# Problem Two students take an exam. The student with the higher score will receive a grade ‘Excellent’ and the one with the lower score will receive…

Problem

Two students take an exam. The student with the higher score will receive a grade ‘Excellent’ and the one with the lower score will receive ‘Good’.

Student 1’s score equals x1 + 1.5, where x1 is the amount of eﬀort she invests in studying. (we assume that the greater the eﬀort, the higher is the score.).

Student 2’s score equals x2, where x2 is the amount of eﬀort (measured in full days dedicated to studying the subject) he exerts.

It is assumed that student 1 is the smarter of the two, i.e. if the amount of eﬀort is held ﬁxed, student 1 has a higher score by an amount of 1.5.

Assume that x1 and x2 can take any integer value in 0, 1, 2, 3, 4, 5 . A student receives an award of 10 chocolate bars if she gets ‘Excellent’, and 8 chocolate bars if she gets ‘Good’. Both students’ invested eﬀort of one study day has a negative eﬀect on their well-being that corresponds to forgone consumption of one chocolate bar. Thus, the payoﬀ to student i is (10 − xi) if she gets ‘Excellent’, and (8 − xi) if she gets ‘Good’, i = 1, 2.

Question:

1. What are the possible strategies of the students? What are their payoﬀs for each strategy combination? Represent this game in the normal (strategic) form.

2. Derive the strategies that survive the iterated elimination of strictly dominant strategies.

3. From the remaining strategies, which strategies are weakly dominated?

4. After the deletion of weakly dominated strategies, ﬁnd a solution (equilibrium) of this game. What are the equilibrium eﬀorts of the students?