Abstract
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.
Citation
Luciano Campi. Markus Fischer. "$N$-player games and mean-field games with absorption." Ann. Appl. Probab. 28 (4) 2188 - 2242, August 2018. https://doi.org/10.1214/17-AAP1354
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