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.
"Convergence to the mean field game limit: A case study." Ann. Appl. Probab. 30 (1) 259 - 286, February 2020. https://doi.org/10.1214/19-AAP1501