Behavior-Dependent Contexts for Repeated Plays of the Prisoner's Dilemma

Abstract

This article analyzes the game-theoretic stability of three strategies, Tit-For-Tat (TFT), all-Defect (all-D), and all-Cooperate (all-C), that actors might use for repeated plays of the Prisoner's Dilemma (PD). The probability that there will be a next play is assumed to depend on the current behavior of one of the actors—it is w after cooperation and u after a defection—and two cases are examined. The first case is where an actor assumes that the continuation probability depends on its own behavior, and the second is where the continuation probability is assumed to depend on the other actor's behavior. It is shown that the potential for mutual cooperation is higher in the first case than in the second. A detailed examination of the first case reveals that when the ratio (1 - w)/(1 - u) is sufficiently extreme for certain classes of PD, the “cooperative” strategy TFT is stable and the “noncooperative” strategy all-D is unstable. For these classes of PD, it is thus possible both for cooperation to be maintained once it is established, and for cooperation to become established in a world of defectors. The sensitivity of these results to the precision in measurement of payoffs and probabilities is discussed.