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X^I^CH^ Digitized by the Internet Archive in 2011 with funding from Boston Library Consortium IVIember Libraries http://www.archive.org/details/reputationequiliOOschm mMu¥^i ^ . ^fdmfj:i.\j « '''<JJ±r^ ^c^Mli»T/;^'%^vr-> HB31 .M415 no. ^^ -1-1 forking paper department of economics REPUTATION AND EQUILIBRIUM CHARACTERIZATION IN REPEATED GAMES WITH CONFLICTING INTERESTS Klaus M. Schmidt No. 92-7 March 1992 massachusetts institute of technology 50 memorial drive Cambridge, mass. 02139 REPUTATION AND EQUILIBRIUM CHARACTERIZATION IN REPEATED GAMES WITH CONFLICTING INTERESTS Klaus M. Schmidt No. 92-7 March 1992 M.l.T. AUG LIBRARIES 4 1992 RECtlVtO Reputation and Equilibrium Characterization in Repeated Games with Conflicting Interests by Klaus M. Schmidt^ Department of Economics, E52-252C Massachusetts Institute of Technology Cambridge, MA Tel: (617) USA 02139, 253-6217 March 1991 revised: February, 1992 A two-person game is of conflicting interests player one would most like to minmax payoff. Suppose there a "commitment type" get at least her game if who will commit is if the strategy to which herself holds player two down to his a positive prior probability that player one always play this strategy. commitment payoff in Then is player one will any Nash equilibrium of the repeated her discount factor approaches one. This result is robust against further perturbations of the informational structure and in striking contrast to the message of the Folk theorem for games with incomplete information. Keywords: Commitment, Folk theorem. Repeated Games, Reputation. ^This paper is based on Chapter 3 of Doctoral Programme at my PhD thesis which was completed within the European Bonn University. I would like to thank David Canning, In-Koo Cho, Benny Moldovanu, Georg Noldeke, Ariel Rubinstein, Avner Shaked, Joel Sobel, Monika Schnitzer, Eric van Damme and in particular Drew Fudenberg for many helpful comments and discussions. Financial support by Deutsche Forschungsgemeinschaft, SFB 303 at Bonn University, is gratefully acknowledged. 1 Introduction Consider a repeated relationship between two long-run players one of private information about her type. A common intuition whom has some that the informed player is may take advantage of the uncertainty of her opponent and enforce an outcome more favourable to her than what she would have got under complete information. This intuition has been and has found considerable attention called "reputation effect" purpose of this paper to formalize this intuition in a general is with "conflicting interests" and to show that the effect in the literature. The model of repeated games robust against perturbations of is the informational structure of the game. The formalizations of the "reputation effect" in first games with incomplete in- formation have been developed by Kreps and Wilson (1982) and Milgrom and Roberts amount (1982). They have shown cient to overcome Selten's (1978) chain-store paradox. that a small may faces a sequence of potential entrants "toughness" if there of incomplete information can be suffi- An incumbent deter entry by maintaining a reputation for a small prior probability that she is monopolist who is who a "tough" type prefers a price war to acquiescence. Recently, this result has been generalized and considerably strenghtened by Fudenberg and Levine (1989, 1992). peated games whom in prior probability of a all previous play. like to commit herself, what she would have obtained this strategy. This result player one's payoff in games, and i.e. it is (iii) it is They show that if there "commitment type", who always plays the strategy and all is if player one if can enforce at least her commitment payoff at least class of all re- which a long-run player faces a sequence of short-run opponents, each of plays only once but observes one would most They consider the a positive which player sufficiently patient, then she any Nash equihbrium, in i.e. she will get she could have committed herself publicly to very powerful, because Nash is to is equilibria, (ii) it (i) it gives a tight lower bound for holds for finitely and infinitely repeated robust against further perturbations of the informational structure, independent of what other types Fudenberg and Levine's analysis is may exist with positive probability. restricted to However, games where a long-run player faces a sequence of short-run opponents. Our paper provides a generalization and qualification of their results for the two long-run player condition for this generalization to hold is case. We that the show that a necessary and game is sufficient of conflicting interests. The who first studied reputation effects in repeated games with two long-run players were again Kreps, Wilson, Milgrom and Roberts (1982). They demonstrated that sequential equilibria of the finitely repeated prisoner's will is be achieved committed if there is dilemma the cooperative outcome a small prior probability that the players are of a type which to always play the "tit-for-tat" strategy. In the beginning of the 1980s repeated games with incomplete information have been very popular to solve of puzzles in game in all all But the theory, industrial organization, macroeconomics etc.. kind initial enthusiasm has been considerably dampened by a Folk-theorem type result of Fudenberg and Maskin (1986). They have shown that exists an e-perturbation of this game (in for any finitely or infinitely game repeated there which each of the players has a different payoff function with an arbitrarily small but positive prior probability) such that any individually rational, feasible payoff vector of the unperturbed game can be sustained as the outcome of a sequential equilibrium of the perturbed and if there are enough repetitions. But if game, if the players are sufficiently patient any outcome can be "explained" by just picking the right perturbation then the predictive power of the theory of repeated games is very limited indeed. However, before this message is we should have a accepted, closer look at Fudenberg and Maskin mean by an e-perturbation. They assume that with an small but positive probability each of the players may be "crazy", arbitrarily she i.e. what may have a completely different payoff function as compared to the original game. For any given payoff vector Fudenberg and Maskin then pick one very specific type of "crazyness" which is used to sustain this payoff vector as an equilibrium outcome. informational structure of a game should capture A perturbation of the the idea that the players may be slightly may include uncertain about what the exact payoff functions of their opponents are. This the possibiHty that the opponent it is actually crazy as in Fudenberg and Maskin should not be restricted to only one (very particular) type of crazyness. the modeller should allow for a broader class of perturbations, such that payoff functions may have positive robustness yields a result which No matter what types may is prior probability. Surprisingly, insisting in striking contrast to the possibly be if the game is argue that many on different this kind of message of the Folk-theorem. drawn by nature (including those considered by Fudenberg and Maskin) and how Hkely they are patient, We (1986), but of "conflicting interests", to occur, and if there if is player one is sufficiently an arbitrarily small but "commitment" type, then we can give a positive probability of a equihbrium outcome of To make all more this which player one would equilibria. precise, consider a repeated commit like to action", in every period. "commitment Nash informational structure of this repeated game always says that if if the herself to take an action aj game that the that playing aj holds player two down game is to his is , called her in "commitment to play aj and call of "conflicting interests" in the sense minimax payoff. Now whom it is may be one of a dominant strategy in the her the "commitment type". commitment type has any Our main theorem arbitrarily small but positive probability player one's discount factor goes to one then her payoff in any bounded below by her commitment suppose that the perturbed such that player one Consider a type for several possible "types". game with complete information player two responds optimally to aj player one gets her If Assume payoff'. tight