Stochastic Learning of Equilibria in Games: The Ordinary Differential Equation Method

Our purpose is to discuss stochastic algorithms to learn equilibria in games, and their time of convergence. To do so, we consider a general class of stochastic algorithms that converge weakly (in the sense of weak convergence for stochastic processes) towards solutions of particular ordinary differential equations, corresponding to their mean-field approximations. Tuning parameters in these algorithms provides several dynamics having limit points related to Nash equilibria, and hence provide means to compute equilibria in a distributed fashion in games. We relate the time of convergence of stochastic dynamics to the time of convergence of their corresponding ordinary differential equation. This gives lower and upper bounds on the time needed to learn equilibria in games through such stochastic dynamics.