  In this paper we consider Dynkin's games with payoffs which are functions of
an underlying process. Assuming extended weak convergence of underlying
processes $\{S^{(n)}\}_{n=0}^{\infty}$ to a limit process $S$ we prove
convergence Dynkin's games values corresponding to $\{S^{(n)}\}_{n=0}^{\infty}$
to the Dynkin's game value corresponding to $S$. We use these results to
approximate game options prices with path dependent payoffs in continuous time
models by a sequence of game options prices in discrete time models which can
be calculated by dynamical programming algorithms. In comparison to previous
papers we work under more general convergence of underlying processes, as well
as weaker conditions on the payoffs.
