Contemporaneous perfect epsilon-equilibria

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1 Games and Economc Behavor ) Contemporaneous perfect epslon-equlbra George J. Malath a, Andrew Postlewate a,, Larry Samuelson b a Unversty of Pennsylvana b Unversty of Wsconsn Receved 8 January 2003 Avalable onlne 18 July 2005 Abstract We examne contemporaneous perfect ε-equlbra, n whch a player s actons after every hstory, evaluated at the pont of devaton from the equlbrum, must be wthn ε of a best response. Ths concept mples, but s stronger than, Radner s ex ante perfect ε-equlbrum. A strategy profle s a contemporaneous perfect ε-equlbrum of a game f t s a subgame perfect equlbrum n a perturbed game wth nearly the same payoffs, wth the converse holdng for pure equlbra Elsever Inc. All rghts reserved. JEL classfcaton: C70; C72; C73 Keywords: Epslon equlbrum; Ex ante payoff; Multstage game; Subgame perfect equlbrum 1. Introducton Analyzng a game begns wth the constructon of a model specfyng the strateges of the players and the resultng payoffs. For many games, one cannot be postve that the specfed payoffs are precsely correct. For the model to be useful, one must hope that ts equlbra are close to those of the real game whenever the payoff msspecfcaton s small. To ensure that an equlbrum of the model s close to a Nash equlbrum of every possble game wth nearly the same payoffs, the approprate soluton concept n the model * Correspondng author. E-mal addresses: gmalath@econ.upenn.edu G.J. Malath), apostlew@econ.upenn.edu A. Postlewate), larrysam@ssc.wsc.edu L. Samuelson) /$ see front matter 2005 Elsever Inc. All rghts reserved. do: /j.geb

2 G.J. Malath et al. / Games and Economc Behavor ) s some verson of strategc stablty Kohlberg and Mertens, 1986). In ths note, we take the alternatve perspectve of an analyst seekng to ensure that no Nash equlbra of the real game are neglected. The approprate soluton concept n the model s then ε-nash equlbrum: It s a straghtforward exercse Proposton 3 below) that a strategy profle s an ε-nash equlbrum of a game f t s a Nash equlbrum of a game wth nearly the same payoffs. When dealng wth extensve-form games, one s typcally nterested n sequental ratonalty. Radner 1980) defned a perfect ε-equlbrum as a strategy profle n whch each player, followng every hstory of the game and takng opponents strateges as gven, s wthn ε of the largest possble payoff. In Radner s perfect ε-equlbra, however, the gan from devatng from the proposed equlbrum strategy s evaluated from the vantage pont of the begnnng of the game. We accordngly refer to ths as an ex ante perfect ε-equlbrum. The choce of vantage pont s relatvely nnocuous under Radner s average payoff crteron. However, f the game s played over tme and the players payoffs are the dscounted present values of future payments, then there may be ex ante perfect ε-equlbra n whch a player has a devaton strategy that yelds a large ncrease n hs payoff n a dstant perod, but a qute small gan when dscounted to the begnnng of the game. At the begnnng of the game, the player s behavor wll then be ex ante ε optmal. But condtonal on reachng the pont where the devaton s to occur, the gan wll be large. As a result, such an ε-equlbrum wll not be a subgame perfect equlbrum of any game wth nearly the same payoffs. We propose an alternatve defnton of approxmate equlbrum that requres every player to be wthn ε of hs optmal payoff after every hstory, where the evaluaton of strateges s made contemporaneously, that s, evaluatons are made at the pont that an alternatve strategy devates from the proposed strategy. We call a vector of strateges that satsfes ths crteron a contemporaneous perfect ε-equlbrum. Followng the prelmnares presented n Sectons 2 4, Secton 5 shows that any subgame perfect equlbrum n a nearby game s a contemporaneous perfect ε-equlbrum n the game n queston, wth the converse holdng for pure strateges. 2. Multstage games wth observed actons We consder multstage games wth observed actons Fudenberg and Trole, 1991, Chapter 4). There s a potentally nfnte number of perods {0, 1,...,}. In each perod, some subset of the players smultaneously choose from nontrval feasble sets of actons, knowng the hstory of the game through the precedng perod. In each perod, the feasble sets may depend upon the perod and the hstory of play, and some players may have no choces to make. Let G denote such a game. Ths class of games ncludes both repeated games and dynamc games lke Rubnsten s 1982) alternatng-offers barganng game. Every nformaton set for a player n perod t corresponds to a partcular hstory of actons taken before perod t. The converse need not be true, however, snce a player may be constraned to do nothng n some perods or after some hstores. In addton, some fnte hstores may correspond to termnal nodes endng the game, as s the case after an agreement n alternatng-offers barganng. The set of hstores correspondng to an

