Learning aspiration in repeated games

We study infinitely repeated symmetric 2×2 games played by bounded rational agents who follow a simple rule of thumb: each agent continues to play the same action if the current payoff exceeds the average of the past payoffs, and switches to the other action with a positive probability otherwise. By applying the stochastic approximation technique, we characterize the asymptotic outcomes for all 2×2 games. In the prisoners' dilemma game, for example, the players cooperate in the limit if and only if the gain from defecting against cooperation is “modest.”