prediction of the This result payoff. is the other possible types and their respective probabilities. we show the case of two-sided uncertainty. Furthermore, and Nash equihbrium is independent of the nature of We generalize the theorem to that "conflicting interests" are a necessary condition for our theorem to hold. Our result highlights the importance of the relative patience of the two Player one has to be sufficiently patient as compared to player two, < discount factor ^2 commitment payoff there exists 1 in all Nash a. < S^{62) equilibria if 1 interests, at the then same it is time. his discount factor satisfies is two-sided uncertainty. most However, is the if one of them is commitment type sufficiently is <5i > ^i(<52). game has conflicting most preferred outcomes more patient sufficiently higher) (or if the prior then the reputation works to her advantage. A complementary class of repeated analysis to ours is Aumann and Sorin (1989). For a different games, coordination games with "common interests", they obtain a similar result. However, they have to restrict the possible perturbations to types like The intuitive in the case of If this clearly not possible that both players get their probability that she effect game with any given such that player one can enforce his importance of the relative patience of the two players a completely symmetric for i.e. plaj'ers. automata with bounded with positive probability then recall. all They show that if all pure strategy equilibria who act strategies of recall zero exist will be close to the cooperative outcome. In contrcist to Aumann and Sorin we allow for any perturbation of player one's Games of "common" and payoff function and for mixed strategy equilibria. interests are two polar We cases. Finally, in a very recent Nash in more detail in Section 5. paper Cripps and Thomas (1991) characterize the equilibria of infinitely repeated which players maximize the them will discuss of "conflicting" set of games with one-sided incomplete information mean limit of the of their undiscounted payoffs. in Following a different method pioneered by Hart (1985) they also find that in games of conflicting commitment payoff interests the informed player can enforce her small prior probability of a commitment type. Since there seems to indicate that the relative patience we will show The in Section 4, this interpretation closely Then we to the class of on how paper rest of the model following results. is all is is if there an arbitrarily no discounting their result not that important after is is all. However, as misleading. organized as follows. In the next section Fudenberg-Levine (1989) and we briefly we introduce the summarize their main give a counterexample showing that their theorem cannot carry over repeated games with two long-run players. This gives some intuition this class has to be restricted. Section 4 contains our main results. There we generalize Fudenberg-Levine's (1989) theorem to the two long-run player case, and show that the games with restriction to this generalization to hold." "conflicting interests" is we a necessary condition for Furthermore we extend the analysis to the case of two-sided incomplete information. In Section 5 we give several examples which demonstrate how restrictive the "conflicting interests" condition is. Section 6 concludes and briefly outlines several extensions of the model. 2 In most of the paper Description of the we consider the Game following very simple model of a repeated game an adaptation of Fudenberg-Levine (1989) and Fudenberg-Kreps-Maskin (1990) which is to the two long-run player In every period they action sets .4,-, i case. The two players are called "one" (she) move simultaneously and choose an G {1,2}. Here we will action assume that the Ai are a,- and "two" (he). out of their respective finite sets.^ As a point of ^See Section 6 for the extension to extensive form stage games, continuous strategy spaces and more than two players. reference consider the unperturbed game denote the payoff function of player i (with complete information) the unperturbed stage in actions taken by both players. Let Ai denote the set of and strategies an abuse of notation) gi{ai,a2) the expected stage game (in The game T-fold repetition of the stage We or infinite. will deal in g denoted by is most of the paper with the game the repeated overall payoff for player from period i q,- where G-^, if the of player i payoffs. T may infinite horizon case the results carry over immediately to finitely repeated games t) is game g depending on mixed all Let 5^,(01,02) first. T is be but finite of all large enough. Li the onwards (and including period t given by 00 = E^r'9i, v^ (1) T=t where < denotes her (his) discount factor (0 <5,- average discounted payoffs u,-, 8i < Our 1). results are stated in terms of where CO vi (2) = = {i-s,)-Y.8igi (i-<5,).v;° T=0 After each period both players observe the actions that have been taken. perfect recall and can condition on the entire past history of the game. Let their play be a specific history of the repeated game out of the histories up game a sequence of is to and including period maps 5' : A t. H^~^ all pure (mixed) strategies Let B game and : ^1 1-^ is denoted by ^2 be the — where (S,- if ct,- A2y for player : there H^~^ is let — > i of all possible in the repeated a,- = (a/,(jf, • • •) Ai. For notational no ambiguity. The set of respectively). best response correspondence of player two in the stage gl "commitment = max min payoff' of player one. guarantee for herself in the stage Note that the minimum over indifferent i, suppressed 5,- {Ai x Correspondingly, Ai. > 5,- = h^ define (3) as the is H^ set pure strategy denote a mixed (behavioral) strategy of player convenience the dependence on history They have all game if 51(01,02) That she could is gl commit is to the most player one could any pure strategy 02 G B{ai) has to be taken since player two between several best responses to ci in which case he may Oi £ Ai. may be take the response "commitment action") player one prefers least. ^ Let aj (her min (4) = Oi(a',a2) g^ satisfy . cr2€S(aJ) Furthermore, let G B{al) denote any strategy Oj two which is a best response and define to Oj (5) 92 So 92 Oj. of player 52(01,^2) the most player two would get in the stage is ^ A2 Suppose B{a1) payoff = is her maxmin (otherwise the = ^2 (6) Then payoff). game is • game "trivial" there exists a 02 g2{a\,~a-i) = max ^ if player one were committed to because player one's commitment -^(^1) such that < ^2(01,02) ^2 • a2gB(oJ) Note that the ^ B{a^). So (22 is maximum 92 the is because exists maximum it is taken over the finite set of player two can get if maximal payoff player two can Clearly, in the repeated game t and Consider (before the € all /i'-^ now first of a countable set payoff function f:sr-92= T^ = "^2 ^ W-\ a perturbation of this complete information stage f) now • must be true that it T=t for all = game is (a;o,<j-'i, • is a (1967-68). In the perturbed game such played) the "type" of player one • •) game with incomplete game a is : that in period drawn by nature out according to the probability measure additionally depends on her type, so 91 perturbed game G^{fi) action. Finally, a2£>l2 ai€^i < V^ (8) commitment all as = max max ^2(^1, 02) 92 (7) get at (pure) actions he does not take an action which a best response against aj, given that player one takes her define the all Ai x A2 fi. >< Cl Player one's -^ H. The information in the sense of Harsanyi strategy of player one may not only depend on history ^Fudenberg and Levine (1989) refer to gl as the "Stackelberg payoff'. However, it is now customary maxo, rnaXa^eBiai) 9ii<ii>0'2), that is for the maximum payoff player one could get if he could publicly commit himself to any action a\ and player two choses the best response player one prefers most. See Fudenberg (1990). The analysis can be extended to the more general case where player one would like to commit himself to a mixed strategy or to a strategy dependent on history. See Fudenberg and Levine (1992) and the remarks in Section 6. to use this expression only for but also on her type, so cr{ //' : Two xQ, —^ Ai. ^ types out of the set Q. are of particular importance: • The "normal" type as in the of player denoted by wq. Her payoff function = giiai,a2,uo) many fi{ujo) • is applications = the same fJ,{uo) will be close to <7i(ai,a2). 