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GAME THEORY 5 cfu Simulated Exam - 07.11.2017 Surname: Name: Matricola: Points: Ex. 1 : /8 Ex. 2 : /8 Ex. 3-7 : /12 Theory: /7. SOLVE THE EXERCISES AND ANSWER THE QUESTIONS USING ONLY THESE PAPERS Exercise 1 Given the following bimatrix game   (a, 3) (2 , 2) (4, 1) (3 , 2) (0, 3) (4 , b)   , 1. find the Nash equilibria in pure strategies for different values of a, b∈ R; 2. find for which values of a and b there is a Nash equilibrium in which the second player plays ( 1 3 , 2 3 ); 3. find all correlated equilibria of the form   x y z 0 0 0   . 1 Exercise 2 Consider the following cooperative game (N, v) in which N ={1, 2, 3, 4} and v(S) = { |s| if{2}∈ S 0 otherwise. 1. Write the constraints to compute the core of the game; 2. find the nucleolus of the game; 3. compute the Shapley value of the game; 4. compute the Banzhaf value of the game. 2 Ex. 3 — How many strategies do players have in a chomp 2x2 game? Who wins the game and how? Ex. 4 — Show that a strictly dominated strategy can never be part of a Nash equilibrium. Ex. 5 — Provide an example of a simple weighted majority game such that the core of the game is empty. Ex. 6 — Given the nim game (12, 13, x, 9), choose x such that the second player has a winning strategy; Ex. 7 — Find the conservative values of the following zero sum game   3 2 5 1 a 2 −1 3 1   , 3 Theory Question 1. The formula of the Shapley value 2. List and illustrate its properties 3. Prove the corresponding theorem, or illustrate the formula by examples. 4