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# Imagine a small town with a corrupt prosecutor, an honorable prosector, a corrupt public defender, and an honorable public defender.

Imagine a small town with a corrupt prosecutor, an honorable prosector, a corrupt public defender, and an honorable public defender. The honorable officials always give a case their best effort, while the corrupt officials choose whether to apply high effort (H) or low effort (E) based on whatever maximizes their payoffs. Each week, there are two cases, and the prosecutor and defender on each side is chosen randomly with equal probability. The only agents who behave strategically in a game-theoretic sense are the corrupt ones, so we’ll concentrate on their behavior in this problem.

a)If a corrupt attorney applies more effort than his opponent, he wins the case but pays a cost of effort, earning payoff 1 − c, whereas if he applies the same amount of effort, they tie, and he earns a payoff of −c if he chose high effort and 0 if he chose low effort. If he puts in less effort he loses and earns −1. Write down the one-shot normal form game when two corrupt attorneys meet.

b)Now suppose that an attorney prepares for a case without knowing whether he’ll face a corrupt attorney or an honorable one. Assume that if he faces a corrupt attorney, they will both put in low effort. What is his expected payoff from a single week’s case? What is the net present value of playing this strategy profile repeatedly if his discount rate equals .5? Do the same assuming that both players always choose high effort.

c)Assuming as before that the attorneys do not know if they are facing a corrupt or honorable opponent, prove the values of c > 4 for which it is possible to sustain cooperation on low effort using the grim trigger strategy. Assume that if one of the attorneys deviates to high effort, the other attorney will immediately find out even if they don’t oppose each other that round. d) Now suppose c equals 3. Suppose that the corrupt attorneys hatch a plan to assassinate the honorable attorneys. They will split the assassination fees, paying K/2 each. Assume that if they assassinate the honorable attorneys, they will henceforth only face each other in court and cooperate on low effort using the grim trigger strategy. Prove the maximum value of K < c for which they would carry out the assassination?

e)The honorable attorneys have been killed and it’s just the corrupt attorneys now. Assume that they have acquired the honorable attorneys staff, so that the cost of effort c has decreased to .5. Now let δ be an unknown parameter. Suppose that the corrupt attorneys have to prepare for 5 weeks of cases at a time, so that if one of them deviates from the cooperative strategy, the other will punish him by pulling the grim trigger, but not until five weeks have passed (i.e., will start using high effort every week starting in the sixth week after deviation occurred). Prove the minimum value of δ is 0.935 for which cooperation can be sustained?