  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 $\epsilon>0$.
