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We analyze a two-player, nonzero-sum Dynkin game of stopping with incompleteinformation. We assume that each player observes his own Brownian motion, whichis not only independent of the other player's Brownian motion but also notobservable by the other player. The player who stops first receives a payoffthat depends on the stopping position. Under appropriate growth conditions onthe payoff function, we show that there are infinitely many Nash equilibria inwhich both players have infinite expected payoffs. In contrast, the onlyequilibrium with finite expected payoffs mandates immediate stopping by atleast one of the players.

A Dynkin Game with Independent Processes and Incomplete Information