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GAME THEORY - January 23, 2023 Last name: First name: ID #: SOLVE THE EXERCISES AND ANSWER THE QUESTIONS ON THESE SHEETS Exercise 1 5 points Four identical items are auctioned among 6 players using a VCG scheme. The marginal valuation for each additional item received by each player are shown in the following table Item P1 P2 P3 P4 P5 P6 1 75 90 95 90 70 60 2 75 40 80 x 50 60 3 50 20 75 40 50 50 4 30 0 30 40 50 20 Thus, the valuation of player 1 if she receives 1 item is v1 = 75, if she gets 2 items is v1 = 150, and so on. Find the smallest value x ∈ N such that P4 gets two items for sure (no tie), and compute how much she has to pay. Answer of exercise 1 In the example the 3 highest marginal valuations are 95,90,90. For x to be the 4-th highest, we need x ≥ 81. So P4 chooses 81 as her second marginal valuation, and she gets two items for a total value 171. If P4 were not there, her items would both go to P1 and P3, for a total value 155. Therefore P4 pays 155. 1 Exercise 2 5 points Given the bargaining problem C = {(x, y) : 0 ≤ x ≤ 1 , 0 ≤ y ≤ 1 , x + ky ≤ 2} , d = (0, 0) , find the Nash solution for all k ≥ 0. Explain your answer. Answer of exercise 2 If 0 ≤ k ≤ 1, then C = {(x, y) : 0 ≤ x ≤ 1 , 0 ≤ y ≤ 1}, which is symmetric, so that the Nash solution is (1, 1). If k > 1, then the solution must be on the segment S defined by x + ky = 2, x ≤ 1, y ≤ 1. We need to find the largest value of c such that the hyperbola xy = c intersects S. Since the line x + ky = 2 has slope strictly larger than −1, then the intersection is on the line x = 1, and therefore the intersection point, that is the Nash solution, is (1, 1/k). 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. Given a symmetric game defined…