Learning in games with bounded memory

The paper studies infinite repetition of finite strategic form games. Players use a backward looking learning behavior and face bounds on their cognitive capacities. We show that for any given belief-probability over the set of possible outcomes where players have no experience, games can be payoff classified and there always exists a stationary state in the space of action profiles. In particular, if the belief-probability assumes all possible outcomes without experience to be equally likely, in one class of Prisoners' Dilemmas where the uniformly weighted average defecting payoff is higher than the cooperative payoff and the uniformly weighted average cooperative payoff is lower than the defecting payoff, play converges in the long run to the static Nash equilibrium while in the other class of Prisoners' Dilemmas where the reverse holds, play converges to cooperation. Results are applied to a large class of 2 × 2 games.