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 subgame perfect equilibrium outcomes of the discounted repeated prisoners' dilemma can be obtained by subgame perfect equilibrium 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. Therefore, players who are restricted to using pure strategies, have to remember, at the most, $M$ periods in order to play any equilibrium outcome of the discounted repeated prisoners' dilemma. This result leads us to introduce the notion of time dependent complexity, and to conclude that in the repeated prisoners' dilemma, restricting attention to finite time dependent complex strategies is enough.