3 128 G.J. Malath et al. / Games and Economc Behavor ) nformaton set for player s denoted H, and the set of all hstores s denoted H. Hstores n H \ H are called termnal hstores. The set of actons avalable to player at the nformaton set h H s denoted by A h). We assume that each A h) s fnte. Let σ be a behavor strategy for player, so that σ h) A h)) assocates a mxture over A h) to h. Endow each A h)) a subset of a Eucldean space) wth the standard topology and endow the set of strategy profles wth the product topology. Let σ h be the strategy σ, modfed at only) s nformaton sets precedng h so as to take those pure actons consstent wth play generatng the hstory h. In multstage games wth observed actons, the actons specfed by σ h and σ h are unque at each of the precedng nformaton sets. The length of the hstory h s denoted th). Snce the ntal perod s perod 0, actons taken at the nformaton set h are taken n perod t h). In a dynamc envronment, players may receve payoffs at dfferent tmes. We are nterested n the dfference between a decson wth mmedate monetary or physcal consequences and a decson wth the same monetary or physcal consequences, but realzed at some pont n the future. To capture ths dstncton, we formulate payoffs n terms of a dscountng scheme and a reward functon. The reward functon s denoted by r : H R, where r h) s the reward player receves after the hstory h. We emphasze that the reward r h) s receved n perod th) 1 recall that the ntal perod s perod 0), and that t can depend on the entre sequence of actons taken n the precedng th) perods. Player dscounts perod t rewards to perod t 1 usng the factor δ t 0, 1]. For t <t, defne δ t,t ) t τ=t +1 δ τ, so that a reward r receved n perod t has value δ t,t ) r n perod t. We sometmes wrte δ t) for δ 0,t).Wesetδ 0 = δ t,t) = 1. Fnally, for notatonal smplcty, f the game ends n perod T,wesetδ t = β for some fxed β 0, 1) for all t>t. We assume that players dscount, n that there exsts D< such that, for all, sup T t t=t τ=t +1 δ τ = sup T t=t T,t) δ D. 2.1) Ths dscountng formulaton s suffcently general as to mpose very lttle restrcton on the payoffs of the game. For example, the possblty of dfferent dscount factors n dfferent perods allows us to capture games lke Rubnsten s alternatng-offers barganng game, where usng our numberng conventon for perods) offers are made n even perods, acceptance/rejectons n odd perods, and δ t = 1 for all odd t. In addton, we have mposed no bounds on the reward functons r. Hence, by allowng rewards to grow suffcently fast, we can model games n whch future payoffs have larger present values than current ones, even wth dscountng. However, the dscountng scheme s essental n capturng the player s relatve evaluaton of rewards receved n dfferent perods, and hence to our study of ex ante and contemporaneous perfect ε-equlbra. The set of pure strategy profles s denoted by Σ, the outcome path nduced by the pure strategy profle s Σ s denoted by a s), and the ntal t + 1 perod hstory s denoted

4 G.J. Malath et al. / Games and Economc Behavor ) by a t s). For notatonal smplcty, f a s) s a termnal hstory of length T, we defne r a t s)) = 0 for all t T. Player s payoff functon, π : Σ R, s gven by t π s) = δ τ )r a t s) ) = δ t) r a t s) ). 2.2) t=0 τ=0 t=0 We assume the reward functon s such that ths expresson s well-defned for all s Σ. We extend π to the set of behavor strategy profles, Σ, n the obvous way. Ths representaton of a game s qute general. In Rubnsten s alternatng-offers barganng game, r h) equals s share f an agreement s reached n perod th)under h, and zero otherwse. In the T -perod centpede game, we let δ t = 1fort<T snce there are only fntely many perods, ths satsfes our dscountng assumpton) and let r h) equal s payoff when the game s stopped n perod th) 1 under h. Defne π σ h) as the contnuaton payoff to player under the strategy profle σ, condtonal on the hstory h. For pure strateges s Σ, we have recall that δ t,t) = 1): π s h) = r a th) ) s h)) t + δ τ r a t s h)) = t=th) t=th)+1 τ=th)+1 δ th),t) r a t s h)). 2.3) Note that a th) s h ) a th) 1 s h ), a th) s h )) s the concatenaton of the hstory of actons that reaches h and the acton profle taken n perod th). 3. Epslon equlbra The strategy profle σ h specfes a unque hstory of length th) that causes nformaton set h to be reached, allowng us to wrte: th) 1 π σ h ) = δ t) r a t σ h)) + δ th)) π σ h h ). t=0 In other words, for a fxed hstory h and strategy profle σ, π σ h, ) s a player payoff functon on the space of player s strateges of the form σ h that s a postve affne transformaton of the payoff functon π σ, h). Defnton 1. For ε>0, a strategy profle ˆσ s an ε-nash equlbrum f, for each player and strategy σ, π ˆσ) π ˆσ,σ ) ε. A strategy profle ˆσ s an ex ante perfect ε-equlbrum f, for each player, hstory h, and strategy σ, π ˆσ h ) π ˆσ h,σ h ) ε.

