Nonlinear Mean Field Games with Multiple Major Agents and Multiple Populations of Minor Agents

The paper studies a nonlinear mean field game (MFG) model based on McKean-Vlasov (MV) approximation, which involves multiple major agents and multiple populations of minor agents. Since the interactions occur not only between major-minor agents but also among major-major agents and minor-minor populations, complicating equilibrium existence analysis, we first cast the MFG as two sets of adjoint stochastic McKean-Vlasov equations coupled via the mean field behavior. Subsequently, a new norm on the product probability measure space is constructed to prove that one set of equations exists solutions, while the existence of solutions to the other set is proved based on Pontryagin stochastic maximum principle. Then, within the established product normed space, Banach's fixed point theorem is utilized to prove the existence of equilibrium for this MFG, which is manifested as solutions to two sets of coupled equations. Finally, an ϵ-Nash equilibrium is proved for the finite agent situation, in which ϵ →0 while all population sizes go to ∞, and a numerical experiment under certain settings is carried out.