$\varepsilon$-Nash Equilibria of a Multi-player Nonzero-sum Dynkin Game in Discrete Time

TL;DR

This paper establishes the existence of approximate Nash equilibria in multi-player nonzero-sum Dynkin games with stopping strategies in discrete time, providing a constructive algorithm and extending results to finite horizon cases.

Contribution

It introduces a constructive method to find $oldsymbol{ extit{ extepsilon}}$-Nash equilibria for multi-player nonzero-sum Dynkin games, including finite horizon extensions.

Findings

Existence of $ extit{ extepsilon}$-Nash equilibria proved

Constructive algorithm for equilibrium computation provided

Explicit examples illustrating the algorithm included

Abstract

We study the infinite horizon discrete time N-player nonzero-sum Dynkin game ($N \geq 2$) 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.