5 130 G.J. Malath et al. / Games and Economc Behavor ) A strategy profle ˆσ s a contemporaneous perfect ε-equlbrum f, for each player, hstory h, and strategy σ, π ˆσ h) π ˆσ,σ h) ε. Ex ante ε-perfecton appears n Radner 1980) and Fudenberg and Levne 1983). 1 Any contemporaneous perfect ε-equlbra s an ex ante perfect ε-equlbrum, and the two concepts concde n the absence of dscountng or when ε = 0 n whch case they also concde wth subgame perfecton). 2 Radner, 1980) studes ε-equlbra n a repeated olgopoly that are ex ante but not contemporaneous ε-equlbra. We use the fntely repeated prsoners dlemma to capture the sprt of hs analyss, showng that ex ante and contemporaneous perfect ε-equlbra for the same value of ε>0 can be qute dfferent: Example 1. The fntely repeated prsoners dlemma. The stage game s gven by C D C 2, 2 1, 3 D 3, 1 0, 0. Ths game s played N + 1 tmes, wth payoffs dscounted accordng to the common dscount factor δ<1. The unque Nash and hence subgame perfect) equlbrum features perpetual defecton. Consder trgger strateges that specfy cooperaton after every hstory featurng no defecton, and defecton otherwse. If δ s suffcently close to 1, the only potentally proftable devaton wll be to defect n the last perod. As long as N s suffcently large that δ N <ε, 3.1) the beneft from ths defecton s below the ε threshold, and the trgger strateges are an ex ante perfect ε-equlbrum. However, for any ε<1, the unque contemporaneous perfect ε-equlbrum s to always defect. In Example 1, for suffcently small ε n partcular, so that 3.1) s volated), both players must defect n every perod n any ex ante perfect ε-equlbrum of a fntely repeated 1 Radner 1981), whch consders as does Radner, 1980) only players who maxmze average payoffs, used the term robust epslon equlbrum for what we call ex ante perfect ε-equlbrum. Radner 1980, p. 153) also defnes an alternatve noton of perfect ε-equlbrum n whch the utlty of a contnuaton strategy s calculated relatve to the perod at whch the decson s beng made. Whle the resultng focus on payoffs condtonal on hstory s n the sprt of contemporaneous perfect ε-equlbrum, there s a subtlety that arses from the payoff crtera. Snce payoffs are averaged, the mportance of current flow payoffs, relatve to contnuaton payoffs, depends on the horzon relatve to the current perod), wth longer horzons reducng the relatve mportance of current payoffs. In the fntely repeated prsoners dlemma dscussed below), for example, ntal cooperaton, followed by T perods of defecton satsfes Radner s 1980) alternatve noton for any ε for T suffcently large. 2 Lehrer and Sorn 1998) and Watson 1994) consder a concept that requres contemporaneous ε-optmalty condtonal only on those hstores that are reached along the equlbrum path.

