K-Memory Strategies in Repeated Games

In this paper, we study k-memory strategies in two-person repeated games. An agent adopting such a strategy makes his decision only based on the action profiles of the previous k rounds. We show that in finitely repeated games, the best response of a k-memory strategy may not even be a constant-memory strategy. However, in infinitely repeated games, one of the best responses against any given k-memory strategy must be k-memory. Our results are enabled by a graph-structural characterization of the best responses of k-memory strategies. We put forward polynomial algorithms to compute best responses. Also, using this characterization, we show that the best commitments (Stackelberg leader strategy) with k memory (either pure or mixed) in infinitely repeated games have nice structures. CCS Concepts •Theory of computation→ Solution concepts in game theory; Exact and approximate computation of equilibria; Representations of games and their complexity; •Computing methodologies → Multi-agent systems;