1. However, we have to require only that >0. (1° The "commitment" type repeated is unperturbed game: (9) In one game always is denoted by to play Cj. This uj*. is For her it is a dominant strategy in the for exarrlple the case her payoff function if satisfies (10) 5i(ai,a2,u;*) = for all oi 7^ aj, Ci € Ai, and 02)^2 € ^2- repeated game impHes all gi{a'^,a2,uj'') = > fi' will now tical inference is flj s\ = aj for all which is and if /i'~^ € H' < and 7f will < 1. If update has turn implies that in in The set of all lemma such histories is if denoted by H*. of Fudenberg-Levine (1989) about statis- The lemma says that The number which player two of periods in intuition for this result is if u' has which player one has always played aj up to period player two observes Cj being played in his beliefs. t he is will believe the following. Consider any player two believes that the probability of aj being played in period 7f, uj' player two observes Cj being played in every period then there "unlikely" to be played. history strategy property in the equihbrium path. This basic to the following analysis. a fixed finite upper bound on the is t. restate an important positive probability The dominant then with positive probability there exists a history in any Nash equilibrium with We 5i(ai, 4,0;') that in any Nash equilibrium player one with type to play Cj in every period along the fi{uj') > i t — is "surprised" to 1. Suppose smaller than some extend Because the commitment type chooses a^ with probability 1 while player two expected aj to be played with a probability it follows bounded away from 1 from Bayes' law that the updated probability that he faces the commitment type has to increase by an amount bounded away from arbitrarily often because the 0. However, this cannot happen updated probability of the commitment type cannot become This gives the upper bound on the number of periods in which player two bigger than 1. may a'[ expect Note that to be played with a probability less than W. argument this is independent of the discount factors of the two players. To put it more formally: probability distribution Each (possibly mixed) strategy profile {cri,a2) induces a Given a history h^~^ over {Ai x ^2)°° x ^- tt probability attached by player two to the event that the played in period variable 7r'(aj) is 7r'(aj) t, i.e. a random history h induced by = Along (possibly infinite) of the random a random variable, so n. Lemma < W < 1 Let such that Prob{h G H' Prob n (11) Furthermore, for any all Fix any 1. this history let n(7r'(aj)) \uj') Suppose = (7r'(a') ^ l. = h{uj') converges to 1, i.e. ^^ <¥) > ^ > and 0, < to play aj. Lemma \ h E is one with type Lemma 1 = H' 0. always played, fi{uj' it of course to \ ht h^) is 1 mimic the commit- if is bigger than if 8 ht E h') facing u' he is W be played. only interested in his is a tt < 1 such that then a short-run player player one mimics the her short-run opponents will take 02 is \ he observes aj being played Oj is "unlikely" to completely myopic, that is fi{uj' become convinced that he says that he cannot continue to believe that is ujq is does not say that in this case that player two will gradually Lemma is 1. choose a best response against Oj. Thus, type, then by W. Again, since h h such that the truncated histories the probability that player one will play a] will be the number that (cri,<72) are payoff of the current period. Fudenberg and Levine show that there two a ra/.dom and consider any 7f) < is being t. Suppose that player two if /i'~^ is logTT he observes aj always being played. Rather in every period h' 1, strategy Then infinite history feasible strategy for player ment type and always < F < tt, variables 7r'(aj) for which T^\a.l) Proof: See Fudenberg-Levine (1989), One Note that since /i'~^). | have positive probability and such that a^ nondecreasing in if Oj variable as well. (crijCr^). is = Prob{s\ commitment be the let 7r'(aj) B{a'^) in at commitment most k = °^Jt, periods. The worst beginning of the happen that can game and to player one that these k periods occur in the is that in each of these periods she gets (12) g, —1 = min gi(a',Q2). 026.42 This argument provides the intuition for the following theorem. Theorem 0. Then (Fudenberg-Levine) Let 1 there (13) in If 61 least her If goes to 1 > (l-^f'"'')£, + «?"* The result payoff no matter what other types is discussed in 1 is more crucially based detail in One if he may he believes that paper, however, well is may pay Even off in may if may be around to show that with positive is completely be played with a probability arbiis not a short-run best response be that he might invest this yields losses in the in screening the beginning of the game the the future. This leads Fudenberg and Levine to conclude for the returns from an investment point of our well. Since player two's discount factor is smaller may not be delayed to far to the future. type arbitrarily often two long-run player games. The main a more restricted class of games a similar result holds arbitrarily close to one. This idea will for > n' trade off short-run losses against take an action 02 which two long-run players case as "test" player one's 1 = '-»r that their result does not apply to two long-run player games. 1 }i{u') Fudenberg and Levine (1989). Note Cj will intuitive reason for this could different types of player one. in the and on the assumption that player two he cares about future payoffs then he trarily close or equal to 1, investment 0, the "normal" type of player one can guarantee herself on average at long-run gains. Thus, even against Gj. > any Nash equilihrium ofG°°{fi). however that Theorem myopic. 0, /x(u;°) any average equilibrium payoff of player one with type fi';u>°) is commitment probabilty. = a constant ^(/x*) otherwise indepedent of {p.,[i), such that i-iC*,,,.-;^") where Vi{Si, uq is 62 if He the probability that she will play a\ be used in Section 4 to prove an analog of than will is not always Theorem A Game 3 not of Conflicting Interests Before establishing our main result us show that let repeated games with two long-run players. We Theorem 1 does not carry over to commitment the normal type of player one cannot guarantee herself almost her all Nash The example equilibria. instructive for two reasons. is game give a counterexample of a First, it in all which payoff in shows how to construct an equilibrium in which the normal type of player one gets strictly less than her commitment which survives and payoff. This equilibrium is not only a Nash but a sequential equilibrium standard refinements. Second, the construction leads to a necessary all sufficient condition on the games cla^s of for which Theorem can be generalized to 1 the two long-run player case. Consider an infinite repetition of the following stage game with three types of player one: R L u 1 ' D 1 = 0.8 H' Figure Player one chooses between each cell. to play U 1 'commitment" type "normal" type fi° 1 ° 1 U — A game and D = with U ' D ' ' /i' common is ' 10 = given in the upper which would give her a commitment payoff of 10 per period commitment type game always to play U The . it is 0.1 interests. Clearly the normal type of player one would like to publicly equilibrium. For the 1 "indifferent" type 0.1 and her payoff R L '" U ' D R L left corners of commit always in every Nash indeed a dominant strategy in the repeated indifferent type, however, is indifferent between U and D no matter what player two does. If 0.75 < 5i < 1 and 0.95 < <52 < 1 then the following strategies and beliefs form a 10 sequential equilibrium of G°° which gives the normal type of player one = lim vi{u}°) (14) Normal • type of player one: < 9.5 "Play U. = 10 g^ . you ever played D, switch to playing If D forever." type of player one: "Always play [/." • Commitment • Indifferent type of player one: "Always play U • along the equilibrium path. has been any deviation by any player in the past switch to playing Player two: • path. If "Alternate between 19 times L and R player one ever played D, switch to deviated in the last period, play L 1 times R forever. D L there forever." along the equilibrium If player two himself in the following period. If player the deviation by playing U, go on playing If one reacted to forever. If she reacted with D, play R forever." • Beliefs: D Along the equilibrium path to be played he puts