6 G.J. Malath et al. / Games and Economc Behavor ) prsoners dlemma. More generally, n fnte horzon games, for suffcently small ε, ex ante perfect ε-equlbra and contemporaneous perfect ε-equlbra concde: Proposton 1. Suppose G s a fnte game so that t has fnte horzon and fnte acton sets). For suffcently small ε, the sets of ex ante perfect pure-strategy ε-equlbra and of contemporaneous perfect pure-strategy ε-equlbra concde, and they concde wth the set of pure-strategy subgame perfect equlbra. Proof. Observe that any subgame perfect equlbrum s necessarly both an ex ante and a contemporaneous perfect ε-equlbrum. Suppose then that ŝ s not a subgame perfect equlbrum. We wll show that for ε suffcently small but ndependent of ŝ), ŝ s nether an ex ante nor a contemporaneous perfect ε-equlbrum. Snce ŝ s not subgame perfect, there s some player, hstory h and strategy s such that π ŝ,s h) π ŝ h) > 0. Snce the game s fnte, there exsts an ε suffcently small such that, for all such ŝ, h,, and s, π ŝ,s h) π ŝ h) > ε. But, ŝh π,s h ) ŝh ) π = δ th)) [ π ŝ,s h) π ŝ h) ] >δ th)) ε and consequently, the profle ŝ s not an ex ante perfect δ th) ε -equlbrum. Choosng ε = mn {δ T ) ε }, where T s the length of the game, shows that the profle s not an ex ante hence, nor a contemporaneous) perfect ε-equlbrum. Ths result appears to conflct wth Radner s demonstraton that there exst ex ante perfect ε-equlbra featurng cooperaton n the fntely repeated prsoners dlemma for arbtrarly small ε. However, Radner s result s acheved by allowng the number of perods T to grow suffcently rapdly, as ε falls, that the ex ante value of foregong defecton n perod T remans always below ε. Proposton 1 mples that the sets of ex ante perfect ε-equlbra and contemporaneous perfect ε-equlbra can dffer for arbtrarly small ε only n nfnte horzon games. The next example llustrates ths possble dfference: Example 2 An nfnte game. Consder a potental surplus whose contemporaneous value n tme t s gven by 2δ t for some δ 0, 1). In each perod, two agents smultaneously announce ether take or pass. The game ends wth the frst announcement of take. If ths s a smultaneous announcement, each agent receves a contemporaneous payoff of 1 2 2δ t 1). We can thnk of ths as the agents splttng the surplus, after payng a cost of 1. If only one agent announces take, then that agent receves 1 2 2δ t, whle the other agent receves nothng. Hence, a sngle take avods the cost, but provdes a payoff only to the agent dong the takng. The agents common and constant) dscount factor s gven by δ. Players receve zero rewards n each perod before take s announced.

7 132 G.J. Malath et al. / Games and Economc Behavor ) Ths game has a unque pure-strategy contemporaneous perfect ε-equlbrum, n whch both players take n the frst perod, for any ε<δ/2. To verfy ths, suppose that both agents strateges stpulate that they take n perod t>0. Then the perod t 1 contemporaneous payoff gan to playng take n perod t 1 s gven by 1 2 2δ t 1) ) δ ) 1 2 2δ t 1) = δ >ε. 3.2) 2 Hence, a smultaneous take can appear only n the frst perod. If the frst play of take occurs n any perod t>0and s a take on the part of only one player, then t s a superor contemporaneous) response for the other player to take n the prevous perod, snce 1 2 2δ t 1) 0 >ε. The only possble pure-strategy contemporaneous equlbrum thus calls for both agents to take n every perod. It remans only to verfy that such strateges are a best reply, whch follows from the observaton that 1 2δ 0 1 ) > 0. 2 A straghtforward varaton on ths argument shows that the only pure or mxed) Nash and hence subgame perfect) equlbrum outcome of the game also calls for both agents to take n the frst perod. In contrast, let τ satsfy δ τ < 2ε. Gven any such τ, there exsts a pure-strategy ex ante perfect ε-equlbrum n whch both players pass n every perod t<τand take n every perod t τ. In partcular, the most proftable devaton for ether player s to choose take n perod τ 1, for an ex ante payoff ncrement of δ τ δτ 1 δ τ 1 2 2δ τ 1 ) = 1 2 δτ, whch s, by constructon, smaller than ε. In contrast to Proposton 1, Example 2 shows that n nfnte games, ex ante and contemporaneous perfect ε-equlbra can be qute dfferent for arbtrarly small ε. Fudenberg and Levne 1983, p. 261) ntroduce a condton under whch suffcently dstant future perods are relatvely unmportant, makng nfnte games approxmately fnte: Defnton 2. The game s contnuous at nfnty f for all, [ π s h) π s h) ] = 0. lm t sup s,s,h s.t. t=th) δ th)) Equvalently, a game s contnuous at nfnty f two strategy profles gve nearly the same payoffs when they agree on a suffcently long fnte sequence of perods. A suffcent condton for contnuty at nfnty s that the reward functon r a t s)) be bounded and the players dscount.

