Reputation without commitment in finitely repeated games

Transcription

1 Theoretcal Economcs 11 (2016), / Reputaton wthout commtment n fntely repeated games Jonathan Wensten Department of Economcs, Washngton Unversty n St. Lous Muhamet Yldz Department of Economcs, Massachusetts Insttute of Technology 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. Then classc reputaton results can be acheved wth uncertanty concernng only the stage payoffs. Keywords. Reputaton, repeated games, commtment. JEL classfcaton. C72, C Introducton 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 et al. (1982) (henceforth, the 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, so as 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. For example, the tt-for-tat types used by the Gang of Four n the analyss of the fntely repeated prsoner s dlemma could be assgned payoff 1 f they follow tt-for-tat and 0 otherwse. However, such payoffs cannot arse n a standard repeated game,.e., as a dscounted sum of stage-game payoffs. The only commtment types that arse drectly from modfed stage-game payoffs, wthn a standard repeated-game structure, are those who commt to playng the same acton Jonathan Wensten: j.wensten@wustl.edu Muhamet Yldz: myldz@mt.edu We thank the edtor, the referees, and semnar partcpants at Harvard, Koc, Prnceton, and Yale Unverstes and the SITE, especally Dlp Abreu and Stephen Morrs, for useful comments. 1 We refer to the textbook of Malath and Samuelson (2006) for a revew. Throughout the paper, we also refer to the same textbook for the exstng repeated-games results that we menton wthout a ctaton. Copyrght 2016 Jonathan Wensten and Muhamet Yldz. Lcensed under the Creatve Commons Attrbuton-NonCommercal Lcense 3.0. Avalable at DOI: /TE1893

2 158 Wensten and Yldz Theoretcal Economcs 11 (2016) throughout the game. In some models, such types have very sgnfcant effects, 2 but unlke tt-for-tat types, they have no effect on the repeated prsoner s dlemma game. The form of commtment types s mportant for the nterpretaton of reputaton results. When commtment types must be restrcted by fat to follow a certan plan or to have payoffs that are not a dscounted sum of stage-game payoffs, the lterature has sometmes referred to them as crazy types. A more generous characterzaton, more n keepng wth the current tone of the lterature, would say that commtment types reflect psychologcal anomales and motvatons that le outsde the game, such as mantanng reputaton n the context of a supergame. Alternatvely, f commtment types arse solely from heterogenety n stage-game payoffs and belefs about these payoffs, then reputaton formaton can occur wth full ratonalty and wthout resortng to such supergame concerns. 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 utlty-maxmzng type, whch we call a twn. The twn knows t s common knowledge that they play a repeated game (.e., he comes from a type space n whch only the stage-game payoff functons can vary by type), but 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, canbeconvertedto a strategcally equvalent model n whch they form a reputaton for certan belefs about the stage-game payoffs. Ths constructon requres that we allow suffcent varatons n stage-game payoffs and consder a rch set of nformaton structures. Of course, one may also wsh to restrct the stage-game payoff functons. For example, n a standard prsoner s dlemma game, one mght want to assume that t s common 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 so as to have any reputatonal effect. We show that f the stage game s domnance-solvable and the stage-game payoffs are restrcted to a suffcently 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 completenformaton verson), regardless of the length of the game. Therefore, one needs to allow some substantal amount of varaton n stage-game payoffs so as 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 2 For example, the exstence of such commtment types s suffcent for the semnal analyses of the repeated entry-deterrence models by Kreps and Wlson (1982) and Mlgrom and Roberts (1982), and for the Fudenberg and Levne (1989) result that the nformed player s payoff s wthn hs Stackelberg payoffs when the unnformed player s short-lved (best replyng myopcally).

