(3) Imagine a small seminar course with three students. All three students think that the final assignment for the course is unreasonably demanding. To voice their discontent, the students are
considering staging a protest in the final class. The students have two possible actions: protest (P)
or stay quiet (Q). If at least two students protest, the professor will adjust the final assignment to
the students' satisfaction; if less than two students protest, the professor will leave the assignment as is. If the professor adjusts the assignment, all students receive a benefit B; if the professor
leaves the assignment as is, they receive nothing (0). Any student who engages in protest pays a cost c (which corresponds to the risk of provoking the professor's anger). We assume that B > c > 0.
(a) Chat-GPT says that a profile in which all protest (P, P, P) is a Nash equilibrium, explaining "in this case, all students receive the benefit B, and no one can improve their payoff by unilaterally
(b) Suppose players 2 and 3 protest with probability p. What is player 1's expected payoff for protesting, expressed as a function of p? (1 point)
(c) Still supposing players 2 and 3 protest with probability p. What is player 1's expected payoff for staying quiet, expressed as a function of p? (1 point)
(d) Suppose B = 2 and c = 1. Characterize the Nash equilibrium mixed strategy profiles in which each player protests with the same probability p > 0 (i.e., specify the value(s) for p that
correspond to mixed strategy Nash equilibrium profiles, assuming all three players protest
with probability p). (2 points)
