4. The three musketeers, Aramis, Athos and Porthos (players 1,2 and 3 respectively), decide that they...

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4. The three musketeers, Aramis, Athos and Porthos (players 1,2 and 3 respectively), decide that they are no longer "all for one, one for all" and agree to a truel (a truel is a duel among three opponents). According to the rules of the truel, each opponent has a musket loaded with one bullet. The musketeers act sequentially, in alphabetic order starting with Aramis. When his turn comes, a musketeer chooses his target and shoots. In normal circumstances, a musketeer i = {1,2,3} hits a target with probability p = [0, 1] (assume that a musketeer cannot miss a target on purpose). Once a musketeer is wounded in the truel, he cannot shoot and cannot be chosen as a target by another musketeer. A wounded musketeer has to pay his own medical expenses that are equal to C > 0 (the three musketeers are so tough that they cannot be killed). Assume that the musketeers are risk-neutral and that they minimise their expenses. (a) Formalize this situation as a game. (b) Assuming that a musketeer who is indifferent between two targets picks either with probability 0.5, find a subgame perfect Nash equilibrium. (c) For the equilibrium that you found, provide ex ante probabilities of becoming wounded for all three duelists. Explain intuitively what would change if you drop the assumption that a musketeer cannot miss a target on purpose. 4. The three musketeers, Aramis, Athos and Porthos (players 1,2 and 3 respectively), decide that they are no longer "all for one, one for all" and agree to a truel (a truel is a duel among three opponents). According to the rules of the truel, each opponent has a musket loaded with one bullet. The musketeers act sequentially, in alphabetic order starting with Aramis. When his turn comes, a musketeer chooses his target and shoots. In normal circumstances, a musketeer i = {1,2,3} hits a target with probability p = [0, 1] (assume that a musketeer cannot miss a target on purpose). Once a musketeer is wounded in the truel, he cannot shoot and cannot be chosen as a target by another musketeer. A wounded musketeer has to pay his own medical expenses that are equal to C > 0 (the three musketeers are so tough that they cannot be killed). Assume that the musketeers are risk-neutral and that they minimise their expenses. (a) Formalize this situation as a game. (b) Assuming that a musketeer who is indifferent between two targets picks either with probability 0.5, find a subgame perfect Nash equilibrium. (c) For the equilibrium that you found, provide ex ante probabilities of becoming wounded for all three duelists. Explain intuitively what would change if you drop the assumption that a musketeer cannot miss a target on purpose.