3 Theoretcal Economcs 11 (2016) Reputaton wthout commtment 159 such a foundaton as long as there s enough varaton n allowable stage-game payoff functons. One lmtaton s worth emphaszng here. Our constructon makes fundamental use of players who do not know ther own payoffs. Some of the lterature has focused on models wth common knowledge that each player knows hs own payoffs; Fudenberg et al. (1988) call ths a model wth personal types. We beleve an mportant future step s to determne the extent to whch our results can be recovered n a model wth personal types. 2. Prevew of results In ths secton, we prevew our man result more carefully on the example analyzed by the 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: Cooperate Defect Cooperate Defect (PD) 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. The 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 τ1, whose payoffs and avalable moves are as n the repeated prsoner s dlemma game above, and a commtment type τ1 T4T, who 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 τ1 T4T s some small ε>0. Player2stllhas one type τ2, whch s ratonal as n the orgnal game. The Gang of Four shows that n any sequental equlbrum of the new game, each ratonal type τ must play Cooperate at all but a few perods. As we mentoned n the Introducton, one can replcate the above equlbrum behavor wth payoff uncertanty by assgnng the payoff functon of τ1 T4T 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 τ1 off the path. Such a payoff functon s ncompatble wth the repeatedgame payoff structure, and one cannot replcate the commtment to tt-for-tat by smply modfyng the stage-game payoff functon for τ1 T4T. Indeed, such modfcatons can lead to only two commtment types: the type that plays Cooperate throughout and the

4 160 Wensten and Yldz Theoretcal Economcs 11 (2016) type that plays Defect throughout. Commtment to cooperaton can be justfed by the stage-game payoff functon Cooperate Defect Cooperate 1 1 Defect 0 0 (CC) for example. The ncluson of such smple commtment types cannot affect the behavor of ratonal types n ths game. However, the austere nformaton structure above s not the only structure we can consder. Our man result (Proposton 1) uses rcher type spaces to replcate arbtrary commtment types by payoff types. For any ε >ε, applyng our man result to the game G n the Gang of Four generates a game G wth the followng propertes. Ex Ante Proxmty. The pror probablty of the ratonal type profle (τ1 τ 2 ) s at least 1 ε,andeachτ 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. The games G and G are strategcally equvalent n the followng sense: 1. The game G contans types τ1, τ 2,atwn ˆτT4T 1 of the tt-for-tat type τ1 T4T n G, and a number of other new types (of both players) that we use to encode the belefs of type ˆτ 1 T4T. 2. Though ˆτ T4T 1 s allowed to play any plan of acton, tt-for-tat s hs unque ratonalzable plan. 3. The nterm belefs of ratonal types are equvalent n G and G : ratonal type τ1 s certan that he faces the ratonal type τ2, and the ratonal type τ 2 n turn puts probablty 1 ε on τ1 and probablty ε on the twn ˆτT4T 1 of τ1 T4T. By the strategc equvalence property, the strategc stuaton the ratonal types face s the same as n G, exceptnowτ2 thnks that ˆτT4T 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 (τ1 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 (τ1 τ 2 ) and ε(1 ε )/(1 ε) on ( ˆτ 1 T4T τ2 ), and the remanng small probablty (ε ε)/(1 ε) on the newly constructed types. Two ponts about ths constructon are worth emphaszng. Frst, when ε ε s small compared to ε, the pror probabltes of (τ1 τ 2 ) and ( ˆτT4T 1 τ2 ) are approxmately 1 ε

5 Theoretcal Economcs 11 (2016) Reputaton wthout commtment 161 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 1 T4T assgns much larger probablty to the standard type τ2 than to the new types. Despte ths, ˆτT4T 1 has a unque ratonalzable plan because ˆτ 1 T4T beleves that hs own plan has a nonneglgble 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. Our results buld on our prevous work on nonrobustness 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 that mantans common knowledge of the repeatedgame 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. Asde from the obvous dfferences n motvaton and applcatons, there are two major techncal dstnctons from our work n Wensten and Yldz (2013). Frst, extendng the above lemma from nfntely repeated games to fntely repeated games requres a more dffcult constructon, as we cannot use future ncentves n the last perod of a fntely repeated game. Second, the perturbatons allowed here are more constraned. Here, as n the tradtonal reputaton lterature, we create a perturbed model that assgns hgh ex ante probablty to the orgnal model. Ths s also smlar to the perturbatons n Kaj and Morrs (1997) and other papers on robustness. Ths ex ante noton of perturbaton commonly gves very dfferent results from our nterm framework n Wensten and Yldz (2013) and earler papers, where we allow arbtrary perturbatons of nterm belefs n the unversal type space. 3 One reason the results here can be acheved wth ex ante perturbatons s that our constructon centers around perturbng the commtment types,whodonothavesetbelefs.themandffcultyturnsouttobeembeddngtypes constructed n the lemma nto a common-pror model wthout affectng the types ratonalzable actons, whle keepng the ex ante probabltes of the new types arbtrarly small. We ntroduce the basc defntons and formulatons n Secton 3. InSecton 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. In Secton 6, we generalze our result to n-player games n whch all players may 3 The key dfference s that the ex ante perturbatons under a common pror mpose addtonal commonbelef restrctons (Kaj and Morrs 1997), whch are crucal n extendng the equlbra of the orgnal game to the perturbed one (Monderer and Samet 1989).

