Finite Mean Field Games: Fictitious play and convergence to a first order continuous mean field game

Abstract

In this article we consider finite Mean Field Games (MFGs), i.e. with finite time and finite states. We adopt the framework introduced in [1] and study two seemly unexplored subjects. In the first one, we analyze the convergence of the fictitious play learning procedure, inspired by the results in continuous MFGs (see [2] and [3]). In the second one, we consider the relation between some finite MFGs and continuous first order MFGs. Namely, given a continuous first order MFG problem and a sequence of refined space/time grids, we construct a sequence finite MFGs whose solutions admit limit points and every such limit point solves the continuous first order MFG problem.