Learning by Fictitious Play in Large Populations

We consider learning by fictitious play in a large population of agents engaged in single-play, two-person rounds of a symmetric game, and derive a mean-filed type model for the corresponding stochastic process. Using this model, we describe qualitative properties of the learning process and discuss its asymptotic behavior. Of the special interest is the comparative characteristics of the fictitious play learning with and without a memory factor. As a part of the analysis, we show that the model leads to the continuous, best-response dynamics equation of Gilboa and Matsui (1991), when all agents have similar empirical probabilities.