# Dynamics of Strategy Distribution in Iterated Games

Stéphane Airiau, Sabyasachi Saha, and Sandip Sen

Evolutionary tournaments have been used as a tool for comparing game-playing strategies. For instance, in the late 1970’s, Axelrod organized tournaments to compare strategies for playing the iterated prisoner’s dilemma (PD) game. While these tournaments and later research have provided us with a better understanding of successful strategies for iterated PD, our understanding is less clear about strategies for playing iterated versions of arbitrary single-stage games. While solution concepts like Nash equilibria has been proposed for general-sum games, learning strategies like fictitious play may be preferred for playing against sub-rational players. In this paper, we discuss the relative performance of both learning and non-learning strategies in different population distributions including those that are likely in real-life. The testbed used to evaluate the strategies includes all possible structurally distinct 2$\times$2 conflicted games with ordinal payoffs. Plugging head-to-head performance data into an analytical finite-population evolution model allows us to evaluate the evolutionary dynamics of different initial strategy distributions. Two key observations are that (a) the popular Nash strategy is ineffective in most tournament settings, (b) simple strategies like best response benefit from the presence of learning strategies and we often observe convergence to a mixture of strategies rather than to a single dominant strategy. We explain such mixed convergence using head-to-head performance results.