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Game Theory 5 cfu - 24/02/2016 Tot = Surname: Name: Matricola: Exercise 1 (2+3+4) Given the following zero-sum game: ✓ 224 24 a ◆ , 1. find the conservative values of the players and say if there exists a such that there are equilibria in pure strategies; 2. find all the optimal strategies for the players for a 2; 3. find all the optimal strategies for the players for 0  a< 2. Exercise 2 (2+2+2+2) Let ( N, v) be the a TU-game where N = {1, 2, 3}, v({1})= v({2}) = 1, v({3}) = 0, v({1, 2}) = 2, v({1, 3})= v({2, 3}) = 3, v(N )= a. 1. Say for which a the core is nonempty. 2. Find and draw the core for a = 5. 3. Find the nucleolus for a = 5. 4. Compute the Banzhaf value for a = 5. 9 Excercise 3 (3+3+4) 1. Consider the Nim game with starting position (10 , 4, 12, 2). Suppose that player I can only take chips from the third pile when he makes the first move. Which player has a winning strategy? How many winning moves does he have when he plays for the first time? 2. Given the bimatrix game: 0 @ (1, 0) (3 , 1) (0, 1) (2 , 0) (0, 0) ( a, 1) 1 A , (a) find the equilibria in pure strategies for di ↵erent values of a; (b) find the best reply of the first player to the strategy ¯ y =( 1 3 , 2 3 ) of the second one for di ↵erent values of a. 3. Consider the following game in extensive form. d e a f g b 1/3 2/3 c I II II N (1,2) (1,-1) (1,5) (5,4) (-2,2) (4,1) (a) Write all the strategies of the two players. (b) Solve the game using backward induction and say which are the optimal strategies for the two players. First theoretical question (4 points) Discuss the notion of correlated equilibrium and its relation with the concept of Nash equilibrium profile. Second theoretical question (4 points) Prove the characterization of the Shapley value for TU-games. 10