On a filtered probability space (Omega, F, P, F = (F-t) t=0 ,..., T), we consider stopping games (V) over bar := inf rho is an element of T-ii sup(tau is an element of T) E[U(rho(tau), tau)] and (V) under bar := sup (i)(tau is an element of T) inf rho is an element of T E[U(rho, tau (rho))] in discrete time, where U(s, t) is F-svt -measurable instead of F-s<^>t - measurable as is assumed in the literature on Dynkin games, T is the set of stopping times, and T-i and T-ii are sets of mappings from T to T satisfying certain nonanticipativity conditions. We will see in an example that there is no room for stopping strategies in classical Dynkin games unlike the new stopping game we are introducing. We convert the problems into an alternative Dynkin game, and show that (V) over bar = (V) under bar = V, where V is the value of the Dynkin game. We also get optimal rho is an element of T-ii and tau is an element of T-i for (V) over bar and (V) under bar respectively.