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.
a. Assume that student 2 plays the following strategy: "I offer to exchange for every grade I get." What is the best response of student 1?
b. Define a weak exchange Bayesian Nash equilibrium (WEBNE) as a Bayesian Nash equilibrium in which each student i choosessi(gi) = X whenever
That is, given his grade gi and his (correct belief about his) opponent's strategy s-i, choosing X is as good as or better than H. In particular a WEBNE is a pair of strategies (s1, s2) such that given s2 student 1 offers to exchange grades if exchange gives him at least as much as holding, and vice versa. Find all the symmetric (both students use the same strategy) WEBNE of this game. Are they Pareto ranked?
c. Now assume that the professor suggests modifying the game: everything works as before, except that the students must decide if they want to exchange before opening their envelopes. Using equilibrium analysis, would the students prefer this game or the original one?
d. From your conclusion in (c), what can you say about the statement "more information is always better"?