Finite mean field games: fictitious play and convergence to a first order continuous mean field game

TL;DR

This paper investigates finite mean field games, analyzing the convergence of fictitious play algorithms and establishing links between finite and continuous first-order MFGs through grid refinement.

Contribution

It provides the first analysis of fictitious play convergence in finite MFGs and connects finite models to continuous MFGs via grid refinement and limit analysis.

Findings

Fictitious play converges in finite MFGs under certain conditions.

Finite MFG solutions have limit points that solve continuous first-order MFGs.

A method to approximate continuous MFGs using refined finite models.

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 Gomes Mohr and Souza in 2010, 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. In the second one, we consider the relation of 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 limits points and every such limit point solves the continuous first order MFG problem.