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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 therepresentative minor player is standard while the major player faces anoptimization over conditional McKean-Vlasov stochastic differential equations.The definition of this limiting game is justified by proving that its solutionprovides approximate Nash equilibriums for large finite player games. Thisproof depends upon the generalization of standard results on the propagation ofchaos to conditional dynamics. Because it is on independent interest, we provethis generalization in full detail. Using a conditional form of the Pontryaginstochastic maximum principle (proven in the appendix), we reduce the solutionof the mean field game to a forward-backward system of stochastic differentialequations of the conditional McKean-Vlasov type, which we solve in the LinearQuadratic setting. We use this class of models to show that Nash equilibriumsin our formulation can be different from those of the formulations contemplatedso far in the literature.

A Probabilistic Approach to Mean Field Games with Major and Minor Players