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Dynamic Games and Applications

30-03-2023

\(\varepsilon \)-Nash Equilibria of a Multi-player Nonzero-Sum Dynkin Game in Discrete Time

Authors: Said Hamadène, Mohammed Hassani, Marie-Amélie Morlais

Published in: Dynamic Games and Applications

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Abstract

We study the infinite horizon discrete time N-player nonzero-sum Dynkin game (\(N\ge 2\)) with stopping times as strategies (or pure strategies). We prove existence of an \(\varepsilon \)-Nash equilibrium point for the game by presenting a constructive algorithm. One of the main features is that the payoffs of the players depend on the set of players that stop at the termination stage which is the minimal stage in which at least one player stops. The existence result is extended to the case of a nonzero-sum game with finite horizon. Finally, the algorithm is illustrated by two explicit examples in the specific case of finite horizon.

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We refer to the beginning of Sect. 3.

For sake of clarity, \((B_n)_n\) is assumed to be a Brownian motion sampled at discrete time. Among its desired features, B is independent of N, it is a martingale with respect to its own filtration \(\mathcal {F}^B = (\mathcal {F}_n^B)\) and \(B_{n+1}-B_n\) is centered and independent of \(\mathcal {F}_n^B \).

From now, we omit both symbols \(\varepsilon \) and \(\omega \): contrary to the first example, all stopping times \(\theta _m\), \(\mu _m\) and \(\tau _m\) are a priori random and so it is for the \( {{\mathbf{NEP}}} \).

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Title: -Nash Equilibria of a Multi-player Nonzero-Sum Dynkin Game in Discrete Time
Authors: Said Hamadène
Mohammed Hassani
Marie-Amélie Morlais
Publication date: 30-03-2023
Publisher: Springer US
Published in: Dynamic Games and Applications
Print ISSN: 2153-0785
Electronic ISSN: 2153-0793
DOI: https://doi.org/10.1007/s13235-023-00500-3

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