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This paper studies the repeated prisoner’s dilemma with private monitoring for arbitrary number of players. It is shown that a mixture of a grim trigger strategy and permanent defection can achieve an almost efficient outcome for some range of discount factors if private monitoring is almost perfect and the number of players is large. This result also holds when the number of players is two for any prisoner’s dilemma as long as monitoring is almost perfect and symmetric. A detailed characterization of this sequential equilibrium is provided. Download Paper
We describe the maximum efficient subgame perfect equilibrium payoff for a player in the repeated Prisoners' Dilemma, as a function of the discount factor. For discount factors above a critical level, every efficient, feasible, individually rational payoff profile can be sustained. For an open and dense subset of discount factors below the critical value, the maximum efficient payoff is not an equilibrium payoff. When a player cannot achieve this payoff, the unique equilibrium outcome achieving the best efficient equilibrium payoff for a player is eventually cyclic. There is an uncountable number of discount factors below the critical level such that the maximum efficient payoff is an equilibrium payoff. Download Paper
We analyze infinitely repeated prisoners' dilemma games with imperfect private monitoring, and construct sequential equilibria where strategies are measurable with respect to players' beliefs regarding their opponents' continuation strategies. We show that, when monitoring is almost perfect, the symmetric efficient outcome can be approximated in any prisoners' dilemma game, while every individually rational feasible payoff can be approximated in a class of prisoner dilemma games. We also extend the approximate efficiency result to n-player prisoners' dilemma games and to prisoner's dilemma games with more general information structure. Our results require that monitoring be sufficiently accurate but do not require very low discounting. Download Paper