On non-uniqueness in mean field games

Abstract

We analyze an N + 1 N+1 -player game and the corresponding mean field game with state space { 0 , 1 } \{0,1\} . The transition rate of the j j th player is the sum of his control α j \alpha ^j plus a minimum jumping rate η \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 η > 1 2 \eta > \frac {1}{2} . We also prove that although multiple solutions exist, only the one coming from the entropy solution is charged (when η = 0 \eta =0 ), and therefore resolve a conjecture of Hajek and Livesay.