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Game Theory 5 cfu - 11/07/2016 Tot = Surname: Name: Matricola: Exercise 1 (2+3+4) Consider the following bimatrix game: 0 @ (1, 0) (3 ,b )( 3 , 2) (2, 4) (3 , 0) (2 , 3) (0, 1) (3 , 0) ( a, 3) 1 A , where a, b are real numbers. 1. Find the Nash equilibrium profiles for all a, b in pure strategies; 2. Find the best reaction of the second player to the mixed strategy (0 ,p , 1 p) of the first player; 3. Find when (0 , 2/3, 1/3) is part of a Nash equilibrium profile as a strategy for the first player. Exercise 2 (1+2+3+3) Let ( N, v) be the TU-game defined as follows: N = {1,...,n } and for S ✓ N : v(S)= ⇢ |S| if S \ {1, 2} 6= ; and |S|  4 0 otherwise . 1. Are there any symmetric players? 2. Compute the Shapley value for n = 3. 3. Find the core for n = 3. 4. Say for which n the core of the game is empty. 15 Excercise 3 (3+3+3) 1. Consider the Take-Away game in which each player can take 1,3 or 5 chips and the winner is the last person who moves. Find the P-positions of this game. Who wins if there are 21 chips? 2. Given the following zero-sum game: ✓ 113 a 21 ◆ , • say for which a the second player’s optimal strategy is to play a combination of the second and third column; • how many optimal strategies does the first player have when a = 1? 3. Consider the following game in extensive form: e f c g h d a c d b I II II I I (3,1) (2,1) (0,0)(1,1) (3,0) (-1,2) (a) Write all the strategies of the two players. (b) Solve the game by using backward induction. First theoretical question (4 points) Define the concept of correlated equilibrium and show that a Nash equilibrium gives rise to a correlated equilibrium. Second theoretical question (4 points) Illustrate by examples the idea of TU game. 16