6 162 Wensten and Yldz Theoretcal Economcs 11 (2016) have commtment types. After presentng our contnuty result n Secton 7, we offer further remarks on the lterature n Secton 8 and conclude n Secton 9. Some of the more complcated proofs are relegated to the Appendx. 3. Basc defntons We begn wth a standard two-player fntely repeated game wth perfect montorng and normal-form stage games; see Secton 6 for the n-player 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 ). 4 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 s denotes the stage-game strategy profle played at date s T.Wewrteh 0 for the empty ntal hstory and wrte H for the set of all nontermnal hstores. An outcome path, or 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 a repeated game s smply the sum 5 of the stage-game payoffs, u(a 0 a 1 a t g) = g(a 0 ) + g(a 1 ) + +g(a t ) (1) 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. 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 g 2 ). The payoff functon n the repeated game s u( g ), gven by the formula n (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 nontermnal 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, reservng 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 for the set of mxed acton plans σ : H (A ) for player. 4 Followng the conventon n game theory, we wrte for the player j and drop the subscrpt to denote profles, e.g., x = (x 1 x 2 ) X = X 1 X 2 and X 1 = X 2. 5 Dscountng would not affect our results; settng the dscount rate to 1 smplfes our dervatons.

7 Theoretcal Economcs 11 (2016) Reputaton wthout commtment 163 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 elaboraton uses commtment types,assstandardnthereputaton lterature. Defnton 1. An ε-elaboraton wth (one-sded) commtment types (C π) s a Bayesan game such that the followng statements hold: The sets of types for players 1 and 2 are {τ1 } C and {τ 2 }, respectvely, where C S 1. Player 2 s belef π about player 1 s type satsfes π(τ 1 ) = 1 ε. The set of plans avalable to τ s as n the repeated game above, whle the only avalable plan for type c C s c. The payoffs are as n the complete-nformaton game. 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 τ1 who can play all plans s called theratonal type, whle the types c C who canplay only accordngto one plan of acton are called commtment types. Observe that snce C S 1,weconfneourselvestopure commtment types. 6 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 (1). Toward statng ths noton formally, we defne a type space as a lst (G T π), where G G s a fnte set of payoff functon profles g, T = T 1 T 2 s the set of type profles τ = (τ 1 τ 2 ),andπ (G T ) s the common pror. 7 We defne a Bayesan repeated game (wthout commtment types) as a lst (N A (G T π)). 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 (1). Defnton 2. An ε-elaboraton wthout commtment types of a complete-nformaton game g s a Bayesan repeated game (N A (G T π)) wth dstngushed types τ 1, τ 2, where 1. (g τ ) G T 2. π(g τ ) = 1 ε 3. π(g τ ) = 1. 6 Ths s wthout loss of generalty because a belef n a commtment type that plays a mxed strategy n the repeated game s equvalent to a belef n a mxture of pure commtment types (see Secton 9 below). 7 Here, (X) denotes the set of all probablty measures on the fnte set X.

8 164 Wensten and Yldz Theoretcal Economcs 11 (2016) 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 (τ1 τ 2 ) know ther payoffs, and ther payoffs are as n the orgnal complete-nformaton game. The novelty n ths defnton s that the elaboraton s requred to be a Bayesan repeated game,.e., the structure gven by the formula (1) s common knowledge. In that sense, all the types n an elaboraton wthout commtment types are ratonal, although we reserve the term ratonal for types (τ1 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 (1) s common knowledge. Whle ε-elaboratons wth commtment types were presented above n terms of uncertanty about the set of avalable strateges, they could also be represented as Kaj Morrs elaboratons wth a specfc smple type space n whch the formula (1) fals. Fnally, we revew a couple of standard concepts n game theory. Frst, a strategy of a player n a Bayesan repeated game (N A (G T π)) s a mappng σ : T. Second, 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 et al. (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. Thrd, 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 f and only f s (h t ) = s (ht ) 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 from the hstory of other players play to own stagegame actons. We wrte S for the set of reduced-form acton plans s ; these map each (a l ) 0 l<t to some acton a A n the stage game. Fnally, we ntroduce the followng notaton for sets of equvalent acton plans: Gven any two sets X, Y of acton plans, we wrte X Y f for every x X, thereexstsy Y that s equvalent to x, andforevery y Y,thereexstsx X that s equvalent to y. In partcular, X {x } means that X conssts only of strateges that are equvalent to x. 4. Irrelevance of commtment types In ths secton, we state and outlne the proof of our man result: any ε-elaboraton wth commtment types can be transformed, for any ε >ε, nto an ε -elaboraton wthout commtment types, where each commtment type c s replaced wth a payoff type τ c 1 for whch c s unquely ratonalzable. These payoff 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

