On Solutions of Mean Field Games with Ergodic Cost

TL;DR

This paper investigates the existence, long-term behavior, and convergence of solutions in mean field games with ergodic cost, including their relation to finite horizon problems and N-player games.

Contribution

It establishes existence results for mean field game solutions, analyzes their asymptotic behavior, and demonstrates convergence from N-player Nash equilibria to mean field solutions.

Findings

Existence of mean field game solutions under various conditions.

Convergence of finite horizon solutions to ergodic solutions.

Nash equilibria in N-player games converge to mean field game solutions.

Abstract

A general class of mean field games are considered where the governing dynamics are controlled diffusions in $\mathbb{R}^d$. The optimization criterion is the long time average of a running cost function. Under various sets of hypotheses, we establish the existence of mean field game solutions. We also study the long time behavior of mean field game solutions associated with the finite horizon problem, and under the assumption of geometric ergodicity for the dynamics, we show that these converge to the ergodic mean field game solution as the horizon tends to infinity. Lastly, we study the associated $N$-player games, show existence of Nash equilibria, and establish the convergence of the solutions associated to Nash equilibria of the game to a mean field game solution as $N\to\infty$.