2 The basic setting of MFG methodology and the strategy for its implementation

In this paper, we investigate the mean field games with K classes of agents who are weakly coupled via the empirical measure. The underlying dynamics of the representative agents is assumed to be a controlled nonlinear Markov process associated with rather general integro-differential generators of Lévy-Khintchine type (with variable coefficients), with the major stress on applications to stable and stablelike processes, as well as their various modifications like tempered stable-like processes or their mixtures with diffusions. We show that nonlinear measure-valued kinetic equations describing the dynamic law of large numbers limit for system with large number N of agents are solvable and that their solutions represent 1/N -Nash equilibria for approximating systems of N agents. Mathematics Subject Classification (2000): 60H30, 60J25, 91A13, 91A15.

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