Minimax Approach to First-Order Mean Field Games

The paper is devoted to the first-order mean field game system in the case when the distribution of players can contain atoms. The proposed definition of a generalized solution is based on the minimax approach to the Hamilton-Jacobi equation. We prove the existence of the generalized (minimax) solution of the mean filed game system using the Nash equilibrium in the auxiliary differential game with infinitely many identical players. We show that the minimax solution of the original system provide the $\varepsilon$-Nash equilibrium in the differential game with finite number of players.