Open Access
June 2016 A probabilistic approach to mean field games with major and minor players
René Carmona, Xiuneng Zhu
Ann. Appl. Probab. 26(3): 1535-1580 (June 2016). DOI: 10.1214/15-AAP1125


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.


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René Carmona. Xiuneng Zhu. "A probabilistic approach to mean field games with major and minor players." Ann. Appl. Probab. 26 (3) 1535 - 1580, June 2016.


Received: 1 September 2014; Revised: 1 May 2015; Published: June 2016
First available in Project Euclid: 14 June 2016

zbMATH: 1342.93121
MathSciNet: MR3513598
Digital Object Identifier: 10.1214/15-AAP1125

Primary: 93E20
Secondary: 60H10 , 60K35

Keywords: McKean–Vlasov diffusion , Mean field games , mean-field forward–backward stochastic differential equation , Mean-field interaction , Stochastic control , stochastic Pontryagin principle

Rights: Copyright © 2016 Institute of Mathematical Statistics

Vol.26 • No. 3 • June 2016
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