  In this work, we systematically investigate mean field games and mean field
type control problems with multiple populations using a coupled system of
forward-backward stochastic differential equations of McKean-Vlasov type
stemming from Pontryagin's stochastic maximum principle. Although the same cost
functions as well as the coefficient functions of the state dynamics are shared
among the agents within each population, they can be different population by
population. We study the mean field limit for the three different situations;
(i) every agent is non-cooperative; (ii) the agents within each population are
cooperative; and (iii) the agents in some populations are cooperative but those
in the other populations are not. We provide several sets of sufficient
conditions for the existence of a mean field equilibrium for each of these
cases. Furthermore, under appropriate conditions, we show that the mean field
solution to each of these problems actually provides an approximate Nash
equilibrium for the corresponding game with a large but finite number of
agents.
