On the connection between symmetric $N$-player games and mean field games

TL;DR

This paper investigates the convergence of Nash equilibria in symmetric N-player games to solutions of mean field games, focusing on finite time horizon problems with Itô dynamics and using probabilistic solution concepts.

Contribution

It establishes conditions under which sequences of approximate Nash equilibria converge to mean field game solutions for finite horizon problems with Itô dynamics.

Findings

Limit points of Nash equilibria are solutions to mean field games.

Weak convergence of occupation measures characterizes the limit behavior.

Probabilistic solutions are used to identify the limits.

Abstract

Mean field games are limit models for symmetric $N$-player games with interaction of mean field type as $N\to\infty$. The limit relation is often understood in the sense that a solution of a mean field game allows to construct approximate Nash equilibria for the corresponding $N$-player games. The opposite direction is of interest, too: When do sequences of Nash equilibria converge to solutions of an associated mean field game? In this direction, rigorous results are mostly available for stationary problems with ergodic costs. Here, we identify limit points of sequences of certain approximate Nash equilibria as solutions to mean field games for problems with It{\^o}-type dynamics and costs over a finite time horizon. Limits are studied through weak convergence of associated normalized occupation measures and identified using a probabilistic notion of solution for mean field games.