Playing with ghosts in a Dynkin game

We study a class of optimal stopping games (Dynkin games) of preemption type, with uncertainty about the existence of competitors. The set-up is well-suited to model, for example, real options in the context of investors who do not want to publicly reveal their interest in a certain business opportunity. We show that there exists a Nash equilibrium in randomized stopping times which is described explicitly in terms of the corresponding one-player game.