In the classic Rob Reiner movie The Princess Bride, there is a scene at the end where Wesley (the protagonist) confronts the evil prince Humperdinck. The interaction can be modeled as the following game: Wesley is one of two types: weak or strong. Wesley knows whether he is weak or strong, but the prince only knows that he is weak with probability 1/2 and strong with probability 1/2. Wesley is lying in a bed in the prince's castle when the prince enters the room.
Wesley decides whether to get out of bed (O) or stay in bed (B). The prince observes Wesley's action but does not observe Wesley's type. The prince then decides whether to fight (F) or surrender (S) to Wesley. The payoffs are such that the prince prefers to fight only with the weak Wesley, because otherwise the prince is an inferior swordsman. Also, the weak Wesley must pay a cost c to get out of bed. The extensive-form representation of the game follows the questions.
(a) What conditions on c guarantee the existence of a separating PBE? Fully describe such an equilibrium.
(b) For what values of c is there a pooling equilibrium in which both strong and weak Wesleys get out of bed? Fully describe such an equilibrium.
