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