Learning Processes, Mixed Equilibria and Dynamical Systems

Fudenberg and Kreps consider adaptive learning processes, in the spirit of fictitious play, for infinitely repeated games of incomplete information having randomly perturbed payoffs. They proved the convergence of the adaptive process for 2 X 2 games with a unique completely mixed Nash equilibrium. Kaniovski and Young proved the convergence of the process for generic 2 X 2 games subjected to small perturbations. We extend their result to games with several equilibria--- possibly infinitely many, and not necessarily completely mixed. For a broad class of such games we prove convergence of the adaptive process; stable and unstable equilibria are characterized.For certain 3-player, 2-strategy games we show that almost surely the adaptive process does not converge. We analyze coordination and anticoordination games.The mathematics is based on a general result in stochastic approximation theory. Long term outcomes are shown to cluster at an attractor-free set for the dynamics of a vector field F canonically associated to an infinitely repeated many player game with randomized payoffs, subject to the long-run adaptive strategy of fictitious play.The phase portrait of F can in some cases be explicitly described in sufficient detail to yield information on convergence of the learning process, and on stability and location of equilibria