Discrete Mean Field Games: Existence of Equilibria and Convergence

TL;DR

This paper studies discrete mean field games across various settings, proving the existence of equilibria under general conditions and analyzing the convergence of finite-player equilibria to mean field equilibria, revealing cases of non-convergence.

Contribution

It establishes the existence of equilibria in discrete mean field games under broader conditions than previous work and characterizes convergence and non-convergence phenomena of finite-player equilibria.

Findings

Existence of mean field equilibria under general continuity conditions.

Identification of strategy classes where finite-player equilibria converge to mean field equilibria.

Demonstration of non-convergence cases and implications for the Folk theorem in discrete time.

Abstract

We consider mean field games with discrete state spaces (called discrete mean field games in the following) and we analyze these games in continuous and discrete time, over finite as well as infinite time horizons. We prove the existence of a mean field equilibrium assuming continuity of the cost and of the drift. These conditions are more general than the existing papers studying finite state space mean field games. Besides, we also study the convergence of the equilibria of N -player games to mean field equilibria in our four settings. On the one hand, we define a class of strategies in which any sequence of equilibria of the finite games converges weakly to a mean field equilibrium when the number of players goes to infinity. On the other hand, we exhibit equilibria outside this class that do not converge to mean field equilibria and for which the value of the game does not converge. In discrete time this non-convergence phenomenon implies that the Folk theorem does not scale to the mean field limit.