9 Theoretcal Economcs 11 (2016) Reputaton wthout commtment 165 pont of vew of the ratonal types, these are the only types wth postve probablty, mrrorng the elaboraton wth commtment types. From the pont of vew of ratonal types who beleve n the ICR concept, the two elaboratons are dentcal. Hence, under ICR (as well as a broader set of soluton concepts), the set of solutons for each ratonal type s dentcal n the two elaboratons. Proposton 1. For any ε ε (0 1) wth ε >εand for any ε-elaboraton G wth commtment types (C π), there exsts an 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, meanng 1. π (g τ 2 τ 1 ) = 1 and π (g τ 1 τ 2 ) = π(g τ 1 τ 2 ) = 1 ε 2. for every c C, there exsts τ c 1 T 1 such that S [τ c 1 ] {c} and π (τ c 1 τ 2 ) = π(c τ 2 ) = π(c). Here, the frst condton states 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 states that each commtment type c s replaced by a type τ1 c for whch followng c s unquely ratonalzable n reduced form (.e., S [τ1 c] {c}), and that the ratonal type of player 2 n G assgns the same probablty to the type τ1 c as the ratonal type n G assgns to the commtment type c (.e., π (τ 1 c τ 2 ) g 1 π ((g 1 g2 ) τc 1 τ 2 ) = π(c τ 2 ) = π(c)). The equvalence of G wth G s establshed despte the followng constrants: 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 τ1 c 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. 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; weprovdeadetaledntutonfortlater n ths secton.) Lemma 1. For any s S, there exsts a Bayesan repeated game G s (G s T s π s )) wth a type τ s T s such that = (N A 1. π s (g τ) > 0 for every (g τ) G s T s 2. S [τ s G s ] {s }.

10 166 Wensten and Yldz Theoretcal Economcs 11 (2016) By relabelng f necessary, we take all of the types n the type spaces T s 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 g2 )} G c c C where T ={τ } T c ( N) c C 1 ε f (g τ) = (g τ ) 1 ε π 1 ε (g τ) = π(c) f (g τ) = ((0 g 2 ) (τc 1 τ 2 )) ε ε (1 ε) C πc (g τ) f (g τ) G c T c 0 otherwse, 0(a) = 0 ( a A) We now observe that G satsfes the propertes n the proposton. Indeed, ratonal type τ1 of player 1 assgns probablty 1 on (g τ2 ). Lkewse, we have π (G {τ 2 }) = 1 ε + 1 ε 1 ε c C π(c) = 1 ε + 1 ε 1 ε ε = (1 ε )/(1 ε) and, therefore, n the nterm, τ2 assgns probablty 1 ε to (g τ1 ) and probablty π(c) to τ1 c for each c. Snce the belefs of type τc 1 altered substantally when (Gc T c π c ) was ncorporated n G,tsnotclearthatτ1 c 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 ];npartcular, S 1 [τc 1 G ]=c. Ths lemma completes the proof of the proposton; ts proof s gven n the Appendx. Our proof has two man steps. The frst, found n Lemma 1, stoconstructatype 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 have 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 the deas and the results developed by Lpman (2003)andWensten and Yldz (2007). The man economc deas nvolved n these constructons come from socal learnng and reward/punshment mechansms n repeated games. Our frst constructon nvolves types who know ther stage-game payoff s a functon of ther own acton alone,

