Question

3 Static game II

Imagine a game with a Professor and Students (who all act together as one player). TheProfessor is giving a final exam and has to decide whether to make it easy or hard. Studentshave to decide whether to put low effort, medium effort, high effort, or max effort intostudying for the exam. Both players decide simultaneously. Payoffs are as follows:
Students
Max effort High effort Medium effort Low effort
Professor Easy 4,3 3,4 2,5 1,2
Hard 6,4 5,3 1,2 0,1
1. Does Professor have a dominant strategy Why or why not?
2. Does Students have a dominant strategy? Why or why not?
3. What is the Nash Equilibrium of this game?
4. If there are multiple Nash Equilibria, which one will be se- lected in the end? If there is no Nash Equilibrium, how will the game end?

Answer 1

1) The professor does not have any dominant strategy. There is no strategy which will be the best response by the professor, irrespective of the strategy used by the students

If the students choose maximum or high effort, the best response is for the professor to choose hard, but if the students choose medium effort or low effort, the professor is to choose easy.

2) The students do not have any dominant strategy. If the professor chooses easy,the students are better off using medium, If the professor chooses hard, the best response for the students is to choose maximum effort

3) We have 2 Nash equilibra for this game.

i) When the professor chooses easy, the students should choose medium.

<Easy, Medium>

ii) Whe the professor chooses hard, the students choose maximum.

<Hard, Max>

4) There are multiple Nash equilibra. So the one that will be selected is the

<Hard, Max>

This is because, the payoff is much higher for the professor if he selects hard, if the professor selects hard he will have a payoff of 6. If the students select maximum and professor selects hard their payoff would be 2, but if the professor by any chance selects easy, their payoff would be 3. It is better the students take the risk by selecting max.

Because no dominant strategy exists, the selected Nash equilibrium at the end would be the professor choosing hard and the student choosing Maximum effort.