  In this paper, we investigate the interaction of two populations with a large
number of indistinguishable agents. The problem consists in two levels: the
interaction between agents of a same population, and the interaction between
the two populations. In the spirit of mean field type control (MFC) problems
and mean field games (MFG), each population is approximated by a continuum of
infinitesimal agents. We define four different problems in a general context
and interpret them in the framework of MFC or MFG. By calculus of variations,
we derive formally in each case the adjoint equations for the necessary
conditions of optimality. Importantly, we find that in the case of a
competition between two coalitions, one needs to rely on a system of Master
equations in order to describe the equilibrium. Examples are provided, in
particular linear-quadratic models for which we obtain systems of ODEs that can
be related to Riccati equations.