probability to a deviation of player beliefs don't change. If player on the commitment type. U two by playing both cases the respective two other types up to Why would is this about her type it is the indifferent type gets probability may is is she U is the normal or the In add commitment type. Since all three types of along the equilibrium path the only way to transmit information to play D. However, playing D "kills" a dominant strategy always to play U. He expects U always is 0. get arbitrary probabilities which the commitment type, because But without the commitment type impossible to get rid of the "bad" equilibrium {D,R). which player one reacts an equihbirum? Consider the normal type of player one. Clearly she player one always play it If 1. like to signal that for her two ever observes What about player two? to be played along the equilibrium path. Nevertheless he plays R, not a short-run best response, in every twentieth period. His problem faces the indifferent type with positive probability. to play R, then this If he choses L when he is is that he supposed might trigger a continuation equilibrium against the indifferent type 11 which gives him far less commitment type than what he would have got from playing against the normal or of player one. this risk It is Note that there are very few which sustains the equilibrium outcome. restrictions imposed on the updating of The example only information sets which are not reached on the equilibrium path. that if D played for the is first time the commitment type gets probability perfectly reasonable given that for her it beliefs in 0, requires which is a dominant strategy in the repeated game is always to play U. To what extend does the example Without the indifferent type it gives player one less than her is still rely possible to construct a commitment normal and the commitment type of player After any deviation they switch to playing period L and one period R. If on the existence of the indifferent type? payoff. Actually, this one always play D plays D off What it is is It requires for is The very simple: along the equilibirum path. two alternates playing one there has been any deviation, he plays not sequential. sustains both equilibria if only 5. R forever. This clearly is Note a Nash example that the commitment type already hold down is the possibility of a continuation equilibrium which he plays his short-run best response against a^ supposed not to do in this case is the equilibrium path.'' punishes player two he U forever. Player that the average payoff of the normal type of player one equilibrium, but Nash equilibrium which to his so. Note that this construction does not in periods work minimax payoff by the commitment strategy if when player two is of player one, since nothing worse can happen to him. In the next section we show that this is the only case in which Fudenberg and Levine's result can be generalized to the two long-run player case. ''Whether there exists a sequential equilibrium in which player one gets substantially less than 10 if commitment type around is an open question. Note, however, that we want to characterize equilibrium outcomes which are robust to general perturbations of the informational there are only the normal and the structure of the game. From this perspective it makes types only. 12 little sense to ristrict attention to two possible Main Results 4 4.1 The Theorem Suppose that the commitment strategy of player one holds player two down to In this case there payoff. is no his minimax "risk" in playing a best response against Cj because minimax payoff player two cannot get less than his in any continuation equilibrium. This motivates the following definition: A game Definition 1 one to player to his if the minimax g called a commitment i.e. payoff, (15) is gl = game o/ conflicting interests with respect strategy of player one holds player two down if g2{a'i,a2) = minmax5'2(Q;i,Q;2)- "Conflicting interests" are a necessary and sufficient condition for our main result. Note that the definition puts no restriction on the possible perturbations of the payoffs of player one. a restriction only on the It is We player two. Section in 5. commitment will discuss this class of game with Clearly, in a games extensively and give conflicting interests player any continuation equihbrium after any history V,' (16) This is strategy and on the payoff function of = several examples in two can guarantee himself ht at least Y^^-g;. crucial to establish the following result: Lemma and let fJ.{oj') tory Qj. 2 Let g be a /i' = Suppose (17) < > 0. of conflicting interests with respect to player one Consider any Nash equilibrium ((7i,(72) and any his- consistent with this equilibrium in which player one has always played that, scribes to take 62, /x* game 62 < M given this history, the equilibrium strategy of player two pre- .Sj"^ I, > ^ -^(^i) ^^^^ positive probability in period there exists a finite integer + I. M, N = H^-^^) + H92-97)-H92-92) In ^2 13 t ^ ^^ For any and a positive number = e (18) t, (^-'^^/•(f2--g2) ^2-52 such that in one of the periods at least _^M.(^_^^) ^ 0^ < + l,i+2, •••,f + M the probability that player one does not take a^, given that he always played a^ before, be at least must e. Proof: See appendix. Because g Let us briefly outline the intuition behind this result. interests player two can guarantee himself librium after any history ^ action s*^^ B{a'[) in period for the rest of the equal to Therefore, ht. game. t + this period. Recall that ^2 is - may get in any period in of course and 82 < - \ On M periods (1 — e). ^ maximal payoff player two The numbers M B{a'{) in period is gets if — 92 > in he does not an upper bound on what player she plays a\. But if and t -\- \ e loss argument this if future payoffs are bounded today must not be delayed too are constructed such that then Otherwise player two would get Note that of at least ^2^^ which player one does not take her commitment action, and it less if player two takes cannot be true that in each of the next commitment action the probability that player one takes her contradiction. type and takes an B{a'[) yields a "loss" of at least g2 then the compensation for an expected s':^^ any continuation equi- tries to test player one's the other hand, ^2 he cannot get more than g^ far to the future. a strategy ^ defined as the take a best response against a^. two of conflicting player one chooses Cj with a probability arbitrarily close or If then choosing an action 02 1, he in must give him an expected payoff this I if = r-^ Vj^^ at least is than minimax payoff his also holds for finitely repeated is bigger than in equilibrium, a games if T is large enough. Lemma in the history in n • 2 holds in any proper subform of up to that subform. M periods, then in at least not play a\ must be at least Theorem /z(u;°) > 0, e. Thus n if G as long as player one always played a\ player two chooses actions G2 ^ B{a\) along of these periods the probability that player one does Together with Lemma 1 this imphes our main theorem: 2 Let g be of conflicting interests with respect to player one and and n{u}') = /j.' > h' 0. Then 14 there is let a constant k{jj.',62) otherwise independent of{Q.,fi), such that where Vi{6i,62, uq > «,(SuS2,i>-;^°) (19) in fJ.*;uj°) is (l -<;"*') 2. + '"9-: ''!"' any average equilibrium payoff of player one with type any Nash equilibrium of G°°{pL). Take Proof: Consider the strategy for the normal type of player one of always playing a\. the integer N where M= and e [N] + 1, where are defined in is [A''] Lemma an action 02 ^ B(aJ) then there is the integer part of By Lemma 2. we know 2 one period at least N, and a (call it real that ti) if (1 — e). Lemma 1 we know among < In tt) > In than more , Tr*^ ) is M smaller ~ fi' . W in more than = TT commitment action cannot be smaller the probability that player one takes her '-j^ periods. Therefore, player two cannot choose actions 02 ^ B{a'[) often than s , , times. Substituting M = [N] + k{^',6,) (23) = 1 and ([7V] + e In u" , from Lemma In the worst case player 2, we get In ^' 1) In (l V - two chooses these actions '^-^!'