Growth of strategy sets, entropy, and nonstationary bounded recall

The paper initiates the study of long term interactions where players' bounded rationality varies over time. Time dependent bounded rationality, for player i, is reflected in part in the number ψi(t) of distinct strategies available to him in the first t-stages.We examine how the growth rate of ψi(t) affects equilibrium outcomes of repeated games. An upper bound on the individually rational payoff is derived for a class of two-player repeated games, and the derived bound is shown to be tight.As a special case we study the repeated games with nonstationary bounded recall and show that, a player can guarantee the minimax payoff of the stage game, even against a player with full recall, by remembering a vanishing fraction of the past. A version of the folk theorem is provided for this class of games.