  On a filtered probability space
$(\Omega,\mathcal{F},P,\mathbb{F}=(\mathcal{F}_t)_{t=0,\dotso,T})$, we consider
stopper-stopper games $\overline
V:=\inf_{\Rho\in\bT^{ii}}\sup_{\tau\in\T}\E[U(\Rho(\tau),\tau)]$ and
$\underline V:=\sup_{\Tau\in\bT^i}\inf_{\rho\in\T}\E[U(\Rho(\tau),\tau)]$ in
discrete time, where $U(s,t)$ is $\mathcal{F}_{s\vee t}$-measurable instead of
$\mathcal{F}_{s\wedge t}$-measurable as is often assumed in the literature,
$\T$ is the set of stopping times, and $\bT^i$ and $\bT^{ii}$ are sets of
mappings from $\T$ to $\T$ satisfying certain non-anticipativity conditions. We
convert the problems into a corresponding Dynkin game, and show that $\overline
V=\underline V=V$, where $V$ is the value of the Dynkin game. We also get the
optimal $\Rho\in\bT^{ii}$ and $\Tau\in\bT^i$ for $\overline V$ and $\underline
V$ respectively.
