Time Dependent Bounded Recall Strategies Are Enough to Play the Discounted Repeated Prisoners ’ Dilemma∗

We show that for any discount factor, there is a natural number M such that all sub-game perfect outcomes of the discounted repeated prisoners ’ dilemma (pure strategies) can be obtained by subgame perfect strategies with the following property: current play depends only on the number of the time-index and on the history of the last M periods. This strong result, however, does not suffice to establish the claim of the title of this study, because we also establish that for any natural number K, there exist a discount factor and an equilibrium outcome which can only be obtained by equilibrium strategies with recall strictly higher than K. On the other hand, for cases with high discount factors, we prove that payoffs that are bounded away from the common minmax return can be approximated in equilibrium with a bounded recall strategy whose order of recall is independent of the discount factor and the fineness of the approximation. Therefore, noticing that payoffs not bounded away from the minmax return can each be approximated by the minmax payoff (sustainable in equilibrium with 0-recall) enables us to conclude that bounded recall strategies are enough to play the prisoners ’ dilemma