12.4 Grade Gambles: Two students, 1 and 2, took a course with a professor who decided to allocate grades as follows: Two envelopes will each include a grade gi∈ {A, B, C, D, F}, where each of the five options is chosen with equal probability and the draws for each student i ∈ {1, 2} are independent. The payoffs of each grade are 4, 3, 2, 1, and 0, respectively. Assume that the game is played as follows: Each student receives his envelope, opens it, and observes his grade. Then each student simultaneously decides if he wants to hold on to his grade (H) or exchange it with the other student (X). Exchange happens
if and only if both choose to exchange. If an exchange does not happen then each student gets his assigned grade. If an exchange does happen then the
grades are bumped up by one. That is, if student 1 had an initial grade of C and student 2 had an initial grade of D, then after the exchange student 1 will get a C (which was student 2's D) and student 2 will get a B (which was
student 1's C). A grade of A is bumped up to an A+, which is worth 5.
12.8 Jury Voting: Consider the jury voting game in Section 12.4.
a. Show that both players choosing CC is a Bayesian Nash equilibrium of the two-player game