  The purpose of this paper is to provide a complete probabilistic analysis of
a large class of stochastic differential games for which the interaction
between the players is of mean-field type. We implement the Mean-Field Games
strategy developed analytically by Lasry and Lions in a purely probabilistic
framework, relying on tailor-made forms of the stochastic maximum principle.
While we assume that the state dynamics are affine in the states and the
controls, our assumptions on the nature of the costs are rather weak, and
surprisingly, the dependence of all the coefficients upon the statistical
distribution of the states remains of a rather general nature. Our
probabilistic approach calls for the solution of systems of forward-backward
stochastic differential equations of a McKean-Vlasov type for which no
existence result is known, and for which we prove existence and regularity of
the corresponding value function. Finally, we prove that solutions of the
mean-field game as formulated by Lasry and Lions do indeed provide approximate
Nash equilibriums for games with a large number of players, and we quantify the
nature of the approximation.
