Win-stay, lose-shift strategies for repeated games-memory length, aspiration levels and noise

Win-stay, lose-shift, the principle to retain a successful action is a simple and general learning rule that can be applied to all types of repeated decision problems. In this paper I consider win-stay, lose-shift strategies with diverse memory sizes and strategies that adapt their aspiration levels, i.e. the payoff level considered as "success". I study their evolution for the Prisoner's Dilemma, as well as in a rapidly changing environment, where a randomly selected game is assigned to the players. For win-stay, lose-shift strategies with memory one the average payoffs are computed and their evolutionary stability is discussed. Using computer simulations I show that the win-stay, lose-shift strategies with longer memory are very successful both for the Prisoner's Dilemma, where cooperation dominates even for high noise levels, and the randomly assigned games, where the players achieve nearly the expected Pareto optimal payoffs. I discuss the impact of noise and show that the memory length of the players increases with the noise level. These results indicate that the win-stay, lose-shift principle is a very successful strategy in repeated games with noise.