Let (X,Y,Z) be a triple of payoff processes defining a Dynkin game (R) over tilde(sigma,tau) = E[X(sigma)1({tau>sigma}) + Y(tau)1({tau<σ}) + Z(τ)1({τ=σ})], where and are stopping times valued in [0,T]. In the case Z=Y, it is well known that the condition X&LE;Y is needed in order to establish the existence of value for the game, i.e., inf(τ)sup(σ) <(R)over tilde>(sigma,tau)=sup(sigma)inf(tau)($) over tilde(sigma,tau). In order to remove the condition Xless than or equal toY, we introduce an extension of the Dynkin game by allowing for an extended set of strategies, namely, the set of mixed strategies. The main result of the paper is that the extended Dynkin game has a value when Zless than or equal toY, and the processes X and Y are restricted to be semimartingales continuous at the terminal time T.