Discrete Approximation Of Stationary Mean Field Games

TL;DR

This paper introduces a discrete approximation scheme for stationary mean-field games, proving existence, uniqueness, and convergence of solutions to the classical stationary MFG solutions.

Contribution

It presents a novel discrete approximation method for stationary MFGs, extending previous Hamilton-Jacobi schemes and establishing convergence results.

Findings

Existence and uniqueness of discrete stationary MFG solutions

Convergence of discrete solutions to classical solutions

Uniform convergence in the nonlocal case and weak convergence in the local case

Abstract

In this paper, we focus on stationary (ergodic) mean-field games (MFGs). These games arise in the study of the long-time behavior of finite-horizon MFGs. Motivated by a prior scheme for Hamilton-Jacobi equations introduced in Aubry-Mather's theory, we introduce a discrete approximation to stationary MFGs. Relying on Kakutani's fixed-point theorem, we prove the existence and uniqueness (up to additive constant) of solutions to the discrete problem. Moreover, we show that the solutions to the discrete problem converge, uniformly in the nonlocal case and weakly in the local case, to the classical solutions of the stationary problem.