19 06 2023 E TS

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GAME THEORY - June 19, 2023 Last name: First name: ID #: SOLVE THE EXERCISES AND ANSWER THE QUESTIONS ON THESE SHEETS Exercise 1 5 points Two players I and II have to split one asset, so that P1 gets x ≥ 0 and P2 gets y ≥ 0, with x + y ∈ [0, 1]. Their utilities are u(x) = x2 and v(y) = √y respectively, and the disagreement point is d = (0, 0). Find the Nash solution. Answer of exercise 1 The bargaining set is characterised by the relation √u + v2 ≤ 1, with u, v ≥ 0. So finding the maximum of uv on this set is equivalent to finding the maximum of v(1 − v2)2, with 0 ≤ v ≤ 1, which is given by v = 1√ 5, that is y = 1 5, and then x = 4 5. The utilities for the Nash solution are  16 25 , 1√ 5  . 1 Exercise 2 5 points In a school board there are 10 professors that rank four students: Anne, Beth, Cecil and Dan. Their ranks are: 4 professors : D ≻ C ≻ B ≻ A 3 professors : A ≻ B ≻ D ≻ C 3 professors : B ≻ D ≻ C ≻ A Find the best student according to the simple majority rule, the Condorcet rule and the Borda rule. Explain your answer. Answer of exercise 2 By the simple majority rule Dan is the best student, since he is the first one according to 4 professors. The Condorcet winner is Beth (7 professors think he is better than Anne, and 6 professors think he is better than Cecil and Dan). The Borda winner is Dan (assuming that 4 points are assigned to the best student, 3 to the second best and so on, Ann gets 19 points, Beth 29, Cecil 21 and Dan 31). 2 Theory Questions Answer one and only one question (7 points). Only the question 2 may lead to the top grade 30 e lode. 1. State Arrow’s theorem and explain the assumptions. 2. State and prove Arrow’s theorem (no need to explain the assumptions). The first step of the proof suffices. 3