# On non-unique solutions in mean field games

## TL;DR

This paper investigates the non-uniqueness of solutions in mean field games, focusing on a simple symmetric two-state model, and explores the relationship between finite-player Nash equilibria and mean field game solutions.

## Contribution

It characterizes all equilibria in a symmetric two-state mean field game and links finite-player Nash equilibria to mean field game solutions through fluid limits.

## Key findings

- All equilibria in the symmetric two-state game are identified.
- Finite-player Nash equilibria converge to mean field game equilibria as N increases.
- Stable fixed points of the mean field best response are likely the limits of finite-player equilibria.

## Abstract

The theory of mean field games is a tool to understand noncooperative dynamic stochastic games with a large number of players. Much of the theory has evolved under conditions ensuring uniqueness of the mean field game Nash equilibrium. However, in some situations, typically involving symmetry breaking, non-uniqueness of solutions is an essential feature. To investigate the nature of non-unique solutions, this paper focuses on the technically simple setting where players have one of two states, with continuous time dynamics, and the game is symmetric in the players, and players are restricted to using Markov strategies. All the mean field game Nash equilibria are identified for a symmetric follow the crowd game. Such equilibria correspond to symmetric $\epsilon$-Nash Markov equilibria for $N$ players with $\epsilon$ converging to zero as $N$ goes to infinity. In contrast to the mean field game, there is a unique Nash equilibrium for finite $N.$ It is shown that fluid limits arising from the Nash equilibria for finite $N$ as $N$ goes to infinity are mean field game Nash equilibria, and evidence is given supporting the conjecture that such limits, among all mean field game Nash equilibria, are the ones that are stable fixed points of the mean field best response mapping.