8 G.J. Malath et al. / Games and Economc Behavor ) Fudenberg and Levne s 1983) Lemma 3.2 can be easly adapted to gve 3 : Proposton 2. In a game that s contnuous at nfnty, every convergng n the product topology on the set of strategy profles) sequence of ex ante perfect εn)-equlbra and hence every convergng sequence of contemporaneous perfect εn)-equlbra) wth εn) 0 converges to a subgame perfect equlbrum. Proof. We argue to a contradcton. Suppose {σn)} s a sequence of ex ante perfect εn)-equlbra, where εn) 0, convergng to a strategy ˆσ that s not a subgame perfect equlbrum. Because ˆσ s not a subgame perfect equlbrum, there exsts an nformaton set h for player, strategyσ and γ>0such that π ˆσ h,σ h ) = π ˆσ h ) + γ 3.3) whle σn)must be an ex ante perfect γ/4-equlbrum for all suffcently large n, requrng π σ h n), σ h ) π σ h n) ) + γ ) Because the game s contnuous at nfnty, we can fnd n suffcently large that 4 π ˆσ h,σ h ) π σ h n), σ h ) γ < 4 and π ˆσ h ) π σ h n) ) < γ 4. Combnng wth 3.4), ths gves π ˆσ h,σ h ) π ˆσ h ) + 3γ 4, contradctng 3.3). Example 2 shows that n games that are not contnuous at nfnty, Proposton 2 does not hold for ex ante perfect ε-equlbra. The followng example shows that, wthout contnuty at nfnty, t also need not hold for contemporaneous perfect ε-equlbra: Example 3. A sngle player, after every nontermnal hstory, chooses between L and R. The player dscounts future payoffs at constant rate δ t = δ 0, 1). A choce of R n perod t ends the game wth a perod-t reward of δ t δ t. A choce of L leads to the next 3 See Fudenberg and Levne 1986) for extensons of Fudenberg and Levne s 1983) results and Börgers 1989) and Harrs 1985) for related work. Fudenberg and Levne 1983, Theorem 3.3) go further to show that an equlbrum σ s subgame perfect n an nfnte-horzon game f and only f there s a sequence σn)of ex ante perfect εn)-equlbra n fnte-horzon truncatons of length Tn) of the orgnal game, wth σn) σ, εn) 0, and Tn). The fact that, for suffcently small ε, the fntely-repeated prsoners dlemma of any length) has a unque contemporaneous perfect ε-equlbrum, regardless of length, shows that the same s not true for contemporaneous perfect ε-equlbrum. 4 Intutvely, by choosng n suffcently large, we can make all behavor dfferences arbtrarly small except those that are dscounted so heavly as to have an arbtrarly small effect.

9 134 G.J. Malath et al. / Games and Economc Behavor ) perod, wth zero reward ths perod. For any ε, t s a contemporaneous perfect ε-equlbra to choose L n every perod t for whch t ln ε/ln δ.e., every perod n whch δ t ε) and R n every perod for whch t>ln ε/ln δ. 5 However, as ε goes to zero, the sequence of such equlbra converges to always choosng L, whch s not a subgame-perfect equlbrum. Indeed, ths game has no subgame perfect equlbrum. In a fnte game, ex ante and contemporaneous perfect ε-equlbra concde for suffcently small ε Proposton 1). The observaton that any contemporaneous perfect ε-equlbra s also an ex ante perfect ε-equlbra, together wth Proposton s 2 convergence result for both concepts, rases the possblty that the followng counterpart of ths fnte-horzon equvalence mght hold for nfnte games that are contnuous at nfnty: for every ε there s an ˆεε) ε such that every ex ante perfect ε equlbrum s a contemporaneous perfect ˆεε) equlbrum, wth lm ε 0 ˆεε) = 0. However, ths s not the case, as the followng example llustrates. Example 4. As n Example 3, a sngle player, after every nontermnal hstory, chooses between L and R. The player dscounts future payoffs at constant rate δ t = δ 0, 1). A choce of R n perod t ends the game wth a perod-t reward of 1. A choce of L leads to the next perod, wth zero reward ths perod. Snce δ<1 and payoffs are bounded, the game s contnuous at nfnty. For any ε and τ ln ε/ln δ, t s an ex ante perfect ε-equlbrum to choose L for all t<τand R for all t τ. However, for ε<1, the only contemporaneous perfect ε-equlbrum s to choose L at every opportunty whch s also the subgame perfect equlbrum). We thus have ex ante perfect ε-equlbra for arbtrarly small ε that are contemporaneous perfect ˆε-equlbra only for large > 1) values of ˆε. 4. Nearby games For the remander of the paper, we fx the game form and the dscountng scheme and dentfy games wth ther assocated sequence of reward functons. In ths vew, two games are close f the reward functons are close. 6 Formally, we defne two metrcs on games: and d ) N G, Ĝ = sup,h r h) ˆr h) th) d ) P G, Ĝ = sup r h) ˆr h). 4.1),h 5 For any perod τ t, choosng R n perod τ gves a payoff evaluated n perod τ )ofδ τ δ τ, whch s no larger than the payoff δ t τ δ t δ t ) of adherng to the equlbrum strategy. In any perod τ>t, choosng R also gves a payoff δ τ δ τ, whle watng untl some later perod t to choose R gves a payoff of δ t τ δ t δ t ), whch exceeds the former by less than ε when δ τ ε. 6 More generally, we mght defne two games to be close f ther game forms, dscountng schemes, and reward functons are close. Börgers 1991, p. 95) ntroduces such a measure, defned n terms of the game form and the payoffs πσ). Gven our nterest n the mplcatons of dfferent tmng of rewards for ε-optmzaton, t s most revealng to fx the game form and dscountng scheme whle examnng perturbatons of the reward functon.

