Learning Dynamics in Games with Stochastic Perturbations

Consider a game that is played repeatedly by two populations of agents. In fictitious play, agents learn by choosing best replies to the frequency distribution of actions taken by the other side. We consider a more general class of learning processes in which agents' choices are perturbed by incomplete information about what the other side has done, variability in their payoffs, and unexplained trembles. These perturbed best reply dynamics define a non-stationary Markov process on an infinite state space. We show that for 2x2 games it converges with probability one to a neighborhood of the stable Nash equilibria, whether pure or mixed. This generalizes a result of Fudenberg and Kreps, who demonstrate convergence when the game has a unique mixed equilibrium