A Probabilistic Approach to Mean Field Games with Major and Minor Players

By René A. Carmona, Xiuneng Zhu. 2014

119 citations recorded by the source corpus.

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 on 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 of the formulations contemplated so far in the literature.

Subfields: Finance, Modeling and Simulation, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/ea4f30808e225783336f53024001971d242d917c

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Citations in the corpus

An Alternative Approach to Mean Field Game with Major and Minor Players, and Applications to Herders Impacts (2016, methodology): “Identifying Yt with [P † t , P 0† t ] † we recognize the FBSDE used in [4].”
Splitting methods and short time existence for the master equations in mean field games (2020, background): “Concerning the existence of a solution, [15] shows the existence of an equilibrium in short time for the case of a finite state space, [26] proves the existence of a solution to the master equation still in the finite state space framework and notes that the Hilbertian techniques described in [29] c”
From the master equation to mean field game limit theory: a central limit theorem (2018, background): “, [22, 41, 44] for the case without common noise, or [11, 15, 17, 28] and [9, Chapter 2, Section 2.”
A mean-field stochastic control problem with partial observations (2017, unknown): “On the other hand, when G is a given filtration, this is the so-called conditional mean-field SDE (CMFSDE for short) studied in [11].”
McKean–Vlasov optimal control: The dynamic programming principle (2019, methodology): “The stochastic maximum principle approach was also used in a context involving conditional distributions in Buckdahn, Li, and Ma [20] and Carmona and Zhu [26].”