30 08 2022 E TS

Pagina 1

GAME THEORY - August 30, 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 4 7 1 0 4 2 4   find a pair of optimal strategies for the players and the value of the game. Answer of exercise 1 First of all, notice that vI ̸= vII , therefore there are no optimal pure strategies. We observe that no strategy is strictly dominated. Then we can look for a solution where the strategies of both players are full mixed, using the indifference principle. This leads to strategy (7/12, 1/12, 1/3) for Pl1 and (1/2, 1/6, 1/3) for P2. The value of the game is 11/3. 1 Exercise 2 5 points Given the game (N, v) with N = {1, 2, 3} and v({i}) = 0 , v({1, 2}) = v({1, 3}) = 1 , v({2, 3}) = 0 , v(N ) = 3 , represent the core of the game. Answer of exercise 2 The core is the convex hull of the set of vectors {(1, 0, 2), (0, 1, 2), (0, 2, 1), (3, 0, 0), (1, 2, 0)} 2 Theory Questions Answer one and only one question. Only the question 2 may lead to the top grade 30 e lode. 1. Define the core for a TU game and explain its meaning. (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