  The goal of the paper is to introduce a formulation of the mean field game
with major and minor players as a fixed point on a space of controls. This
approach emphasizes naturally the role played by McKean-Vlasov dynamics in some
of the players optimization problems. We apply this approach to linear
quadratic models for which we recover the existing solutions for open loop
equilibria, and we show that we can also provide solutions for closed loop
versions of the game. Finally, we implement numerically our theoretical results
on a simple model of flocking.
