  We study a robust Dynkin game over a set of mutually singular probabilities.
We first prove that for the conservative player of the game, her lower and
upper value processes coincide (i.e. She has a value process $V $ in the game).
Such a result helps people connect the robust Dynkin game with second-order
doubly reflected backward stochastic differential equations. Also, we show that
the value process $V$ is a submartingale under an appropriately defined
nonlinear expectations up to the first time $\tau_*$ when $V$ meets the lower
payoff process $L$. If the probability set is weakly compact, one can even find
an optimal triplet. The mutual singularity of probabilities in causes major
technical difficulties. To deal with them, we use some new methods including
two approximations with respect to the set of stopping times. The mutual
singularity of probabilities causes major technical difficulties. To deal with
them, we use some new methods including two approximations with respect to the
set of stopping times
