How complex are networks playing repeated games?

This paper examines implications of complexity cost in implementing repeated game strategies through networks with finitely many classifiers. A network consists of individual classifiers that summarize the history of repeated play according to a weighted sum of the empirical frequency of the outcomes of the stage game, and a decision unit that chooses an action in each period based on the summaries of the classifiers. Each player maximizes his long run average payoff, while minimizing the complexity cost of implementing his strategy through a network, measured by its number of classifiers. We examine locally stable equilibria where the selected networks are robust against small perturbations. In any locally stable equilibrium, no player uses a network with more than a single classifier. Moreover, the set of locally stable equilibrium payoff vectors lies on two line segments in the payoff space of the stage game.