In this talk I will consider two types of Dynkin game with non-standard information structures. The first one is a zero-sum game between two players who observe a geometric Brownian motion but in which the minimiser knows the drift of the process whereas the maximiser doesn't know it. We construct an explicit Nash equilibrium in which the uninformed player uses a pure strategy and the informed player uses a randomised strategy. The second game is a non-zero sum game between two agents interested in the purchase of the same asset. Neither of the two players knows with certainty whether their competitor is `active' and in that sense that they have uncertain competition. Also in this case we construct explicitly a Nash equilibrium in which both players randomise their strategy.