Convergence to the Mean Field Game Limit: A Case Study

TL;DR

This paper investigates the convergence of Nash equilibria in optimal stopping games to mean field game limits, highlighting conditions for convergence and presenting cases where some equilibria do not emerge as limits, thus challenging their interpretation.

Contribution

It demonstrates that mean field equilibria satisfying a transversality condition are limits of finite-player equilibria, but also identifies equilibria that are not limits, questioning their role as large $n$ limits.

Findings

Transversality condition ensures convergence to mean field equilibria.

Existence of mean field equilibria that are not limits of finite-player equilibria.

Challenges in interpreting certain mean field equilibria as large $n$ limits.

Abstract

We study the convergence of Nash equilibria in a game of optimal stopping. If the associated mean field game has a unique equilibrium, any sequence of $n$-player equilibria converges to it as $n\to\infty$. However, both the finite and infinite player versions of the game often admit multiple equilibria. We show that mean field equilibria satisfying a transversality condition are limit points of $n$-player equilibria, but we also exhibit a remarkable class of mean field equilibria that are not limits, thus questioning their interpretation as "large $n$" equilibria.