On the eve of a problem-set due date, a professor receives an e-mail from one of

Question

On the eve of a problem-set due date, a professor receives an e-mail from one of her students, who claims to be stuck on one of the problems after working on it for more than an hour. The professor would rather help the student if he has sincerely been working at the problem, but she would rather not render aid if the student is just fishing for hints. Given the timing of the request, she could simply pretend not to have read the e-mail until later. Obviously, the student would rather receive help whether or not he has been working on the problem. But if help isn't coming, he would rather be working instead of slacking, since the problem set is due the next day. Assume the payoffs are as follows: (a) What is the mixed-strategy Nash equilibrium to this game? (b) What is the expected payoff to each of the players?

Explanation / Answer

1. Let the professor play Help Student with prob p and Ignore email with prob (1 - p)

Let the student play Work with prob q and Slack with prob (1 - q)

As this is a mixed-strategy equilibrium:

Professor's pay-off from playing Help Student = Professor's pay-off from playing Ignore email
=> 3q - (1 - q) = -2q
=> 6q = 1
=> q = 1/6

Also, Student's pay-off from playing Work = Student's pay-off from playing Slack
=> 3p + (1 - p) = 4p
=> 2p = 1
=> p = 1/2

Hence, mixed-strategy NE of the game is:
Professor playing Help Student with prob 1/2 and Ignore Email with prob 1/2
Student playing Work with probability 1/6 and Slack with prob 5/6

2. Expected pay-off of professor = 3*(1/6) -1*(5/6) = -1/3
Expected pay-off of student = 3*(1/2) + 1*(1/2) = 2

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