## A probabilistic approach to mean field games with major and minor players.(English)Zbl 1342.93121

Summary: We propose a new approach to mean field games with major and minor players. Our formulation involves a two player game where the optimization of the representative minor player is standard while the major player faces an optimization over conditional McKean-Vlasov stochastic differential equations. The definition of this limiting game is justified by proving that its solution provides approximate Nash equilibria for large finite player games. This proof depends upon the generalization of standard results on the propagation of chaos to conditional dynamics. Because it is of independent interest, we prove this generalization in full detail. Using a conditional form of the Pontryagin stochastic maximum principle (proven in the appendix), we reduce the solution of the mean field game to a forward-backward system of stochastic differential equations of the conditional McKean-Vlasov type, which we solve in the linear quadratic setting. We use this class of models to show that Nash equilibriums in our formulation can be different from those originally found in the literature.

### MSC:

 93E20 Optimal stochastic control 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis) 60K35 Interacting random processes; statistical mechanics type models; percolation theory 49K45 Optimality conditions for problems involving randomness
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