4 Princess Bride In the classic movie The Princess Bride, there is a scene at the end where the hero, Wesley, confronts the evil Prince Humperdinck. This interaction can be modeled as the following game. Wesley can either be strong or weak, as randomly picked by nature with equal probability. Wesley, knowing his state, can then choose to stand or continue laying on the bed. The prince observes this decision but does not observe Wesley's type and chooses whether to fight or sur- render. The prince can beat a weak Wesley (Wesley was "mostly" dead only a few hours earlier Nature Strong (.5) Weak (.5) Wesley Wesley Stand Bed Stand Bed Prince Prince Prince Prince Fight Not Fight Not Fight Not Fight Not N -1-c 1-c 0 after all) and can really beat up a weak Wesley who stays in bed. If a weak Wesley stands and the Prince fights, the the base payoffs are 1 for the Prince and -1 for Wesley while the Prince receives 2 in the event Wesley stays in bed and Wesley still gets -1. If Wesley is weak and the prince surrenders, regardless of whether Wesley has stood or not the base bayoffs are 1 for Wesley and 0 for the Prince. A strong Wesley, however, will destroy the Prince and enjoy doing it should the Prince try to fight. It is easier for Wesley to beat the Prince up if he stands, though, so in that case he gets a payoff of 3 while the Prince receives -2 while if Wesley stays in bed and the Prince chooses to fight, the payoffs are 1 and -1. In the event that the Prince chooses not to fight, Wesley gets 2 if he stands 1 if he stays in bed while in either case the Prince gets 0. The final complication to the payoffs is that if Wesley is weak, standing costs him some extra amount c, while it costs a strong Wesley nothing. The game form and payoffs are seen in a. Find a pooling Bayes-Nash equilibrium where p represents the Prince's belief that Wesley is weak. Verify that it is and in doing so, find the range of values for c that makes it valid. b . Derive a separating Bayes-Nash equilibrium for this game as well as the range of c over which it is valid