  We study the existence of optimal actions in a zero-sum game
$\inf_{\tau}\sup_PE^P[X_{\tau}]$ between a stopper and a controller choosing a
probability measure. This includes the optimal stopping problem
$\inf_{\tau}\mathcal{E}(X_{\tau})$ for a class of sublinear expectations
$\mathcal{E}(\cdot)$ such as the $G$-expectation. We show that the game has a
value. Moreover, exploiting the theory of sublinear expectations, we define a
nonlinear Snell envelope $Y$ and prove that the first hitting time
$\inf\{t:Y_t=X_t\}$ is an optimal stopping time. The existence of a saddle
point is shown under a compactness condition. Finally, the results are applied
to the subhedging of American options under volatility uncertainty.
