A class with 30 students enrolled is given a homework assignment with five questions. The first four are the usual kinds of problems, totaling to 90 points. But the fifth is an interactive game for the class. The question reads: "You can choose whether to answer this question. If you choose to do so, you merely write ‘I hereby answer Question 5.’ If you choose not to answer Question 5, your score for the assignment will be based on your performance on the first four problems. If you choose to answer Question 5, then your scoring will be as follows: If fewer than half of the students in the class answer Question 5, you get 10 points for Question 5; i. e. 10 points will be added to your score on the other four questions to get your total score for the assignment. If half or more than half of the students in the class answer Question 5, you get – 10 points; i. e. 10 points will be subtracted from your score on the other questions." (a) Draw a diagram illustrating the payoffs from the two possible strategies, "Answer Question 5" and "Don’t Answer Question 5," in relation to the number of other students who answer it. Find the Nash equilibrium of the game.
(b) What would you expect to see happen in this game if it were actually played in a college classroom? Why? Consider two cases: (i) the students make their choices individually with no communication; and (ii) the students make their choices individually but can discuss these choices ahead of time in a discussion forum available on the class Web site.