11 Theoretcal Economcs 11 (2016) Reputaton wthout commtment 167 but do not ntally know ther optmal acton. They wll learn ther optmal acton from the actons of others. Only some plans are consstent wth such belefs; for nstance, no such player could play move a 0 n perod 0 and the same move a 1 a 0 n all contnuatons. For nfnte-horzon games, we extended the result to all acton plans (ncludng plans that contradct the condton for ndvdual learnng) usng a reward and punshment mechansm n Wensten and Yldz (2013). It 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. In our constructon, the player s stage payoffs are addtvely separable n hs acton and others actons. In all perods before the last, hs ncentves are domnated by the desre to be rewarded by the other players, whle n the last perod, he has learned hs own optmal acton and acts accordngly. The second man step s to ncorporate the above type spaces nto one commonpror 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 the common-pror model. Condtons () and () oppose each other, makng the constructon more dffcult. 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). Whenε 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 Gc. In our constructon, type τ c 1 n game G assgns probablty p c = C (1 ε )π(c) C (1 ε )π(c) + (ε ε)π c (τ) (2) on type τ2. Note that, for fxed π(c), whenε ε approaches 0, p c approaches 1. 8 In contrast, τ1 c n game Gc assgns zero probablty on τ2. Consequently, the belef herarchesofthetypesng can be qute dfferent from the belef herarches of the types n G c wth the same label, whch could lead to a dstnct set of ICR actons. We crcumvent ths problem wth the followng trck. We set the belefs such that whenever player 2hastypeτ2, the payoff of type τc 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 τ2, thereby replcatng the best responses of hs twn n G c. Snce ths was the only dfference between the two 8 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 pc near 1 even when ε and ε are far apart.

12 168 Wensten and Yldz Theoretcal Economcs 11 (2016) 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 to the same set of possble behavor. We wll next establsh such strategc equvalence formally. 5. Strategc equvalence In ths secton, we show that the elaboratons G wth commtment types and G wthout commtment types descrbed n Proposton 1 are strategcally equvalent for ratonal types. By ths we mean that, for a broad set of soluton concepts, the sets 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. 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 nonratonalzable 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 equvalence for sequental equlbrum; ths requres an addtonal off-path belef restrcton commonly mposed n the reputaton lterature. 5.1 Strategc equvalence under nvarant solutons The followng defntons are standard: A soluton concept maps every Bayesan game G to a set (G) of mxed strateges n game G. ForanytypespacesT 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 s sad to be nvarant to elmnaton of nonratonalzable strateges f (G) = (G ) for any two games G and G wth dentcal type spaces satsfyng () f an acton plan s s avalable for a type τ n game G, thens s avalable for τ n G, and () f s s not avalable for τ n G,thens / S [τ G ].

13 Theoretcal Economcs 11 (2016) Reputaton wthout commtment 169 Defnton 4. A soluton concept s sad to be nvarant to trval enrchments of the type spaces f (G) ={σ T σ (G )} for any two games G and G wth type spaces T and T such that () T T, () every type n T has an 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 nonratonalzable 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. Proposton 2. Let be a soluton concept that s nvarant to elmnaton of nonratonalzable strateges and to trval enrchment of the type spaces. Then, for any ε ε (0 1) wth ε >εand 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 (G)}={σ(τ ) σ (G )}.e., the sets 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 () ntroducng new types such that only commtted types beleve n the new types, and () 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 step undoes an elmnaton covered by Defnton 4, so the concluson follows. 5.2 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. 9 An assessment s a par ( σ μ) of strategy profle σ : T 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 h n the contnuaton game. An assessment ( σ μ) s sad to be consstent f there exsts a sequence ( σ n μ n ) ( σ μ) such that σ n assgns postve probablty to 9 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.

14 170 Wensten and Yldz Theoretcal Economcs 11 (2016) each avalable move at every hstory and μ n s derved from Bayes rule and σ n.anassessment ( 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 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 >εand 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 the same n games G and G. Proof. We take G as n Proposton 1. 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 condton for τ2 s μ 2 τ 2 h(c) = μ σ( τ 1 ) h (c) { π(c) Pr(h σ( τ1 ))(1 ε)+ c C h π(c f c C ) h 0 otherwse ( h c) (C)

