A queen is looking for a bodyguard. A swordsman nominates himself, but the queen is unsure if

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Question:

A queen is looking for a bodyguard. A swordsman nominates himself, but the queen is unsure if this man is a hero or a coward. The swordsman knows whether he is a hero or a coward, but the queen’s prior belief is that both possibilities are equally likely. Give the importance of the job, the queen will hire him if and only if the probability that he is a hero is at least 0.9. If the swordsman is hired, he gains 16 utils from this honourable job.

To learn more about the truth, the queen lays three fiery hot peppers in front of the swords-man. The peppers are not poisonous, but eating them will cause an extremely painful burning sensation. If the swordsman is a hero, he loses 5 utils for each pepper that he eats. If the swordsman is a coward, he loses 10 utils for each pepper that he eats. The queen observes whether the swordsman eats some or all of the peppers and updates her belief accordingly.

Recall that a hero loses 5 utils for each pepper he eats and a coward loses 10 utils for each pepper he eats. The queen is indifferent between hiring and not hiring when Pr(hero) = 0.9, strictly prefers to hire if Pr(hero)>0.9, and strictly prefers not to hire if Pr(hero) < 0.9. The common prior belief is Pr(hero) = 0.5. For the next few questions in this section, suppose that the swordsman gains X utils if he is hired.

Part 1: If the swordsman gains X = 41 utils if he is hired, is there a perfect Bayesian equilibrium of the following kind? A hero always eats 1 pepper. A coward sometimes eats 1 pepper, and sometimes eats none. If yes, calculate the probability Pr(Queen hires | swordsman eats 1 pepper).

Part 2: If the swordsman gains X = 15 utils if he is hired, is there a perfect Bayesian equilibrium of the following kind? A hero always eats 2 peppers. A coward sometimes eats 2 peppers and sometimes eats none. If yes, calculate the probability Pr(eat 2 peppers | coward).

Part 3: If the swordsman gains X = 14 utils if he is hired, is there a perfect Bayesian equilibrium of the following kind? A hero sometimes eats 1 pepper and sometimes eats none. A coward never eats any pepper. If yes, calculate the probability Pr(eats 1 pepper | hero).