Optimal Retaliation for Optimal Cooperation

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Abstract
There has been a growing number of theoretical as well as experimental investigations on the emergence of evolutionarily stable cooperative strategies in the iterated Prisoner's Dilemma game. From a methodological viewpoint, investigations of this sort suffer so far from two shortcomings. The phenomenon of noise, that is, any deviation from the assumption of perfect information among the players, has been given only unsystematic consideration, if any. Furthermore, only insufficient notice has been taken of a recent major development within game theory, dynamical population games, which allow the analysis of games with an infinite number of players in a much more concise and compact way than before. The model presented in this article allows us to study the interactions among conditionally cooperative strategies—reactive strategies—in iterated Prisoner's Dilemma population games in noisy environments with this powerful new analytical tool. The main results are: Under noise, the best strategy to establish cooperation in a world of noncooperators is GRIM, the strategy that begins with a cooperative move, but never cooperates again, once a defection occurs; the best strategy to maintain already established cooperation, however, is a more or less restrained version of TIT-FOR-TAT (how much restrained depends on the necessary safety margin against an eventual reinvasion of noncooperators); cooperating with unconditional cooperators—ALL COOPERATE—destroys the social order.

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