15 Theoretcal Economcs 11 (2016) Reputaton wthout commtment 171 where C h s the set of commtment plans c C that s consstent wth hstory h. Of course, μ σ( τ 1 ) h (τ1 ) = 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( τ1 ) and, hence, the followng sequental ratonalty condtons are solely on σ( τ ). (SR1) The term σ( τ 1 ) s a sequental best response to σ( τ 2 ) under g 1. (SR2) At each hstory h, σ( τ2 ) s condtonal best response to the mxed strategy σ μ σ( τ 1 ) h (τ1 )σ( τ 1 ) + μ σ( τ 1 ) h (c)c c C under g 2. 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. Frst, snce type τ1 puts probablty 1 on (g τ2 ), the sequental ratonalty condton for that type s (SR1). Second, snce c s the unque ratonalzable acton plan of τ1 c n G (by Lemma 2) onall hstores h consstent wth c, Hence, by Assumpton 1 and consstency, σ(c(h) h τ c 1 ) = 1 ( c Ch h) (4) μ 2 τ 2 h(τc 1 ) = μσ( τ 1 ) h (c) ( h c) (5) whch of course also mples that μ 2 τ2 h(τ 1 ) = μσ( τ 1 ) h (τ1 ). By (4) and(5), under the belef of type τ2, player 1 plays accordng to σ above, and the sequental ratonalty condton for type τ2 s (SR2). To show that (SR1) and (SR2) are suffcent for σ( τ ) SE (G ),takeanyσ( τ ) that satsfes (SR1) and (SR2). We wll construct a sequental equlbrum (σ μ ) of G that satsfes Assumpton 1. Setμ 1 τ1 h(g τ2 ) = 1 and μ 2 τ2 h = μσ( τ 1 ) h.foreachc C, consder a sequental equlbrum (σ c μ c ) of the game n whch the acton plan of type τ2 s fxed as σ( τ 2 ) as moves of nature and the type space s T c wth the nterm belefs n G.Setσ( 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, σ c (c(h) h τ c 1 ) = 1 ( c Ch h) (6) Hence, μ 2 τ2 h s consstent and satsfes Assumpton 1. The sequental ratonalty condtons for ratonal types are (SR1) and (SR2) by constructon and (6). The sequental ratonalty and consstency for types n T c mmedately follow from the constructon and the fact that (σ c μ c ) s a sequental equlbrum n the auxlary game.

16 172 Wensten and Yldz Theoretcal Economcs 11 (2016) 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 τ1 c, who plans to follow c, devatesfromc, 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-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 τ2 smultaneously. Assumpton 1 ensures that player 2 assgns zero probablty to τ1 c 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. Remark 1. 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 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 wth some nose. 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 other gven the samplng nose (see the supplementary fle on the journal webste, for a formal result along these lnes). 6. General case 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, wth common pror π, such that the set of types for each player s {τ } C,whereC S can be empty, type τ can play any acton plan whle a type c C canplayonlyc, and the probablty π(τ ) of the ratonal type profle s 1 ε. Note that when ε>0, somec s nonempty. 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. Fnally, we wrte for the set of soluton concepts that are () nvarant to the elmnaton of nonratonalzable plans, () nvarant to trval enrchments of the type spaces, and () nclude all solutons generated by the sequental equlbra that satsfy Assumpton 1. The result s generalzed to ths case as follows.

17 Theoretcal Economcs 11 (2016) Reputaton wthout commtment 173 Proposton 4. For any ε ε (0 1) wth ε >εand 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 T such that all ICR acton plans of τ c are equvalent to c,andπ (τ c τj ) = π(c τj ) for every j. 3. For every, {σ(τ ) σ (G)}={σ(τ ) 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 put probablty 1 on ratonal types off the path. An outlne of the proof for ths result can be found n the Appendx. 7. Necessty of commtment under common knowledge of approxmate payoffs 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 complete-nformaton 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 profle of the stage game s played throughout. Here, ε s unform over all type spaces and number of repettons. For example, n the repeated prsoner s 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: d(g g)= max a g (a) g(a) Proposton 5. Fx a complete-nformaton stage game g that has unque ratonalzable profle a. Then there exsts ε>0 such that for any ε > 0 and any t, everyε - elaboraton (N A (G T π)) wthout commtment types, satsfyng the addtonal requrement that d(g g )<εfor all g G, has a unque sequental equlbrum n whch a s played by all types at all hstores.

