  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.
