Mathieu Faure

Stochastic Approximations of Set-Valued Dynamical Systems: Convergence with Positive Probability to an Attractor

A successful method to describe the asymptotic behavior of a discrete time stochastic process governed by some recursive formula is to relate it to the limit sets of a well-chosen mean differential equation. Under an attainability condition, Benaim proved that convergence to a given attractor of the flow induced by this dynamical system occurs with positive probability for a class of Robbins Monro algorithms. Benaim, Hofbauer, and Sorin generalised this approach for stochastic approximation algorithms whose average behavior is related to a differential inclusion instead. We pursue the analogy by extending to this setting the result of convergence with positive probability to an attractor.

Stochastic Approximation, Cooperative Dynamics and Supermodular Games ⁄

This paper considers a stochastic approximation algorithm, with decreasing step size and martingale difference noise. Under very mild assumptions, we prove the non convergence of this process toward a certain class of repulsive sets for the associated ordinary differential equation (ODE). We then use this result to derive the convergence of the process when the ODE is cooperative in the sense of [Hirsch, 1985]. In particular, this allows us to extend significantly the main result of [Hofbauer and Sandholm, 2002] on the convergence of stochastic fictitious play in supermodular games.