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 the jth player is the sum of his control αj plus a minimum jumping rate η. 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 η < 1 2 . We also prove that although multiple solutions exist, only the one coming from the entropy solution is charged (when η = 0), and therefore resolve a conjecture of Hajek and Livesay.