t5. |13] Two French aristocats {great Disney film] IChevalier Chagrin and Marquis de Renard, fight a duel. Each is armed with a pistol that contains one bullet. They begin 6 paces apart and walk toward each other. After each step, each simultaneously decides whether to sheet [5} or not shoot [NS], wiTh the probability ofkilling the other equal to U3 where k is The number of steps that each has taken. That is, initially, the probability of killing the other ifhe shoots equals zero, after the rst step, This probability equals N3, after two steps, Ell, when They are nose to nose, it equals 1. Their code of aristocracy compels each to continue stepping forward even if he has chosen 3 and missed but the other chose NS to fail to do so would bring ignominy to their family. Each gets a payoff of I] if each is killed. Each gets a payoff of [I if neither is killed. If one is killed but The other is not, the one who is not killed gets I and the one who is killed gets -1. If there are two Nash equilibria in pure strategies but one in mixed strategies, find The mixed strategy Nash equilibrium and assume that this is the equilibrium that will be played. After all, their code of honor is such that They cannot play an equilibrium in which one is favored This game is {almost} guaranteed to be played only oncel a] [7"] Let o. = 4. Find The subgame perfect Nash equilibrium. Hint: Start, of course, by graphing the normal form alter Three steps, when S is guaranteed to kill. Then, use backward induction and graph the normal lorm after two steps. The payofls now will be an expected payoff since 3 may or may not kill, and reason through what happens if both are still alive and proceed to the third step. b} |3IShow that it a = 3, then after their second step, they each prefer to choose 5 in response to D5. Explain why making This change makes them less likely to wait to choose 3. c} I3} Suppose That Chevalier is a crack shot. Initially, at six paces apart, his probability of killing de Renard if Chevalier chooses S, is H2, and then increases to 233 at four paces apart, 5H5 at two paces apart, and I when they are nose-tonose. Mssr. de Renard knows this; in lact, it is common knowledge between them, and even all of France, for Chevalier is the reigning national champion. In the new subgame perlect Nash equilibrium, neither chooses S at the initial stage, when they are six paces apart, but Chevalier chooses S at the second stage and de Renard chooses DS. Explain why de Renard does not choose 5 when Chevalier does, and Then use this to explain why Chevalier will not choose 5 initially, or wait until they are each only two steps apart If The third stage}