05 09 2023 E TS

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GAME THEORY - September 5, 2023 Last name: First name: ID #: SOLVE THE EXERCISES AND ANSWER THE QUESTIONS ON THESE SHEETS Exercise 1 5 points Find a value of x ∈ N such that the following game has a unique Nash equilibrium.   (6, 3) ( x, 6 − x) (4 , 1) (2, 1) (6 , 4) (4 , 2) (5, 4) (7 , 8) (6 , 2)   Answer of exercise 1 The second row is strictly dominated by the third one. Then, the last column is strictly dominated by the first one. Then, if 6 − x > 3, that is x < 3, the second column strictly dominates the first one, and finally the last row dominates the first one. If x = 0, 1, 2, then (7, 8) is the only NE. 1 Exercise 2 5 points Consider the game (N, v) with N = {1, 2, 3} and v({i}) = 0 , v ({1, 2}) = 1 , v ({1, 3}) = v({2, 3}) = 4 , v (N ) = a . Let C(v) and ν(v) be its core and nucleolus. 1. Find all values of a such that C(v) = ∅. 2. Find ν(v) when a = 8 Answer of exercise 2 1. The core is C(v) = {(x1, x2, x3)} with x1 + x2 + x3 = a, 0 ≤ x1 ≤ a − 4, 0 ≤ x2 ≤ a − 4, 0 ≤ x3 ≤ a − 1; then, if a < 4 the core is empty. On the other hand, if a ≥ 4 the game is super-additive: therefore the core is empty if and only if v({1, 2}) + v({1, 3}) + v({2, 3}) > 2v(N ), that is, if and only if a < 9/2. 2. As player 1 and 2 are symmetric, the nucleolus is an imputation of the form (x, x, 8 − 2x), with 0 ≤ x ≤ 4. The excesses are −x, 2x − 8, 1 − 2x, x − 4. The point that minimises the maximum excess lies at the intersection between the straight lines −x and x − 4. The two lines cross at x = 2 . The nucleolus is then ν(v) = (2 , 2, 4). 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 Bouton’s theorem, explaining what the Nim sum is. 2. Prove that the set of natural numbers with the Nim sum is a…