This paper is the first step in the proof of existence of equilibrium payoffs for two-player stochastic games with finite state and action sets. It reduces the existence problem to the class of so-called positive absorbing recursive games. The existence problem for this class is solved in a subsequent paper

Let (X,Y,Z) be a triple of payoff processes defining a Dynkin game \tilde R(\sigma,\tau) &=& E\left[ X_\sigma\1_{\{\tau > \sigma\}} +Y_\tau \1_{\{\tau < \sigma\}} +Z_\tau \1_{\{\tau=\sigma\}}\right] , where $\sigma$ and $\tau$ are stopping times valued in [0,T]. In the case Z=Y, it is well known that the condition X $\leq$ Y is needed in order to establish the existence of value for the game, i.e., $\inf_{\tau}\sup_{\sigma}\tilde R(\sigma,\tau)$ $=$ $\sup_{\sigma}\inf_{\tau}\tilde R(\sigma,\tau)$. In order to remove the condition $X$ $\leq$ $Y$, 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 $Z\leq Y$, and the processes X and Y are restricted to be semimartingales continuous at the terminal time T

We study a general model of dynamic games with purely informational externalities. We prove that eventually all motives for experimentation disappear, and provide the exact rate at which experimentation decays. We also provide tight conditions under which players eventually reach a consensus. These results imply extensions of many known results in the literature of social learning and getting to agreement.