In this paper we investigate a game of optimal stopping with incomplete information. There are two players, but only one is informed about the precise structure of the game. Observing the informed player, the uninformed player is given the possibility of guessing the missing information. We show that these games have a value which can be characterized as a viscosity solution to a fully nonlinear variational PDE. Furthermore, we derive a dual representation of the value function in terms of a minimization procedure. This representation allows us, under some additional assumptions, to determine optimal strategies for the informed player.