Three students are enrolled in a class. At the end of the class, the professor announces that 1 point extra credit will be awarded to all students if at least two students complete the course evaluation. The students decide whether or not to complete the evaluation independently and simultaneously. Evaluations are a hassle and cost each student 1 jolly (payoff) to complete. Each student values the extra credit at 10 jollies (payoff).

(a) Draw the strategic form of the game among the students.

(b) Find all pure strategy Nash equilibria of this game.

(c) Find one mixed strategy Nash equilibrium.

(d) Consider the same setup as above, but except now the value of the extra credit to each student is their private information. In particular, if a student expects to get an A in the course without the extra credit, the extra credit is worthless. If the student does not expect to get an A, the extra credit is worth 10 jollies (payoff). Each student believes that each student (including herself) expects to get an A with probability p. These probabilities are independent. Each student's type is drawn by nature and revealed only to that student. Then the students simultaneously decide to complete the course evaluation or not. An course evaluation costs 1 jolly (payoff) to complete. Find conditions on p such that it is a pure strategy Bayes Nash equilibrium for students complete the evaluation if and only if they do not expect to get an A.