Abstract
In this chapter we study whether repetition can lead to cooperation. Specifically, it is investigated which outcomes can be sustained by means of subgame perfect (or Nash ) equilibria when a game is repeated finitely or infinitely many times. The main result is the Perfect Folk Theorem, which states that, for almost all games, every outcome that is feasible and individually rational in the one-shot game can be approximated by subgame perfect equilibrium outcomes of the discounted supergame as the discount rate tends to zero, and that, for almost all games with more than one Nash equilibrium, any such outcome can be even approximated by a subgame perfect equilibrium payoff of the finitely repeated game as the number of repetitions tends to infinity.1
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Notes
The theory on repeated games with complete information is also surveyed in Sorin [1988]. At the advanced level, a comprehensive overview is provided in Mertens, Sorin and Zamir [1990].
For the unraveling to occur it is not sufficient that both players know what the last stage of the game is, there has to be a finite upper bound on the game that is common knowledge, see Neyman [1987].
The equilibria constructed in Friedman [1971] are not subgame perfect. In that paper, Friedman considers equilibria in reaction functions, i. e. at each time t each player conditions his actions only on the opponents’ actions at time t — 1. In repeated duopoly games with discounting, such equilibria can be subgame perfect only if they are trivial, i. e. if they prescribe to play the static Cournot equilibrium in every stage, see Robson [1986] and Stanford [1986a]. In Stanford [1986b] it is shown that in undiscounted supergames such strategies can give rise to nontrivial SPEa. Friedman and Samuelson [1988] obtained SPEa in continuous reaction functions that specify the action of a player in each period as a function of the actions of all players in the previous period (i. e. a player also looks at what he himself did in the previous period).
Giith, Leininger and Stephan [1988] criticize folk theorems. They argue for the imposition of subgame consistency (Harsanyi and Selten [1988]), i.e. the requirement that two isomorphic games should be played in the same way. Notice that all subgames of the discounted supergame Γ (δ) are isomorphic, hence, subgame consistency implies stationarity. For a critique on stationarity see Rubinstein [1988]. Also see Mertens [1989b]. For existence and robustness results concerning Markov equilibria, see Maskin and Tirole [1989].
See Kaneko [1982]. If there are infinitely many players and there is no discounting this result only holds under additional regularity assumptions, see Masso and Rosenthal [1989].
Some recent papers dealing with the issue of how much information the per period outcomes must convey to obtain a folk theorem are Fudenberg and Levine [1989b, c], Fudenberg, Levine and Maskin [1989] and Abreu, Milgrom and Pearce [1990].
Fershtman, Judd and Kalai [1987] show that cooperation can result in static games if players can delegate their decisions to agents (and if contracts between players and agents are binding and observable).
For a behavioral theory of the finitely repeated Cournot duopoly game, see Selten, Mitzkewitz and Ulrich [1988] and Mitzkewitz [1988] as well as the references therein.
Neyman [1987] shows that cooperation may result in the finitely repeated PD if the length of the game is not common knowledge. Neyman’s model also falls under the heading of incomplete information.
In Stahl [1989] it has been argued that the prisoners’ dilemma from Fig. 8.4.1 is a knife-edge case since, irrespective of what the other player does, defecting yields one unit of utility more than cooperating. Stahl describes the set of subgame perfect correlated equilibrium payoffs for each discount factor for general prisoners’ dilemma games. He shows that the set of payoffs is monotonically nondecreasing and upper hemicontinuous in δ.
Other models that force the players to cooperate in some classes of games are described in Aumann and Sorin [1989], Matsui [1989], Fudenberg and Maskin [1990], Binmore and Samuelson [1990] and Robson [1989]. In Aumann/Sorin the driving force is that each player thinks that his opponent may be an automaton playing a perfect recall strategy. In Matsui cooperation is caused by the fact that each player with a small probability may find out the opponent’s supergame strategy. The other papers invoke arguments of evolutionary stability (see Chap. 9).
For another example, see Bernheim and Whinston [1987].
The problem has been solved in a satisfactory way in Benoit and Krishna [1989]. In that paper it is shown that, if R = limt →∞RPEP (T ) exists, then either R is a singleton or R is a subset of the weak Pareto frontier of the feasible set of payoffs. ( RPEP (t) is the set of average payoffs obtainable by renegotiation-proof equilibria in Γ(t ) as defined in (8.8.7).) Benoit and Krishna also give examples to illustrate that, if R is not a singleton it need not be a subset of the strong Pareto frontier, and that even if Γ has multiple equilibria, R may be a single inefficient point.
The issue of renegotiation in repeated games has recently received considerable attention. Papers that employ a definition of renegotiationproofness that is closely related to the one discussed in this section are Bernheim and Ray [1989], Farrell and Maskin [1989], Evans and Maskin [1989], Asheim [1990], Ray [1989] and Van Damme [1989]. Most of these papers make a distinction between internal consistency (i. e. there should be no Pareto dominance relations between two payoff vectors in the same renegotiation proof set (cf. (8.8.9)) and external consistency (i. e. a renegotiation proof set should not be dominated by another one, cf. (8.8.10)). Farrell/Maskin define a set V to be weakly renegotiation-proof if (i) V= V+ and (ii) V ⊂Pδ( V ). (The notation is as in the main text. Note that this condition is weaker than Eq. (8.8.9), instead of equality, set inclusion is required.) They characterize the payoffs that belong to some weakly renegotiation-proof set and then they continue by proposing several concepts of external stability. The necessity for such external conditions was already established in Van Damme [1989]: If δ ≥ 1/4, then in the prisoners’ dilemma from Fig. 8.4.1 any payoff belongs to a weakly renegotiation-proof set. Evans/Maskin show that for generic 2-person games there exists a weakly renegotiation-proof payoff that is Pareto-efficient provided that the discount factor δ is near enough to 1. Bernheim and Ray [1989] adopt the same definition of weak renegotiation-proofness under the name internal consistency. Also these authors propose various notions of external consistency. Asheim [1990] expounds on the theory of social situations as developed in Greenberg [1990] (also see Greenberg [1989a, b]) and defines external stability by using an idea similar to that of Von Neumann and Morgenstern’s stable sets: It is required that a ‘nonviable’ equilibrium is “defeated” by a ‘viable’ one. Asheim also does not insist on the assumption of stationarity that is implicit in the definition of weak renegotiation proofness. Ray [1989] argues that the notion of weak renegotiation-proofness does not fully capture the idea of internal stability and he argues in favor of condition (8.8.9) instead, i. e. Ray insists on the equality of the sets rather than the inclusion of V in P δ(V ). Ray shows that the limits (as δ tends to 1) of sets satisfying (8.8.9) are either singletons or are contained in the weak efficient frontier of the feasible set (cf. the similar result obtained in Benoit and Krishna [1989] for finitely repeated games). The relation between finitely and infinitely repeated games is also investigated in Blume [1987], An entirely different concept of renegotiation-proofness has been proposed in Pearce [1987]. The intuition underlying this concept is also discussed in Abreu and Pearce [1989]. Bergin and MacLeod [1989] present a unifying framework in which the different concepts can be discussed.
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van Damme, E. (1991). 8 Repeated Games. In: Stability and Perfection of Nash Equilibria. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-58242-4_8
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DOI: https://doi.org/10.1007/978-3-642-58242-4_8
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