Self-Refuting Theories of Strategic Interaction: A Paradox of Common Knowledge

Abstract

Game theoretic reasoning is sometimes strikingly inconsistent with observed behavior, or even with evidence from introspection. Famous examples of such inconsistency are the finitely repeated Prisoner’s Dilemma game and Selten’s Chain Store Paradox (Selten, 1978). In both cases, some plausible solutions run counter to game theoretic reasoning and appear to point to the inadequacy of the game theoretic notion of rationality in capturing important features of human behavior. These considerations do not apply to artificial settings only: in a wide range of ordinary social interactions it does not pay to be (or look) too rational. As Goffman puts it, expert poker players sometimes discover that one can lose the game because of playing too well (Goffman, 1969). In international conflicts, it may well pay to be thought of as a ‘mad dog’. And in the above quoted games, all or some of the players involved get a lower payoff than they would were they to play in less than a rational manner. All attempts to explain the emergence of cooperation in a finitely repeated Prisoner’s Dilemma, as well as that of ‘reputation effects’ in the Chain Store story, have required either some version of ‘imperfect rationality’ (Selten, 1978) or a change in the structure of the game, such as assuming altruism (and thus changing the payoffs), or imposing incomplete information at the beginning of the game (Kreps et al., 1982).

I am grateful to Tommy Tan and Philip Reny for helping me appreciate the importance of common knowledge in games, and to Jon Elster and Michael Woodford for many useful comments. Financial support from National Science Foundation Grant SES 87-10209 is gratefully acknowledged.