18 174 Wensten and Yldz Theoretcal Economcs 11 (2016) 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>0, t s never wthn δ of beng a best reply. Choose ε>0 so that 2ε s smaller than the mnmum of these δ. Now suppose there s a sequental equlbrum strategy profle s that 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 that 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 prsoner s dlemma game, f t s common knowledge that payoffs are close to the prsoner s 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, asε 0, the Bayesan equlbrum payoffs n ε-elaboratons of repeated prsoner s dlemma wth commtment types approach the payoffs from defecton throughout the game. Ths s n lne wth the contnuty results for Nash equlbrum payoffs wth respect to the pror dstrbutons. Hence, for a gven t, our result here dffers from the exstng contnuty results only n terms of the perturbatons t consders, makng the stage payoffs approach to the orgnal game nstead of makng the probablty of types wth unrelated payoffs go to zero. Our result has a major strength however: ε s unform wth respect to the number of repettons. In contrast, for any ε probablty of a tt-fortat type, cooperaton prevals whenever the number of repettons are suffcently large, as famously establshed by the Gang of Four. 8. Remarks Contnuty and robustness of equvalence Snce nterm correlated ratonalzablty s upper hemcontnuous (Dekel et al. 2007), each type τ c wth unque ratonalzable acton c has the same unque ratonalzable acton on a open neghborhood of parameters and belefs. Hence, n the elaboraton constructed n Proposton 1, we can perturb parameters such as the stage-game payoff functons and belefs for the newly constructed types as long as the belefs of ratonal types are fxed. So, relatve to the set of elaboratons wth the same set of types, where ratonal players have fxed belefs, we obtan an open set of ε -elaboratons G wthout commtment types that are strategcally equvalent to G. In partcular, type τ c need not be exactly ndfferent between hs actons condtonal on meetng a ratonal type; ths was only a smplfyng aspect n our constructon.

19 Theoretcal Economcs 11 (2016) Reputaton wthout commtment 175 Our result s slent, through, on the other hand, about contnuty wth respect to varaton of the belefs of the ratonal types. Such contnuty s drectly ted to the contnuty propertes of the soluton concept n the orgnal game G, by our strategc equvalence result. Of course, snce ε must be larger than ε (albet arbtrarly close), our result and the reputaton result that t s appled to are relevant only when the soluton concept on the orgnal model G s contnuous wth respect to small varatons n ε when the commtment types and ther relatve probabltes wth respect to each other are fxed. Ths s ndeed the case for most exstng models. 10 Senstvty to the set of commtment types Despte the contnuty n the prevous paragraph, the equlbrum predctons of reputatonal models are hghly senstve to the set of commtment types one consders: by varyng the set of commtment types, one can obtan a rch set of behavor as equlbrum outcomes n long but fntely repeated games. Indeed, Fudenberg and Maskn (1986) obtan a folk theorem n ths way. Once agan, such senstvty to the set of commtment types wll be nherted by our newly constructed reputaton models wthout commtment types, due to strategc equvalence. Short-lved players The above senstvty s muted when the unnformed player s short-lved (.e., she myopcally best responds to her belef about the other player s move at every hstory). In that case, n any Bayesan Nash equlbrum, the payoff of the ratonal player wth commtment types s near hs Stackelberg payoff, provded that he s suffcently patent and has a type that always plays hs Stackelberg move (Fudenberg and Levne 1989). Snce our players are all long-lved, such an ndependence result does not hold n the reputaton models we consder here. For example, n the repeated prsoner s dlemma game, the Stackelberg type always plays Defect, and the presence of such a type would not have any qualtatve mpact on the equlbrum behavor. When there s a tt-for-tat type, we would stll have cooperaton n all but a few rounds. Here, the payoff of the ratonal type exceeds hs Stackelberg payoffs, but hs payoff could be lower than hs Stackelberg payoffs n other games. 11 We must emphasze that our man result for two-player games would stll be true f we assumed that player 2 s short-lved nstead. In that case, for player 2, we could stll generate any acton plan that s consstent wth her stage payoff beng a functon of her own acton only (as n Lemma 3 n the Appendx). Sncethssallweneedforour Lemma 4 n the Appendx, all of our results would go through as s. 10 For example, sequental equlbrum s upper hemcontnuous wth respect to such scalng of the probabltes of commtment types. Bayesan Nash equlbrum behavor of the ratonal types s also upper hemcontnuous wth respect to all varatons of prors (wth possbly varyng commtment types and relatve probabltes) because such varatons can be represented as an ex ante payoff perturbaton. 11 Ths fact has been demonstrated n nfntely repeated games, but we suspect that t can also be shown n long but fntely repeated games.