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 limits of the three different situations; (i) every agent is non-cooperative; (ii) the agents within each population are cooperative; and (iii) only for some populations, the agents are cooperative within each population. We provide several sets of sufficient conditions for the existence of mean field equilibrium for each of these cases.
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