10 G.J. Malath et al. / Games and Economc Behavor ) Let r k and r be player s reward functons n G k and G respectvely. The followng lemma s an mmedate consequence of the defntons: Lemma ) Suppose that, for a sequence of games {G k } and game G, lm k d N G k,g)= 0 and there s M Rsuch that the assocated reward functons {r k } and r take values n [ M,M]. Then sup,σ π Gk σ ) π G σ ) ) Suppose that, for a sequence of games {G k } and game G, lm k d P G k,g)= 0. Then sup π Gk σ h) π G σ h) 0.,σ,h Convergence under d P s equvalent to unform convergence of the reward functons. Gven the assumed bound on payoffs n the Lemma, convergence under d N s equvalent to pontwse convergence of the reward functons. Wthout ths bound, d N mples, but s stronger than, pontwse convergence. 5. Approxmatng equlbra n nearby games It s straghtforward that, for statc games, ε-nash equlbra of a gven game G approxmate Nash equlbra of nearby games. A smlar result holds for multstage games recall that D, the bound from 2.1), does not depend on the reward functon). The only complcaton n extendng the observaton from statc to multstage games s that our notons of closeness for games examne the reward functons, whle optmalty s based on the dscounted sums of rewards. Snce players dscount and the concept of a Nash equlbrum depends only on ex ante payoffs, under a slght strengthenng of 2.1), t s not necessary for the result that the rewards by unformly n t) close as requred by d p ): Proposton 3. Fx a game G. 3.1) If the strategy profle ˆσ s a Nash equlbrum of game G wth d P G,G)<ε/2, then ˆσ s an εd-nash equlbrum of game G. Moreover, f for each t ) 1/t lm sup δ τ < 1, 5.1) t τ=0 then there exsts D ndependent of the reward functon of G) such that f the strategy profle ˆσ s a Nash equlbrum of game G wth d N G,G)<ε/2, then ˆσ s an εd -Nash equlbrum of game G. 3.2) If ˆσ s a pure-strategy ε-nash equlbrum of game G, then there exsts a game G wth d P G,G)<ε/2 and hence d N G,G)<ε/2)forwhch ˆσ s a Nash equlbrum.

