Annals of Applied Probability

A probabilistic approach to mean field games with major and minor players

René Carmona and Xiuneng Zhu

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We propose a new approach to mean field games with major and minor players. Our formulation involves a two player game where the optimization of the representative minor player is standard while the major player faces an optimization over conditional McKean–Vlasov stochastic differential equations. The definition of this limiting game is justified by proving that its solution provides approximate Nash equilibriums for large finite player games. This proof depends upon the generalization of standard results on the propagation of chaos to conditional dynamics. Because it is of independent interest, we prove this generalization in full detail. Using a conditional form of the Pontryagin stochastic maximum principle (proven in the Appendix), we reduce the solution of the mean field game to a forward–backward system of stochastic differential equations of the conditional McKean–Vlasov type, which we solve in the linear quadratic setting. We use this class of models to show that Nash equilibriums in our formulation can be different from those originally found in the literature.

Article information

Ann. Appl. Probab., Volume 26, Number 3 (2016), 1535-1580.

Received: September 2014
Revised: May 2015
First available in Project Euclid: 14 June 2016

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Zentralblatt MATH identifier

Primary: 93E20: Optimal stochastic control
Secondary: 60H10: Stochastic ordinary differential equations [See also 34F05] 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Mean Field Games stochastic control McKean–Vlasov diffusion stochastic Pontryagin principle mean-field interaction mean-field forward–backward stochastic differential equation


Carmona, René; Zhu, Xiuneng. A probabilistic approach to mean field games with major and minor players. Ann. Appl. Probab. 26 (2016), no. 3, 1535--1580. doi:10.1214/15-AAP1125.

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