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.

N-player games and mean-field games with absorption / L. Campi, M. Fischer. - In: THE ANNALS OF APPLIED PROBABILITY. - ISSN 1050-5164. - 28:4(2018), pp. 2188-2242.

N-player games and mean-field games with absorption

L. Campi
;
2018

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.
Absorbing boundary; Martingale problem; McKean–Vlasov limit; Mean-field game; Nash equilibrium; Optimal control; Weak convergence
Settore MAT/06 - Probabilita' e Statistica Matematica
2018
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2434/750958
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