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In this paper, we investigate the interaction of two populations with a largenumber of indistinguishable agents. The problem consists in two levels: theinteraction between agents of a same population, and the interaction betweenthe two populations. In the spirit of mean field type control (MFC) problemsand mean field games (MFG), each population is approximated by a continuum ofinfinitesimal agents. We define four different problems in a general contextand interpret them in the framework of MFC or MFG. By calculus of variations,we derive formally in each case the adjoint equations for the necessaryconditions of optimality. Importantly, we find that in the case of acompetition between two coalitions, one needs to rely on a system of Masterequations in order to describe the equilibrium. Examples are provided, inparticular linear-quadratic models for which we obtain systems of ODEs that canbe related to Riccati equations.

Mean Field Control and Mean Field Game Models with Several Populations