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. 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: Student Work and ask for help | Slack and fish for hints Professor | Help Student | 3, 3 | -1, 4 Ignore e-mail | -2, 1 | 0, 0 a) What is the mixed-strategy Nash equilibrium to this game? b) What is the expected payoff to each of the players? 

          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. 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:

Student
Work and ask for help | Slack and fish for hints
Professor | Help Student | 3, 3 | -1, 4
Ignore e-mail | -2, 1 | 0, 0

a) What is the mixed-strategy Nash equilibrium to this game?
b) What is the expected payoff to each of the players?
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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. 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: Student Work and ask for help | Slack and fish for hints Professor | Help Student | 3, 3 | -1, 4 Ignore e-mail | -2, 1 | 0, 0 a) What is the mixed-strategy Nash equilibrium to this game? b) What is the expected payoff to each of the players?

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Gregory Mankiw 8th Edition
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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. 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?
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Transcript

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00:01 Hello, in this video i will be explaining the following.
00:03 So first we want to find the nash equilibrium of the game.
00:07 So by doing that we need to find out the professor's payoff from helping the student.
00:13 So prof helping, prof playing helping student equals payoff, professor's payoff from playing ignore email.
00:49 So that would be 3q minus 1, 1 minus q equals negative tq.
01:21 So that gives you 6q equals 1, that gives you q equals 1 over 6.
01:29 So next we need to find the student's payoff from playing work from which equals the slack.
01:57 So the payoff for slack, so sorry i'm not supposed to say slack, there we go...
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