Non-zero-sum stopping games in continuous time

TL;DR

This paper studies a continuous-time non-zero-sum stopping game with two players, establishing the existence of approximate Nash equilibria under certain measurability and continuity conditions.

Contribution

It introduces a non-zero-sum stopping game framework with a new measurability assumption and proves the existence of epsilon-Nash equilibria.

Findings

Existence of epsilon-Nash equilibria for the game.

The game framework generalizes Dynkin games to non-zero-sum settings.

Continuity of payoff functions ensures equilibrium existence.

Abstract

On a filtered probability space $(\Omega ,\mathcal{F}, (\mathcal{F}_t)_{t\in[0,\infty]}, \mathbb{P})$, we consider the two-player non-zero-sum stopping game $u^i := \mathbb{E}[U^i(\rho,\tau)],\ i=1,2$, where the first player choose a stopping strategy $\rho$ to maximize $u^1$ and the second player chose a stopping strategy $\tau$ to maximize $u^2$. Unlike the Dynkin game, here we assume that $U(s,t)$ is $\mathcal{F}_{s\vee t}$-measurable. Assuming the continuity of $U^i$ in $(s,t)$, we show that there exists an $\epsilon$-Nash equilibrium for any $\epsilon>0$.