In the classic Rob Reiner movie The Princess Bride, there is a scene at the end...

 

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In the classic Rob Reiner movie The Princess Bride, there is a scene at the end where Westley (the protagonist) confronts the evil prince Humperdinck. The interaction can be modelled as the following game: Westley 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. Westley is lying in bed in the prince's castle when the prince enters the room. Westley decides whether to get out of bed (0) or stay in bed (B). The prince observes Westley's action but does not observe Westley's type. The prince then decides whether to fight (F) or surrender (S) to Westley. The payoffs are such that the price prefers to fight only with the weak Westley, because otherwise the prince is an inferior swordsman. Also, the weak Westley must pay a cost e to get out of bed. The extensive-form game is given below. Prince S Type W [p = 1/2] W B F S Nature. Type S [(1-p) = 1/2] W B F S Prince F (1,0) (0,-2) (1,0) (0, -2) (1,0) (-1,1) (1-c, 0) (-1-c, 1) (a) What conditions on c guarantee the existence of a separating perfect Bayes Nash equilibrium? Fully describe such an equilibrium. (b) For what values of c is there a pooling equilibrium in which both strong and weak Westleys get out of bed. Fully describe such an equilibrium. In the classic Rob Reiner movie The Princess Bride, there is a scene at the end where Westley (the protagonist) confronts the evil prince Humperdinck. The interaction can be modelled as the following game: Westley 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. Westley is lying in bed in the prince's castle when the prince enters the room. Westley decides whether to get out of bed (0) or stay in bed (B). The prince observes Westley's action but does not observe Westley's type. The prince then decides whether to fight (F) or surrender (S) to Westley. The payoffs are such that the price prefers to fight only with the weak Westley, because otherwise the prince is an inferior swordsman. Also, the weak Westley must pay a cost e to get out of bed. The extensive-form game is given below. Prince S Type W [p = 1/2] W B F S Nature. Type S [(1-p) = 1/2] W B F S Prince F (1,0) (0,-2) (1,0) (0, -2) (1,0) (-1,1) (1-c, 0) (-1-c, 1) (a) What conditions on c guarantee the existence of a separating perfect Bayes Nash equilibrium? Fully describe such an equilibrium. (b) For what values of c is there a pooling equilibrium in which both strong and weak Westleys get out of bed. Fully describe such an equilibrium.

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a For there to be a separating perfect Bayes Nash equilibrium c must be greater ... View the full answer