The Annals of Applied Probability

On the connection between symmetric $N$-player games and mean field games

Markus Fischer

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Mean field games are limit models for symmetric $N$-player games with interaction of mean field type as $N\to \infty $. The limit relation is often understood in the sense that a solution of a mean field game allows to construct approximate Nash equilibria for the corresponding $N$-player games. The opposite direction is of interest, too: When do sequences of Nash equilibria converge to solutions of an associated mean field game? In this direction, rigorous results are mostly available for stationary problems with ergodic costs. Here, we identify limit points of sequences of certain approximate Nash equilibria as solutions to mean field games for problems with Itô-type dynamics and costs over a finite time horizon. Limits are studied through weak convergence of associated normalized occupation measures and identified using a probabilistic notion of solution for mean field games.

Article information

Ann. Appl. Probab., Volume 27, Number 2 (2017), 757-810.

Received: May 2014
Revised: April 2016
First available in Project Euclid: 26 May 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 91A06: n-person games, n > 2
Secondary: 60B10: Convergence of probability measures 93E20: Optimal stochastic control

Nash equilibrium mean field game McKean–Vlasov limit weak convergence martingale problem optimal control


Fischer, Markus. On the connection between symmetric $N$-player games and mean field games. Ann. Appl. Probab. 27 (2017), no. 2, 757--810. doi:10.1214/16-AAP1215.

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