On non-uniqueness in mean field games

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.