# On non-uniqueness in mean field games

## TL;DR

This paper investigates non-uniqueness in mean field games with a binary state space, showing multiple solutions can exist under certain conditions and identifying the entropy solution as the relevant one.

## Contribution

It demonstrates the existence of multiple solutions in mean field games with anti-monotone costs and clarifies which solution is physically relevant, resolving a previous conjecture.

## Key findings

- Multiple solutions exist when the jump rate parameter is below 1/2.
- The entropy solution is the unique relevant solution when the jump rate is zero.
- The paper resolves a conjecture about solution selection in mean field games.

## Abstract

We analyze an $N+1$-player game and the corresponding mean field game with state space $\{0,1\}$. The transition rate of $j$-th player is the sum of his control $\alpha^j$ plus a minimum jumping rate $\eta$. Instead of working under monotonicity conditions, here we consider an anti-monotone running cost. We show that the mean field game equation may have multiple solutions if $\eta < \frac{1}{2}$. We also prove that that although multiple solutions exist, only the one coming from the entropy solution is charged (when $\eta=0$), and therefore resolve a conjecture of ArXiv: 1903.05788.