-<^J-^-) S2-52 + <5f' 2 +^) J in the first k{fx', 82) periods. the lower bound of the theorem. If 61 one the next that n(7r* (21) is, 0, 1-e =7:. <, < That > So (20) However, by e player two takes periods in which the probability that player one will play Cj (denoted by than number is — »• 1 This gives Q.E.D. (keeping 62 fixed) then the equilibirum payoff of the normal type of player bounded below by her commitment payoff. Thus, in the limit our theorem gives the same lower bound as Fudenberg and Levine's theorem does for the case of a long-run player facing a sequence of short-run opponents. Their result can be obtained as a special 15 Theorem case of then to A'^ 2 for the class of goes to games with a\ game with if 82 goes So 0. 92-92 J \ In a Note that conflicting interests. \92-92 J conflicting interests a short-run player two will play a best response against if (25) > gl 7r-^2 or, equivalently, if TT (26) + (l -7!") -^2 ^^ =W. > 92-92 Using (26) It is Lemma in 1 immediately impHes Theorem important to note that the lower bound given discount factor of player reduced. 1. 2. If 62 compared Theorem increases, so does A:(/i*,^2), Hence, to obtain his commitment sufficiently patient as in payoff" in The to player two. 2 depends on the and the lower bound is equilibrium player one has to be following corollaries elaborate on the importance of the relative patience of the players. Corollary 1 For any such that for any player one Proof: Choose 8_i Si is at least 62 > gl < I, fJ.' > and e > there exists a ^1(^2,^*1 ^) 1j normal type of 81(62 , fi',t) the average payoff of the — < t. such that g',-t^ (27) (l-^f^'^*''^^)-£,+^?'^" ^^).,r Solving for ^^ yields (28) Clearly, 8_, if ^i > = 8_,{ti\82.e) ^^>^' ^^^±-^-1 < 1 . ^i(/i*,62,e) then vi(<5a,62,/^-;u;°) (29) = <''^'''^-5: > (l-<5f(^*''^)-5, > {i-i^''''')-g_,+A^''''-9\=gl-t + Q.E.D. 16 Corollary 2 Consider any sequence {62}, e > 0. < S2 < there exists a sequence {(5"}, 6" Then = YinXn-^oo^i 1, = lim„_oo ^2 1, 1, for any {^",^2} ^^^ average payoff of the normal type of player one below by gl — where fix such that is bounded t. Proof: Take any {<5^} all n, and ^ and 81(82, (i',e) is > fix e Choose 0. given by (28). -^ {8'^] Then the such that 1 > 8"^ 8^(82, n',e) for immediately from the result follows Q.E.D. previous corollary. Corollary 2 shows that there sequence {8^,82} — is an area — in the 81 82 space such that for any commitment (1,1) in this area player one gets at least her » (up to an arbitrarily small however, that limn~.ooYE^ e) for — any pair of discount factors along *'he limit '^^ 0) '•^- player one is this sequence. Note, infinitely more patient than player two. This observation helps to understand a related result of Cripps and who (1991) the limit of the mean prior probability of a Banach Under of their payoffs. perturbations they show that payoff as the games without discounting, consider repeated if the game has commitment limit of the slightly stronger conditions conflicting interests of her stage game Thomas which players maximize in and if on the possible there type, then player one gets at least her mean payoff payoffs. is a positive commitment However, the case of no We can give examples — 1, lirrin^ao^z^ > 0, discounting obscures the role of the relative patience of the players. of equilibria in games with conflicting interests and player one's equilibrium payoff is {8^,82} along this sequence. Thus, player two our lower If bound does not player one has to be where <5i — > 1, 62 > bounded away from her commitment payoff if player one is for any not patient enough as compared to apply. much more patient than player two the reader might be left with the impression that we are essentially back to Fudenberg and Levine (1989) where a long-run player faces a sequence of short-run players. = However, this may be is not the case. First, Fudenberg and Levine's to Second, we are going to show in the next subsection that whenever the 1. not of conflicting interests result requires 82 it is while here 82 is one player does not care at all game is possible to find an equilibrium which violates Fudenberg and Levine's lower bound no matter how much more patient player one to player two. Thus, there arbitrarily close is as compared a fundamental difference between repeated games in which about her future payoffs and games 17 in which she does care but is of the than her opponent. Finally, the importance of the relative patience less patient two players is very intuitive as will be shown after we have introduced the case of two-sided uncertainty in subsection 4.3. 4.2 Necessity of the "Conflicting Interests" Condition The question arises interests. If the equal to her game minimax Proposition ests. whether Theorem 2 is e > there is an tj game which > and a 62 is commitment payoff is < not of conflicting interI such that the following a perturbation of g, in which the commitment type of player has positive probability and the normal type has probability is of conflicting no: is 1 Let g be a non-trivial is games which are hot not "trivial" in the sense that player one's payoff^ the answer Then for any holds: There also holds for (1 — t), and 1 there a sequential equilibrium of this perturbed game, such that the limit of the average payoff of the normal type of player one for 8\ —^ from her commitment payoff by > 8_2- at least rj for any 82 \ bounded away is Proof: See appendix. Proposition 1 shows that the condition of conflicting interests but also necessary for Theorem 2 to hold, in in is two respects. First, it says that if the fact, game is it a is is not only sufficient stronger than that little bit not of conflicting interests, then it not only possible to find a Nash equilibirum which violates Fudenberg and Levine's lower bound, but also to find a sequential equihbrium. As has been indicated in Section 3, the construction of a Nash equihbrium using threats which are not credible simpler. Secondly, Theorem we could have established 2 only requires that y.{u°) in the perturbed game. So who credibly threatens to punish any deviation two from the equilibrium path we want to sustain. However, applications ^It is well it is known much necessity by constructing a perturbation which gives a high prior probability to an "indifferent" type of player > is natural to assume that ^i{uP) is close to one. This that a player can always guarantee herself at least her equilibrium. 18 is in many economic why we provide minimax payoff in a any Nash- stronger proposition which says that even if /i(w°) is arbitrarily close to to construct a sequential equilibrium in which the payoff of the is bounded away from Note that more in Proposition 1 (5i — > 1 while 62 is games possible normal type of player one fixed, so player Thus, Proposition 1 in one may be arbitrarily shows that there whom between games with two long run players, one of the other, and it is g^. patient than player two. difference one is an important is more patient than which a long-run player faces a sequence of short-run playe:s. the latter Fudenberg and Levine's bound holds games, for for all stage in the In former it only holds for games with conflicting interests. 4.3 If Two-sided Incomplete Information and Two-sided Confiicting Interests there are two long-run players sided uncertainty. the game is Our it is result can most natural be extended to perturbed such that there is what happens to ask measure /x,-, respectively. player i, Without i.e. i game out two- Suppose incomplete information about both the payoff of the countable set fi,- Finally, player z's suppose that the game commitment Lemma 2 we ? = is z's type which is drawn by according to the probability of confiicting interests with respect to strategy holds player j 1. didn't say Now why test player one's matter what the reason down to his minimax pa3'off. consider the normal type of player two. player two might choose an action which a best response against player one's commitment strategy. wants to is G {1,2}. Let uf and m' represent the normal and the commitment types, loss of generality let the proof of there this case in the following way. functions of player one and player two. Let w, denote player nature in the beginning of the if He might do is Lemma expect that player one choses s\ probability. This holds for the ^ 2 states that Cj in if he takes 02 ^ B{a'l), one of the following periods with not so because he type or because he wants to build up a reputation himself. is. In No then he must strictly positive normal type of player two no matter what other possible types of player two exist with positive probability. A possible strategy of player one the normal type of player two, then by which player two still is to play Cj in every period. Theorem 2 there are at will not play a best response against a\. 