(solution) A motorist is given a $100 speeding ticket, and must decide whether to try and bribe the police

(solution) A motorist is given a $100 speeding ticket, and must decide whether to try and bribe the police

A motorist is given a $100 speeding ticket, and must decide whether to try and bribe the police officer with $20. There are two types of police officers. A proportion λ are honest and would never accept the motorist’s money, while the remaining (1 − λ) are dishonest (and lucky); they can get away with accepting the bribe, but would prefer to pocket the $100 fine. Assume that for both types, rejecting a bribe requires reporting the incident; the officer thus receives payoff zero, and the motorist incurs an additional fine of $100. The extensive form is shown in Figure 21.8, where player 1 represents the officer and player 2 represents the motorist. Find the PBE. Now suppose that the motorist receives a call from his sister, who just had a run-in with the same police officer and did attempt to bribe him. Explain why, in the repeated game, the police officer might have an incentive to reject the first bribe. Characterize the PBE; in particular: What would a dishonest police officer do if offered a bribe in the second period? What would the second motorist do if his sister’s bribe were accepted? Find the probability, r , that the second motorist offers a bribe if his sister’s bribe was rejected. [ Hint: Let p 2 = Pr { honest | reject } be the second motorist’s posterior probability that the police officer is honest, given that the first bribe was rejected. Show that the dishonest officer’s Incumbent Prey Accommodate Entrant Enter Stay Out −1, −1 10, 0 3, 3 10, 0 Figure 21.9 Problem 8 Incumbent Prey Accommodate Entrant Enter Stay Out −1, −1 11, 0 3, 3 11, 0 Figure 21.10 Problem 9 first-period strategy must be mixed, and find r such that he is indifferent between rejecting and accepting the bribe.] Find the probability that the dishonest officer rejects a bribe in the first period. Under what circumstance will the first-period motorist offer a bribe?