Question 2 Two classmates in a game theory class collaborate on a problem set (incidentally, recall that collaboration on this exam is not allowed!). Each student i 1, 2 invests an amount of effort ei 0; the cost of effort for student i is ez. Both students receive the same grade: ei + 2e2 + ejez (the effort of student 2 is more valuable because she’s better at calculus). Hence, the payoff function of student i is given by Vilei, (2) = ei +2e2 + eje2 – ez. (a) What’s the best-response of student 1 if student 2 invests ez? What’s the best- response of student 2 if student 1 invests ej? (b) Find the Nash equilibrium of this game. (c) Professor Manea recommends that the students choose ei = e2 = 2. Does this outcome Pareto dominate the Nash equilibrium (i.e., is it better for both students)?
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