11 136 G.J. Malath et al. / Games and Economc Behavor ) The proof of ths proposton follows that of the next proposton. The restrcton to pure strategy equlbra cannot be dropped n Proposton 3.2. For example, n the game, L R T 0, 0 1, 1 B 1, 0 2, 1, the strategy profle ε T + 1 ε) B),R) s an ε-equlbrum. However, n any Nash equlbrum of any game ε/2 close to ths game, player 1 must choose B wth probablty 1. The problem mxed strateges are those that, as n the example, put small probablty on an acton that s far from optmal. 7 Returnng to Example 2, t s straghtforward that n all games that are suffcently close, as measured by d P, the strct nequalty n 3.2) contnues to hold. Hence, there s a unque subgame perfect equlbrum n such games, n whch both players mmedately play take. Contrastng ths observaton wth the varety of ex ante perfect ε equlbra that appear n Example 2, we are led to the concluson that f one seeks an approxmate equlbrum concept capturng subgame perfect equlbra of nearby games, contemporaneous perfecton s the approprate concept: Proposton 4. Fx a game G. 4.1) If the strategy profle ˆσ s a subgame perfect equlbrum of game G wth d P G,G)<ε/2, then ˆσ s a contemporaneous perfect εd-equlbrum of game G. 4.2) If ˆσ s a pure-strategy contemporaneous perfect ε-equlbrum of game G, then there exsts a game G wth d p G,G)<ε/2 for whch ˆσ s a subgame perfect equlbrum. The frst statement of Proposton 4 guarantees that the contemporaneous perfect ε- equlbra of a game nclude all subgame-perfect equlbra of nearby games. The second guarantees that every pure strategy contemporaneous perfect ε-equlbrum s a subgame perfect equlbrum of a nearby game. We emphasze agan, however, that nether Proposton 3 nor Proposton 4 contans an f and only f result, snce the ε s for the two parts are not the same. We could replace εd n the frst statements wth ε, makng the two statements symmetrc, f we had also replaced d P wth the metrc d ) P G, Ĝ = sup s,h, t=th) δ th),t) r a t s h)) t=th) δ th),t) ˆr a t s h)). We have chosen to work wth d P rather than d P, however, because d p yelds a more transparent noton of closeness for games n terms of the reward functon. 7 A referee noted that, f we replace player 1 s mxture n ths game wth a node at whch Nature frst draws a real number from [0, 1] and then player 1 chooses T for draws less than ε and B for hgher draws, then these problem strategy profles are no longer ε-nash equlbra. Ths suggests that Proposton 3 could be formulated as an equvalence f we nssted on purfyng mxed strateges n ths way. Dong so rases the nconvenence of dealng wth nfnte numbers of fnte-length hstores.

12 G.J. Malath et al. / Games and Economc Behavor ) Proof of Proposton ) Let ˆσ be a subgame perfect equlbrum of game G wth d P G,G)<ε/2. It follows from 2.1) and 2.2) that, for any strategy profle σ, player and hstory h, π G σ h) π G σ h) < ε 2 a A h) π t t=th) τ=th) δ τ ε D. 5.2) 2 We then have, for any player, hstory h and strategy σ, π G ˆσ h) π G ˆσ,σ h) = π G ˆσ h) π G ˆσ h) ) + π G ˆσ h) π G ˆσ,σ h) ) + π G ˆσ,σ h) π G ˆσ,σ h) ) εd, gvng the result. 4.2) Let ŝ be a pure strategy contemporaneous perfect ε-equlbrum of G. For notatonal purposes, assume that A h) and A h ) are dsjont for all h and h H, so that the acton a unquely dentfes a hstory. For all nformaton sets h H for player, ŝ a denotes the strategy that agrees wth ŝ at every nformaton set other than h, and specfes the acton a at h. In other words, ŝ a s the one-shot devaton { ŝ h ŝ h ) = ), f h h, a, f h = h. Snce ŝ s a contemporaneous perfect ε-equlbrum, 8 γ h) max ŝ, ŝ a h ) π ŝ, ŝ h) < ε. The dea n constructng the perturbed game s to ncrease the reward to player from takng the specfed acton ŝ h) at hs nformaton set h H by γ h), and then lowerng all rewards by ε/2. Dscountng guarantees that subtractng a constant from every reward stll yelds well-defned payoffs.) However, care must be taken that the one-shot beneft takes nto account the other adjustments. So, we construct a sequence of games as follows. For fxed T, we defne the adjustments to the rewards at hstores of length less than or equal to T, γ T h). The defnton s recursve, begnnng at the longest hstores, and proceedng to the begnnng of the game. For nformaton sets h H satsfyng th)= T, set γ T h) = γ h). Now, suppose γ T h ) has been determned for all h H satsfyng th ) = l T.For h satsfyng th ) = l + 1, defne r T h ) r h ), r h, ŝh )) + γ T h ), f h = h, ŝh )) for some h H such that th ) = l, otherwse. 8 Snce s h) A h), γ h) 0.