19 most If she faces k[ji',82) periods in In the worst case for player one this happens in the first k periods of the face the normal type of player which = /i,(a;°) 3 Let g at least is g if she does not in every period. This expected payoff of the normal type of player for the fi° > be of conflicting interests with respect to player and ki{fi',6j) otherwise /x,(a;*) = fi' > 0, z Then G {1,2}. i and let there are constants independent o/(f2,, Qj,/i,), such that > r;,(6.,<5„;.-,/z;;u;°) (30) (l - type uf in any Nash equilibrium ofG°°{fi). is fJ^S^'^"''^^ g: any average equilibrium payoff of player ti*, fi^,uf) + fi^6^'^'''''% where Vi{6i,S2, if the other hand, given in the following theorem: is Theorem Thus, On two her expected payoff argument establishes a lower bound i game. the probability of the normal type of player two very patient, then the lower bound for her average payoff is close to and 1 if again close to her is with i player one is commitment payoff. What like to course, it is can be said commit if there are two-sided conflicting interests and The kept fixed and in the limit player game with down them gets his i i Si i.e. to his most prefered payoff. minimax is played, and the lower goes to 1 is A:,(/x',^j), i.e. is her lower bound. then this k periods become commitment more patient than the payoff. This the (5,- and number of is less and On the other hand, less important, and very intuitive. In a symmetric if one of the parties other. 3 are in striking contrast to the message of the Folk theorem games with incomplete information by Fudenberg and Maskin (19S6). cis an equilibrium outcome of the perturbed game 20 for The Folk theorem says that any feasible payoff vector which gives each of the players at least his payoff can be sustained Of payoff. must expect that a strategy other than the best response against will get his Theorems 2 and each player would But suppose that conflicting interests reputation building has an effect only sufficiently if both players are equally patient if bigger player j's discount factor the bigger commitment strategy if Sj is interests, to a strategy which holds his opponent periods in which player is g has two-sided conflicting impossible that each of 5j differ. her if if minimax the right perturbation is Theorems chosen.® further perturbations. Theorem - and show that 3 one of the players If type has positive probability then probability 2 this result is patient enough and is not robust against her if commitment no matter what other types are around with positive - 2 imposes a tight restriction on the set of equilibrium outcomes in any Nash equihbrium.^ We have to be very precise here in what argues that the Folk theorem is meant by robustness. Fudenberg (1990) robust against a further perturbation of the informal-ional is structure in the following sense: Consider a sequential equilibrium which has been con- structed using an e probability of a "crazy" type as the Folk theorem suggests. given time horizon of the small as compared to game other types are introduced with a probability which then this e, an equilibrium. is still We goes to infinity the equilibrium must break down. conflicting interests and if then the commitment type there will If for is However, a is the time horizon if have shown, that if the game is of an arbitrarily small probability of a commitment type, dominate the play are interested in the set of equilibria for T as T becomes large enough. Thus, —^ oo, the Folk theorem is if we not robust against small perturbations of the informational structure. 5 Examples The Chain Store Game 5.1 Consider the classical chain store game, introduced by Selten (1978), with two long-run players. In every period the enirant may choose to enter a market (/) or to stay out (0), ^Fudenberg and Maskin's Folk theorem for games with incomplete information considers only finitely However, the extension to discounting and an infinite horizon is repeated games without discounting. straightforward. ^Note that even set of this if the Nash equilibrium game suppose that the game a type who mimicks is is not of conflicting interests we can payoffs, although the is bound still to his impose some restriction on the be weaker than Fudenberg and Levine's. To see not of conflicting interests, but that there committed to hold player two down this type, then she will minimax is a positive prior probability of payoff. If the normal type of player one can guarantee herself on average at least the payoff she would have got if she were publicly committed to this strategy. This doesn't give her her most prefered payoff but it may still be more than her minimax payoff and thus reduce the set of Nash equilibrium payoffs as compared to the Folk theorem. and elaborate I am grateful to Drew Fudenberg for this observation. this idea. 21 A companion paper will generalize while the monopolist has to decide whether to acquiesce (A) or to fight (F). game the payoffs of the unperturbed stage Assume that are given as follows: o A ' F ° ' 2 ' -1 3 — The Figure 2 The monopolist would commitment payoff player two's like to commit chain store game. to fight in every period which would give her a and which would hold the entrant down to of 3 minimax payoff the game is interests with resprect to the monopolist. repetitions of this ffi - Since 0. according to our definition - is of conflicting Kreps and Wilson (1982) have analyzed game with some incomplete also finite information about the monopolist's type. For a particular perturbation of player one's payoff function they have shown that there are sequential equilibria in which the monopolist gets on average almost her payoff if her discount factor is close enough to one and if commitment there are enough repetitions. However, Fudenberg and Maskin (1986) demonstrated that any feasible payoff vector which gives each player more than of figure 2, can his minimax payoff, i.e. be sustained as an equihbrium outcome any point it in the shaded area the "right" perturbation is chosen. Thus, our Theorem 2 considerably strenghtens the result of Kreps and Wilson (1982). It says that the only Nash equilibrium outcome against any perturbation gives the monopolist at least her that she cannot get more), ®I am grateful to Eric van average payoff of player two if she Damme is 0. is of this game which commitment payoff sufficiently patient as compared is robust of 3 (note to the entrant.* Theorem 2 does not imply that the more patient than player two. So it may be she gets less than 3 and player two gets more than for the following observation: Recall that player one that in the beginning of the game, say until period L, is but after period L payoffs are always (3,0). For player one the first L periods do not count very much because she is very patient, so her average payoff is 3. However, player two puts more weight on the first L periods and less on everything thereafter, so her average discounted payoff may be considerably bigger 0, than 0. '?2 Furthermore Now it this result carries over to the infinitely repeated shows that suppose that there of the entrant. He would is like to a commitment payoff of 2 while So the game payofT. theorem is applies. If there one sufficiently two is commit which would give him to enter in every period would hold the monopolist down to it minimax her 1, two-sided uncertainty Proposition 2 says that is relative patience of the of player also incomplete information about the payoff function also of conflicting interests with respect to the entrant on the is game. it all two players and the prior probability distribution. more patient than player two and if and our depends If player the probability of the normal type close to one, then player one will get her commitment payoff in any Nash equilibrium, and vice versa. A 5.2 Repeated Bargaining Consider a buyer and a (6) of a perishable good. seller (s) The who bargain is = ^^^ would and only if like to seller is p, = of the buyer PI. commit in G {-,',''-}, and there p,- = = - in every period. The unique perturbed such that with some positive probability she 2 applies any Nash equilibrium down to a is if minimax ^, his and the buyer this will get is Some cooperation example is trade at price a minimal bid - minimax