Mixed Equilibria and Dynamical Systems Arising from Fictitious Play in Perturbed Games

Abstract

Fictitious play in infinitely repeated, randomly perturbed games is investigated. Dynamical systems theory is used to study the Markov process {xk}, whose state vector xk lists the empirical frequencies of player's actions in the first k games. For 2×2 games with countably many Nash distribution equilibria, we prove that sample paths converge almost surely. But for Jordan's 3×2 matching game, there are robust parameter values giving probability 0 of convergence. Applications are made to coordination and anticoordination games and to general theory. Proofs rely on results in stochastic approximation and dynamical systems. Journal of Economic Literature Classification Numbers: C72, C73.