On a filtered probability space $(\Omega,\mathcal{F},P,\mathbb{F}=(\mathcal{F}_t)_{0\leq t\leq T})$, we consider stopper-stopper games $\bar C:=\inf_{\Rho}\sup_{\tau\in\T}\E[U(\Rho(\tau),\tau)]$ and $\underline C:=\sup_{\Tau}\inf_{\rho\in\T}\E[U(\Rho(\tau),\tau)]$ in continuous time, where $U(s,t)$ is $\mathcal{F}_{s\vee t}$-measurable (this is the new feature of our stopping game), $\T$ is the set of stopping times, and $\Rho,\Tau:\T\mapsto\T$ satisfy certain non-anticipativity conditions. We show that $\bar C=\underline C$, by converting these problems into a corresponding Dynkin game.
