Computer Engineering - Game Theory

Full exam

GAME THEORY - August 30, 2022 Last name: First name: ID #: SOLVE THE EXERCISES AND ANSWER THE QUESTIONS ON THESE SHEETSExercise 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 thatv I̸ =v I I, 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 is11/3. 1 Exercise 2 5 points Given the game(N , v)withN={1,2,3}andv({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 Answerone and only onequestion. Only the question 2 may lead to the top grade30 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 TUgames. (7 points) 3