, then n any sequental equlbrum the players defect throughout the game regardless of the number of repettons. At some level ths s a reflecton of general contnuty propertes of Bayesan Nash equlbrum payoffs wth respect to the perturbatons of payoffs. Indeed, t s well known that, for any gven t, as ε 0, the Bayesan equlbrum payoffs n ε-elaboratons of repeated prsoners dlemma