  We analyze a two-player, nonzero-sum Dynkin game of stopping with incomplete
information. We assume that each player observes his own Brownian motion, which
is not only independent of the other player's Brownian motion but also not
observable by the other player. The player who stops first receives a payoff
that depends on the stopping position. Under appropriate growth conditions on
the payoff function, we show that there are infinitely many Nash equilibria in
which both players have infinite expected payoffs. In contrast, the only
equilibrium with finite expected payoffs mandates immediate stopping by at
least one of the players.
