One student (= Jay) in the game theory class class cheats on an exam, and this is observed by 3 other students (Paul, Bob, and Emily). These three students would each like the dean to know about the cheating incident. Reporting the incident costs C. If the cheating is reported (by at least one of Paul, Bob, or Emily), Jay will be punished, and all three of them derive a benefit of B. Suppose 0 < C < B. Thus, this is a three player game in which each player (Paul, Bob, Emily) decides whether or not to report the cheating incident. If nobody reports the cheating, the payoffs to the players are identically 0; if Paul reports, but Bob and Emily do not, then Paul earns B C, whereas Emily and Bob earn B each, etc.

(a) (2 points) What are the Nash equilibria of this game in pure strategies? In particular, show that the game does not have a symmetric Nash equilibrium in pure strategies (a Nash equilibrium is symmetric if every player plays the same strategy).

(b) (4 points) Compute the symmetric Nash equilibrium or equilibria in mixed strategies. (Hint: suppose, in such an equilibrium, every person reports with probability p (0, 1). Use the fact that each player must be indifferent between reporting and not reporting.)

(c) (4 points) For the Nash equilibrium/equilibria in part (b), compute the probability of the incident being reported.

(d) Suppose Jay's cheating is observed by the entire 67 person class. In the equilibrium/equilibria of part (c), do you think the incident is more likely to be reported or less likely to be reported? An informal argument is sufficient.