payoff best reply of the Suppose the payoff function always offer will almost her commitment payoff of ^^^ is close to one. not as clear-cut as the chain store game. > If 0. is the buyer could offer Pb But of 0. that bargaining over a pie of fixed size is payoff. her discount factor possible prices, so he might choose p, point is Consider the commitment strategy of the buyer. She is hold the seller In and the production costs of the which gives him g^ have to assume that there all p,. herself to offer p^ Note however, that between > 1 ^, Then Theorem on average if pt, is a sealed bid double auction in every period: Both players simultaneously submit bids pb and p^, p repeatedly in every period on the sale valuation of the buyer Suppose there seller are 0. Game > if he gets the seller = is We she could indifferent and we end up with no trade. The not quite a needed to ensure that trade takes place game at of confiicting interests. all. Schmidt (1990) we considered a more complex extensive form game of repeated bargaining with one-sided asymmetric information, which confirms the above result that 23 the informed player can use the incomplete information about his type to credibly threaten to accept only offers which are very favourable to him. There, however, we took a approach and interesting to it is not allow for "all possible" function, we assumed € costs, c i.e. [0, 1]. compare the two models. different Schmidt (1990) we did In but only for "natural" perturbations of player one's payoff that there The problem is may be incomplete information about the that in this case there is seller's no commitment type since none of the possible types of the seller has a dominant strategy in the repeated game. However, the game has a finite horizon it a weak Markov property the and only who is can be shown that seller in any sequential equilibrium satisfying with the highest possible type will accept an mimicked by all the other possible types of the try to test the seller's type at most a fixed finite seller. We show that the buyer number of times and seller will get his commitment payoff even if well, consider there is is Theorem much There are not of conflicting interests.^ will 2) we can less patient than Furthermore the bargaining the buyer, so the relative discount factors are not crucial. game we he if that this will happen only in the end of the game. Surprisingly (from the point of view of show that the offer commitment type covers at least his costs. This type plays the role of the if it if common interests as because players have to cooperate to some extend in order to ensure that trade takes place. 5.3 Games with Common and Conflicting Interests "Pure" conflicting interests are a polar case and in most economic applications there are both - common and prisoner's conflicting dilemma depicted interests, but our theorem hcis in no - interests present. Figure bite. 3. one her minimax payoff as in every down Nash equilibrium. to his minimax ®Note that not all In a formal sense this game down like most to commit game payoff, but to of conflicting to his minimax herself to play Z)(efect) in payoff, but well. So, trivially she will get at least her In this is Given that player two takes a best response against her commitment action player one would every period. This holds player two Consider for example the repeated the problem commit is not to it only gives player commitment payoff commit to hold player two to cooperate. possible perturbations are allowed necessary condition for the result in Schmidt (1990). 24 for. This is why conflicting interests are not a D C c ' D ' ° 2 3 1 1 3 — The Figure 3 Another interesting example commit is prisoner's dilemma. a repeated Cournot game. Firm one would like to to choose the "Stackelberg-leader" quantity, which maximizes her stage profit given that firm two chooses a best response against follower" payoff of firm two is it. is not to hold player two interest to conflicting in the sense that each of minimax payoff since he player one gluts the market. So our result does not if Again the problem of player one Both players have the common game However, the "Stackelberg- positive and thus greater than his can be held down to zero profits apply. 51 maximize them would down as far as possible. joint profits, but interests are also like to get more for himself at the expense of the other. 6 To keep the argument stage Extensions and Conclusions as clear as possible games with only two we considered a very simple class of possible players, finite strategy sets, a countable set of possible types, and commitment types who always take the same pure action in every period. these a.ssumptions can be relaxed without changing the qualitative results. and Levine (1989) provide a generalization to n-player games are compact metric spaces and In Fudenberg one.^° to the case ^"If i = n > 2, ...,n 3, in which there is in sets a continuum of possible types of player and Levine (1992) they show that the argument can be extended the definition of a to their Fudenberg which the strategy where the commitment types play mixed strategies and down All of minimax game to games with moral of conflicting interests requires tiiat aj holds payoffs simultaneously. 25 all other players hazard, in which not the action of player one itself but only a noisy signal can be observed by player two. Since the technical problems involved as in our model we refer to their What happens work for commit hermself to a more complex strategy which prescribes to take different actions in different periods and which conditional on player two's past play? It doesn't change anything in the proofs of game with down has to hold player two to her Fudenberg and Levine (1989) game is is easy to check that replacing Oj by and 2 and Theorem 1 The problem 1. buyer has to decide a a'[{ht) demonstrated that the assumption that the stage is that in an extensive form game player two example a repeated bargaining game first whether to buy or not and then the the seller would have produced high quality. This commitment payoff context. The in equilibrium. definition of a strategy of player one holds player two two takes an action 02 more than his player one's minimax commitment payoff player two chooses 02 Theorem To conclude, a repeated If this g is any is defined as g^ finite it commitment in every period the has to choose whether to will not observe whether fail problem does not is to get her arise in our assumes that the commitment paj-off. Therefore, if player a-i 7^ Gj, then player two cannot get = is maXajg/ij Tnma^^B[ai) get less than g^. However 9i{(^i-,o:2)- So if Therefore, following straightforward that our result holds extensive form game. long-run players one of the players the the seller might this paper has shown that "reputation game with two is take an action So 02 must have been an element of B{al). € B{al) player one cannot if seller minimax to his which may which player one's commitment strategy aj 2 of Fudenberg and Levine (1989) without qualification interests. down some other strategy payoff. why conflicting interests in equilibrium after observationally equivalent to is Note however that game with in buy then he deliver high or low quality. If the buyer decides not to in in any period and after any history in which player one has no opportunity to show that her strategy strategy. Consider for is a'[{ht) However, 2. gained by this generalization because minimax payoff also may be simultaneous-move cannot be relaxed without an important qualification of their Theorem after is Lemmata conflicting interests very little same any formal statements and proofs. player one would like to if in these generalizations are the if effects" and only if can explain commitment the game is of conflicting very patient as compared to the other player, then 26 any Nash equilibrium outcome which is robust against perturbations of the information structure gives her on average almost her commitment payoff. message of the Folk theorem may be misleading. However, we the evolution of conflicting - commitment and cooperation interests are present, in which clearly future research. 27 is games in This indicates that the still know very which both - little about common and one of the most important issues of Appendix Lemma Proof of 2: Consider any equilibrium and {a-i,a2) a history fix player one has always played aj, such that Such a history exists because equilibrium strategy probability. It will be shown that Note that e > 0. 