An Alternative Approach to Mean Field Game with Major and Minor Players, and Applications to Herders Impacts

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.