The Annals of Applied Probability

$N$-player games and mean-field games with absorption

Luciano Campi and Markus Fischer

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We introduce a simple class of mean-field games with absorbing boundary over a finite time horizon. In the corresponding $N$-player games, the evolution of players’ states is described by a system of weakly interacting Itô equations with absorption on first exit from a bounded open set. Once a player exits, her/his contribution is removed from the empirical measure of the system. Players thus interact through a renormalized empirical measure. In the definition of solution to the mean-field game, the renormalization appears in form of a conditional law. We justify our definition of solution in the usual way, that is, by showing that a solution of the mean-field game induces approximate Nash equilibria for the $N$-player games with approximation error tending to zero as $N$ tends to infinity. This convergence is established provided the diffusion coefficient is nondegenerate. The degenerate case is more delicate and gives rise to counter-examples.

Article information

Ann. Appl. Probab., Volume 28, Number 4 (2018), 2188-2242.

Received: December 2016
Revised: August 2017
First available in Project Euclid: 9 August 2018

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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

Mean-field game Nash equilibrium McKean–Vlasov limit absorbing boundary weak convergence martingale problem optimal control


Campi, Luciano; Fischer, Markus. $N$-player games and mean-field games with absorption. Ann. Appl. Probab. 28 (2018), no. 4, 2188--2242. doi:10.1214/17-AAP1354.

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