22 06 2022 E TS

Page 1

GAME THEORY - June 22, 2022 Last name: First name: SOL VE THE EXERCISES AND ANSWER THE QUESTIONS ON THESE SHEETS Exercise 1 5 points Given the following game   (a, b) (1 , 2) (2 , 1) (2, 1) (2 , 0) (5 , 3) (0, 2) (0 , 3) (3 , 2)   . 1. find all correlated equilibria when a < 2; (2 points) 2. find a correlated equilibrium, which is not a NE, when a = b = 3 (3 points). Answer of exercise 1 1. By iterated elimination of strictly dominated strategies on e sees that (5, 3) is the only equilibrium, thus the unique CE has probability 1 to the corresponding strategy. 2. In this case also (3, 3) is a NE, and thus CE for the game are   1 − p 0 0 0 0 p 0 0 0   . for 0 < p < 1. Exercise 2 5 points Given the cooperative game (N, v), where v(S) = { s, if 1 /∈ S s − 1, otherwise. 1. Find the Shapley value (2 points); 2. find the nucleolus (2 points); 3. find the core (1 point). Answer of exercise 2 1. Pl1 is a null player the other are symmetric, thus σ(v) = (0 , 1, 1, . . .1); 2. same as Shapley; 3. same as Shapley. Theory Questions Answer one and only one question. Only the question 2 may lead to the top grade 30 e lode. 1. Define the Shapley value for a TU game and explain its meanin g. (7 points) 2. Define the Shapley value for a TU game and prove Shapley’s the orem. (7 points)