0. Suppose that according to the (possibly mixed) Define each of the periods this can't minimax e is in be true ^ B{al) in period + t t + l,t in equilibrium + 2,- • • ,t + M with positive I Oj (given that smaller than is because then player two would payoff. independent of 7r'^(ai) along which t Suppose further that the probability of player one not playing get less than his that > up to any period has positive probability given {(Ji,^^). player two chooses s^^^ a^"^^ he always played aj before) e. fi* /i' h' = t Prob{sl and that = Gi | M has been chosen in a way to guarantee h'^~^) and let V'2^(5[,<72) be the continuation payoff of player two from period t onwards (and including period r) given the strategy profile (sl^a^) in period r. is The expected payoff of pla\-er two from period t + I onwards given by: V^^\a„<T,) (31) + ••• = + ^'+^(aI)-L2(aI,4^^) 5]7r'+^(a:)-yr^(ai,4^^) + + 62-Vr''^']---\\. It will be convenient to substract §2 from both sides of the equation (Recall that §2 is the maximal payoff for player two best response against Oj.) Then we if in every period. he takes an action which get: 92 + ^'^\a',)-\[g2{al4^')-g2] 28 is not a + 92 Vl^\auar)- J^tt'+^K) 8,. 1 -So ai^iaj (32) + <52-:r'+2(ai)-{[^2(«I,^r')-^2] + ••• + J^tt'+^O +6,. 92 )- (ai,cr2 ^2 -(5, 1 + 52-7r'+^K)-{b(at,a^+'')-^2] + By assumption ^' <52 } 1-6, the conditional probability that player one does not take her action given that she always played a^ before 1,- •• ,i + M, ^ and, of course, we can use all, it that 7r'''"'(aJ) V^+\a^,a'+') ,„„„, , ., Furthermore, s^^^ < any period from t + e, Since ^2 1. -^ — ^_^^ 92 < _ is the maximal payoff player two Js and „.._ g2{a',sl+'^) we can use that ^2(01, - V^'-'ia^.a^) <'"2) < 92 -92 -T3^ ^^^''-fzf + 52 92 • 1 -92 —+ -62 ••• +6. = e.{l+82 + --- + -^ ,. + i-{{92-92) , . , + + ••• S2-l-{{g'2-h) + S2-'-f^]-- 1 , Oj, so 52- 82-l-{[g'2-g2) . 92 62- 1- M-\ < -92 92 ^ < rrx 92 + 92 ^_^^ 92- Substituting these expressions yields: + <52-e-^?—^ + 1-62 ^ e n+M+i < _ V^ ., Co supposed not to be a best response against (35) = < T'+'(ai) i. (36) e in has to be true that (34) Finally smaller than so (33) can get at is commitment -92 —+ -82 . 92~ 92 1_62 , ' r 62 . - - X 92) M r-M ^^ 8^'-').^-f:f1—02 29 , (52 '^' ^" ' ^' + 8','.^f^ — 09 1 - 92 1-82 92 + (i_+ 62 92 + • -h Recall from the statement of + 6^_-'). {g; rM 92-92 • , Lemma ,. - easy to check that 92 ^ -92 ^•(ITr^ + (^^) . , 92 -92 .(1-6,) > 0. has been chosen such that e f'iQ\ - h) {gl 2 that -92 92 It is - h) - = il^iil-Mjliil-^f e (37) • rM '^2 92 -92 _ . , - 32-92. -13^ Therefore we get: However, since ^j we is player two's minimax payoff this is a contradiction to the fact that Q.E.D. are in equilibrium. Proof of Proposition The proof is game g such 1: similar to the construction of the counterexample in Section 3. Perturb the that there are three types of player one, the normal type, the type and an indifferent type, whose payoff probabilities (1 — e), |, and |, respectively. is commitment the same for any strategy profile, with Let 62(e) = ^^ < 1 and suppose 62 > ^2(^)- Define - ili^ In \l (*°) and " let m= [n] + 2, where [n] is straightforward to check that n Since the gi ,nS, the integer part of n. Given the restriction on 62 well defined commitment payoff of player one there exists an action that is = - 0.2 such that gi > minmax^i and §2 > = and is ^1(0^,02) minmax^2-^^ We positive. strictly greater < gl will and ^2 now = than her minimax payoff ^2(01,52) < ^2- Suppose construct an equilibrium such that the limit of the average equilibrium payoff of the normal type of player one for is bounded away from her commitment payoff by (41) V at least = --[^r-^i] > m it is 77, 61 —* I where 0. ^^If for any of the players gt < minmaxy,- the construction of the "punishment equilibria" which are used below to deter any deviation form the equilibrium path are slightly more complex. In this case players have to alternate between the outcomes g' and g such that both get on average at least their minimax payoffs. 30 Suppose get at > 1 > ^1 J - J- ^ Z.^\_- ) Along the equilibrium path all. where payoff player one can types of player one play a\ in every period, all while player two plays a^ G B{a\) in the maximum the is g^ first n, then starts again playing a\ for the next m— m— \ periods, then he plays \ periods and so on. If 0.2 in period player one ever deviates from this equilibrium path player two believes that he faces the normal type with probability we In this case 1. where the Folk theorem tells are essentially back in a game with complete information us that any individually rational, feasible payoff vector can be sustained as a subgame perfect equilibrium. So without writing down the strategies we can construct a continuation explicitly is (y^j-^i, 735-^2 )• to check that - commitment and the Clearly, the no incentive to deviate since given m> a\ equilibrium, such that the continuation payoff is at least indifferent type of player weakly dominant 2 and the restriction on - 8\ both of them. for one have It is easy the normal type of player one will not deviate either. Now the suppose player two ever deviates commitment type i +1 any period are supposed to play a\ in period switches to another strategy period in s^-^'^ 7^ a\. If i t. + In this case the 1, normal and while the indifferent type player two does not observe a\ being played in he puts probability one on the indifferent type. Using the Folk theorem we can construct a continuation equilibrium in this subform which gives player two jzfQ'i ^rid which would give the normal type of player one ^zj'9\a\ being played in period i + 1 he puts probability ^^i however, plaj'er two observes on the indifferent type. In the continuation equilibrium of this subform {a\,a'2) are always played along the equilibrium path. If there is any deviation type with probability one and is f Yr£~5i » Tr5~^2 ) • - player one, player two believes that he faces the normal using the Folk theorem again Clearly, always to play Gj always a\ and always a\ It is bj' easy to check that is is - the continuation payoff a best response of player two against a best response for the commitment type against any strategy. it is also a best response for the normal type of player one, given the "punishment" after any deviation. We have already shown that the strategies of the players form an equilibrium after any deviation from the equihbrium path and that given the continuation equilibria player one has no incentive to deviate from strategy is this path. We still a best response along the equilibrium path. 31 have to check that player two's The best point in time for a when deviation is period, never it player two will. is supposed to play If it 0.2- Suppose player two does not deviate. Then m-l = (42) However, is if does not pay to deviate in this his payofT is given by: 2m-l h +Y^^-92 he deviates, the best he can do is - rf^-(^2--^2). to play Cj in period t. In this case his payoff given by v.K) = (43) It is now easy + i,.{(i-5).^.,j + s; to check that e and Thus we have established that We now 6(e) this have been constructed such that payoff of the normal type is is this equilibrium — indeed smaller than g^ ^2(02)- rj path the average payoff of the when 6\ —^ I. The equilibrium 2m-l t=m+l t=\ = > given by: m-l (44) ¥2(0.2) indeed an equilibrium path. is have to show that along normal type of player one f ^.j,} 16"* 1 ^^^-g^ - -.^.[gl-g,]. Therefore the difference between her commitment payoff and her average payoff in this equilibrium is (45) — = — = !!"!'!'S {l-Sl)-Sr' Y:rs^ <5r"^ Taking the limit for 61 —* 1 we . -b--^-' - 1 ^T~^ get cm — (46) r r. -1 1^1-^1] ?^. ~^ 1 ' ^3'i - 9i] = --[^r-^i] Q.E.D. 32 References AUMANN, R. AND S. SORIN Economic Behaviour, AND CRIPPS, M. J. Vol. THOMAS FUDENBERG, 5-39. 1, (1991): of Incomplete Information" Games and "Cooperation and Bounded Recall", (1989): mimeo, University D. (1990): "Explaining Repeated Games Warwick, June 1991. "Learning and Reputation Cooperation of in Commitment and in Repeated Games", forthcoming in Advances in Economic Tlieory, Sixth World Corigress ed. by J. -J. Laffont, Cambridge University Press. FUDENBERG, KREPS, AND D., D. Run and Long-Run FUDENBERG, in D. 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