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Game Theory 5 cfu - 29/09/2016 Tot = Surname: Name: Matricola: Exercise 1 (1+2+3+3) Given the strategic game: ✓ (4, 4) (0 ,a )( 2, 3) (a, 0) (2 , 2) ( 3, 2) ◆ , 1. say if there is a NE with (1 /2, 0, 1/2) as a strategy for the second player; 2. find all a’s such that ((1 , 0), (1, 0, 0)) is a NE; 3. find all NE for a = 4; 4. find all correlated equilibria for a 4. Exercise 2 (3+2+2+2) In a committee of 7 people there are two veto players, and the majority is obtained with at least other two joining the veto players. 1. show that the situation can be described as a weighted majority game; 2. find the Shapley value of the players; 3. find the Banzhaf value of the players, 4. find the nucleolus of the associated game. 21 Excercise 3 (3+3+3) 1. Consider the following game in extensive form: e f c g h d a c d b I II II I I (1,1) (2,-1) (x,2)(1,1) (2,0) (-1,3) (a) Write all the strategies of the two players. (b) Solve the game for di ↵erent values of x by using backward induction. 2 Find the solutions of the following zero sum game: 0 @ 100 020 003 1 A . 3 Given the TU game: N = {1, 2, 3}, v({i}) = 0, v({1, 2})= a, v({1, 3})= v({2, 3}) = 1, v(N ) = 2, find a such that the core of the game is a singleton. In such a case find the nucleolus. First theoretical question (4 points) The Nim game. Second theoretical question (4 points) Existence of Nash equilibria for finite games. 22