Question

Three students form a study group for a Microeconomics class. They share solutions amongst the three of them. They can only solve all the problems on a problem set if at least two of them come to the bi-weekly meeting. The utility from handing in a completed problem set is u = 5. The cost of going to the meeting is c = 1. Find all the pure strategy Nash equilibria of the game.

Answer 1

If the problem set is completed, then each of the three get a utility of 3. (u=1)

Cost of going to the meeting for each person is 1. (c=1)

To complete the problem set, at least two people require to go to the bi-weekly meeting.

Case 1:

Everyone go to the meeting.

In this case, the problem set will be completed and everyone gets a net utility of u-c = 5-1 = 4

Now one of them has incentive to drop from going to the meeting because by doing so, the problem set still gets solved(two people are enough to solve the problem set) and this person gets a net utility of 5 (this person faces no cost of she doesn't go to meeting).

So, one person has an incentive to deviate.

So this is not a Nash Equilibrium.

Case 2:

No one goes to the meeting.

In this case, problem set is not solved and each of them gets a net utility of 0. Here no one has an incentive to go to the meeting because of anyone go to meeting, the problem set will not be solved and the person that went to the meeting gets a net utility of -1.

So no one has an incentive to deviate from this strategy.

Therefore, no one going to the meeting is a Nash equilibrium.

Case 3:

One person going to the meeting.

In this case the problem set doesn't get solved and the person that goes to the meeting gets a net utility of -1 and the other two get a net utility of 0 each.

In this case the person that goes to the meeting has an incentive to drop from going to the meeting because by doing so, she gets a net utility of 0 whereas she is currently getting only -1.

So one of those who don't go to the meeting has an incentive to go for meeting because by doing so, the problem set gets solved and she gets a net utility of 4. So she has an incentive to deviate.

So at least one person has incentive to deviate.

Therefore, this is not a Nash Equilibrium.

Case 4:

Two of them go to the meeting.

In this case, problem set gets solved and those who go to the meeting get a net utility of 4 each and the person that doesn't go will get a net utility of 5. In this case no one has an incentive to deviate.

Those who are going to the meeting will not drop because if one of them drops, there won't be enough people to solve the problem set and the one that dropped gets a net utility of 0 whereas she was getting a net utility of 4 by going to the meeting.

The one that didn't go to the meeting has no incentive to go to the meeting because the problem set is getting solved currently and she's getting a net utility of 5. By going to the meeting she has incur a cost of 1 and she gets a net utility of 4. So, she won't deviate.

So no one has any incentive to deviate.

Therefore, two people going to the meeting is a Nash equilibrium.

RESULT:

NO ONE GOING TO THE MEETING AND TWO PEOPLE GOING TO THE MEETING ARE THE TWO PURE STRATEGY NASH EQUILIBRIA OF THIS GAME.