Probabilistic Analysis of Mean-Field Games

Description

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 in the sense that the private state on which a player bases his own strategy depends upon the empirical distribution of the private states of the other players. We implement the Mean-Field Games strategy proposed and developed analytically by Lasry and Lions in a purely probabilistic framework, relying on tailor-made forms of the stochastic maximum principle. We assume that the state dynamics are affine in the states and the controls and, at the same time, require rather weak assumptions on the nature of the costs. Surprisingly, the dependence of all the coefficients upon the statistical distribution of the states remains of a rather general nature. Reliance on the stochastic maximum principle calls for the solution of systems of forward-backward stochastic differential equations of a McKean-Vlasov type for which no existence result is available, and for which we prove existence, uniqueness and regularity of the corresponding value function. Finally, we prove that the solution of the mean-field game as formulated by Lasry and Lions does indeed provide approximate Nash equilibriums for games with a large number of players, and we quantify the nature of the approximation.

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