A Feedback Stabilization Approach to Fictitious Play

We consider repeated matrix games in which player strategies evolve in reaction to opponent actions. Players observe each other's actions, but do not have access to other player utilities. Strategy evolution may be of the best response sort, as in fictitious play, or a gradient update. Such mechanisms are known to not necessarily converge. We show that the use of derivative action in processing opponent actions can lead to behavior converging to Nash equilibria. We analyze the use of approximate differentiators and reveal a potentially detrimental biasing effect. We go on to provide alternative mechanisms to diminish or eliminate this effect. We discuss two player games throughout and outline extensions to multiplayer games. We also provide convergent simulations throughout to standard counterexamples in the literature.