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GAME THEORY - February 14, 2022 Last name: First name: ID #: SOLVE THE EXERCISES AND ANSWER THE QUESTIONS ON THESE SHEETS Exercise 1 5 points Given the zero-sum game described by the following matrix: A =   3 5 7 2 2 3 4 3 5 2 3 5   find a pair of optimal strategies for the players and the value of the game. Answer of exercise 1 First of all, notice that vI̸= vII , therefore there are no optimal pure strategies. Then we observe that the second row is strictly dominated by the average of the other rows. Thus the game can be reduced to A = (3 5 7 2 5 2 3 5 ) If the first player chooses (p, 0, 1− p), then the outcome for the second player is represented by the following picture: 1 p Then the P2 chooses (0, 1/2, 0, 1/2), by the indifference principle P1 chooses p = 1/2 and the value of the game is 7/2. 1 Exercise 2 5 points Given the TU game: N ={1, 2, 3}, v({i}) = 0 , v({1, 2}) = a, v({1, 3}) = 1 , v({2, 3}) = b, v(N ) = 2 , find a, b such that the core of the game is a singleton, and explain why that is the case. Answer of exercise 2 For (x1, x2, x3) to be in the core we need x1 + x2≥ a, x1 + x3≥ 1, x2 + x3≥ b, and x1 + x2 + x3 = 2. So x3 = 2− x1− x2 and then x2≤ 1, x1≤ 2− b and x1 + x2≥ a. In order to have a unique solution of these inequalities, the corresponding equalities must be satisfied, so b = 3− a and then x1 = a− 1, x2 = 1, x3 = 2− a, with a∈ [1, 2]. 2 Theory Questions Answer one and only one question. Only the question 2 may lead to the top grade 30 e lode. 1. State Nash bargaining theorem. (7 points) 2. Define TU games and Unanimity games. Prove that the set of Unanimity games is a basis for the set of TU games. (7 points) 3