Restoring Uniqueness to Mean-Field Games by Randomizing the Equilibria

TL;DR

This paper introduces a method to restore the uniqueness of solutions in mean-field games by adding an infinite-dimensional Ornstein-Uhlenbeck noise, linking the approach to stochastic differential games with many players.

Contribution

It proposes a novel approach of randomizing equilibria with Ornstein-Uhlenbeck noise to ensure uniqueness in mean-field games derived from large-player differential games.

Findings

Existence and uniqueness of solutions to the noisy mean-field system are established.

The noisy system is shown to be the limit of a stochastic differential game with many players.

The approach connects mean-field game solutions to stochastic differential game limits.

Abstract

We here address the question of restoration of uniqueness in mean-field games deriving from deterministic differential games with a large number of players. The general strategy for restoring uniqueness is inspired from earlier similar results on ordinary and stochastic differential equations. It consists in randomizing the equilibria through an external noise. As a main feature, we choose the external noise as an infinite dimensional Ornstein-Uhlenbeck process. We first investigate existence and uniqueness of a solution to the noisy system made of the mean-field game forced by the Ornstein-Uhlenbeck process. We also show how such a noisy system can be interpreted as the limit version of a stochastic differential game with a large number of players.