Convergence for linear quadratic potential mean field games

TL;DR

This paper investigates the convergence of empirical means of Nash equilibria in linear-quadratic potential mean field games, characterizing the limits as solutions to a deterministic control problem and exploring cases with multiple equilibria and common noise.

Contribution

It introduces a novel approach connecting finite player games to a control problem, providing convergence results even with multiple equilibria and extending to cases with common noise.

Findings

Limit points are characterized as optimal trajectories of a deterministic control problem.

Convergence to the unique minimizer occurs for almost every initial mean.

Symmetry in data can lead to a random limit uniformly distributed among multiple minimizers.

Abstract

This paper studies the limits of empirical means of open-loop Nash equilibria of linear-quadratic stochastic differential games as the number of players goes to infinity, when the corresponding mean field game is of potential type and may have multiple equilibria. Via weak compactness arguments, the limit points are characterized as optimal trajectories of the related deterministic control problem, thus ruling out some of the mean field equilibria. Our result is obtained by first connecting the finite player game to a suitable control problem, whose optimal trajectories are the empirical means of Nash equilibria of the game, and in which the number of players $N$ becomes a parameter. True convergence to the unique minimizer of the limit control problem then holds for almost every initial mean. In cases of multiple optimizers, we focus on examples to show that some symmetry of the data ensures that the sequence admits a random limit which is distributes uniformly among the minimizers of the potential. Multidimensional examples of the convergence result appear here for the first time, which show the flexibility of our method. We also establish a similar convergence results for the corresponding linear-quadratic potential mean field games with common noise, as the noise vanishes.