Question

3 Static game II Imagine a game with a Professor and Students (who all act together as one player). The Professor is giving a final exam and has to decide whether to make it easy or hard. Students have to decide whether to put low effort, medium effort, high effort, or max effort into studying for the exam. Both players decide simultaneously. Payoffs are as follows: Students Max effort High effort Medium effort Low effort 4,3 2,5 1,2 Professor Easy Hard 3,4 6,4 5,3 1.2 0.1 3.1 Does Professor have a dominant strategy? Why or why not? 3.2 Does Students have a dominant strategy? Why or why not? 3.3 What is the Nash Equilibrium of this game? 3.4 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

Answer:

3.1 No, the Professor does not have a dominant strategy. If students are going to put in Maximum or Hard work, Professor's strategy is to set Hard paper. If Students are going to put in Medium or Low efforts, Professor's strategy is to set Easy paper.

3.2 Students does not have a dominanat strategy either. If they anticipate an easy paper, they will put in Medium effort. If they anticipate a Hard paper, they will put in Maximum effort.

3.3 There are 2 Nas equiibria in this game: ((Easy, Medium) and (Hard, Maximum). The highest payoffs of the players coincide on these strategies.  

Professor Easy Hard Students Max effort High effort Medium effort Low effort 4,3 3,4 2,5 1,2 6,4 5,3 1,2 0,1

The preferred strategies by players assuming the other player's strateg have been underlined. The resulting coinciding strategies ((Easy, Medium) and (Hard, Maximum) are Nash equilibrium.

Explanation: If Students assume that the paper will be easy, they will put in medium efforts because that has the highest payoff of 5 which is less than the other payoffs (5>4>3>2). If they assume the paper is going to be Hard, then they will put in Max effort (4>3>2>1).

On the other hand, if Professor assumes Max effort, he will set it Hard (6>4), if Hard effort is anticipated, he will go Hard (5>3), for Medium efforts Easy (2>1) and for Low effort, Easy (1>0). Thus ((Easy, Medium) and (Hard, Maximum) will be Nash equilibria.

3.4 There are multiple Nash equilibra here (2 equilibria). Ultimtely what mattes is the benefit of the Students. Between the payoffs of Students, (Easy, Medium) gives them a better payoff (5>4). So that will be selected.