  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 equilibriums 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 on 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 of the formulations contemplated
so far in the literature.
