## Set Values for Mean Field Games without Monotonicity Condition

#### Jianfeng ZhangUniversity of Southern California (USC)Mathematics

It is well known that a mean field game under the monotonicity condition has a unique mean field equilibrium and the value of the game is the solution to the master equation. For a general game without monotonicity condition, however, there might be multiple equilibriums. In this talk we propose to study the set of values over all equilibriums (in some approximating sense), which we call the set value of the mean field game. We emphasize that, instead of studying the set of all equilibriums, our focus is the set of the values. We establish two main properties of the set value: (i) the set value of the corresponding $N$-player game converges to the set value of the mean field game; (ii) the dynamic set value satisfies the dynamic programming principle. The former property holds for both open-loop and closed-loop controls, but the latter one holds true only for closed-loop controls. The talk is based on an ongoing joint work with Melih Iseri.

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