13 138 G.J. Malath et al. / Games and Economc Behavor ) Ths then allows us to defne for h satsfyng th)= l 1, 9 π T s h) r a th) s h)) t + δ τ )r T a t s h)), and γ T h) max a A h) π T ŝ, ŝ a t=th)+1 τ=th)+1 h ) π T ŝ, ŝ h). Proceedng n ths way determnes r T h) for all h. We clam that for any T, r T h) r h) < ε. To see ths, recall that ŝ s a contemporaneous perfect ε-equlbrum and note that the adjustment at any h can never yeld a contnuaton value under ŝ) larger than the maxmum contnuaton value at h. Moreover, the sequence {r T } T of reward functons has a convergent subsequence there s a countable number of hstores, and for all h H, r T h) [r h), r h) + ε]). Denote the lmt by r. Note that there are no proftable one-shot devatons from ŝ under r,by constructon. As a result, because of dscountng, ŝ s subgame perfect. Fnally, we subtract ε/2 from every reward. Equlbrum s unaffected, and the resultng game s wthn ε/2 under d P. Proof of Proposton 3. The proofs of the statements about d P are the same arguments as n the proof of Proposton 4, but appled only to the ntal hstory. Suppose now that the dscountng scheme satsfes 5.1). From the root test Rudn, 1976, Theorem 3.33), there exsts D such that t t δ τ <D. t=0 τ=0 The proof of the d N result s now agan a specal case of that of Proposton 4.1, wth the excepton that the frst nequalty n 5.2) n the statement of that proposton s now replaced by π G σ h) π G σ h) t <ε t δ τ, whch s less than εd. t=0 τ=0 6. Dscusson The set of contemporaneous perfect ε-equlbra of a game G ncludes the set of subgame perfect equlbra of nearby games. Examnng contemporaneous perfect ε-equlbra thus ensures that one has not mssed any subgame perfect equlbra of the real games that mght correspond to the potentally msspecfed model. 9 Note that the hstory a th)+1 s h ) = a l s h ) s of length l + 1.

14 G.J. Malath et al. / Games and Economc Behavor ) Examples 2 and 3 show that, for games that are not contnuous at nfnty, examnng ether ex ante perfect ε-equlbra or subgame perfect equlbra respectvely) can gve a msleadng pcture of the set of contemporaneous perfect ε-equlbra of a game, and hence subgame perfect equlbra of nearby games, ncludng too many equlbra n the frst case and too few n the second. Suppose, however, that we restrct attenton to games that are contnuous at nfnty. As ε gets small, the set of ex ante perfect ε-equlbra and the set of contemporaneous perfect ε-equlbra of the model converge to the set of subgame perfect equlbra of the model. In lght of ths, why not smply dspense wth ε altogether and examne subgame perfect equlbra of the model? Snce our goal s to ensure that no subgame perfect equlbrum from the real game s neglected, we would need that every subgame perfect equlbrum of the real game s close to some subgame perfect equlbrum of close-by models. That s, suppose a modeler, after fxng ε>0, postulates hs best-guess model of the real game and calculates ts subgame perfect equlbra. Can we be assured that, f the real game s ε-close to the model, all of ts subgame perfect equlbra wll be captured? No, as the followng smple example llustrates: Suppose the model has two choces wth player 1 recevng 0 from L and ε/2 from R. The only subgame perfect equlbrum s to play R. However, f n the true game, player 1 receves ε/2 afterl and 0 after R, the only subgame perfect equlbrum s L. In contrast, from Proposton 4, we n fact know that every subgame perfect equlbrum of the real game s n fact a contemporaneous perfect ε-equlbrum of the model. Acknowledgments We thank Drew Fudenberg, Davd Levne, two referees, and an Assocate Edtor for helpful comments and the Natonal Scence Foundaton for fnancal support under Grants SES , SES and SES ). References Börgers, T., Perfect equlbrum hstores of fnte and nfnte horzon games. J. Econ. Theory 47, Börgers, T., Upper hemcontnuty of the correspondence of subgame-perfect equlbrum outcomes. J. Math. Econ. 20 1), Fudenberg, D., Levne, D.K., Subgame-perfect equlbra of fnte- and nfnte-horzon games. J. Econ. Theory 31, Fudenberg, D., Levne, D.K., Lmt games and lmt equlbra. J. Econ. Theory 38, Fudenberg, D., Trole, J., Game Theory. MIT Press, Cambrdge, MA. Harrs, C., A characterzaton of the perfect equlbra of nfnte horzon games. J. Econ. Theory 37, Kohlberg, E., Mertens, J.-F., On the strategc stablty of equlbra. Econometrca 54, Lehrer, E., Sorn, S., ε-consstent equlbrum n repeated games. Int. J. Game Theory 27 2), Radner, R., Collusve behavour n noncooperatve epslon-equlbra of olgopoles wth long but fnte lves. J. Econ. Theory 22, Radner, R., Montorng cooperatve agreements n a repeated prncpal agent relatonshp. Econometrca 49 5), Rubnsten, A., Perfect equlbrum n a barganng model. Econometrca 50,

15 140 G.J. Malath et al. / Games and Economc Behavor ) Rudn, W., Prncples of Mathematcal Analyss. McGraw-Hll, New York. Watson, J., Cooperaton n the nfntely repeated prsoners dlemma wth perturbatons. Games Econ. Behav. 7 2),