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T eoria Matematica dei Giochi - 5 CFU 14-7-2014 Cognome: Nome: Matricola: Exercise 1 Given the zero sum game 0 @ 84 a 48 a 55 1 1 A with a 2 R, a> 0: 1. prove that all first player’s optimal strategies are in the form ( p, 1 p, 0); 2. find optimal strategies for both players if a> 6; 3. find optimal strateies for both players if a = 6. Solution 1. Since a> 0 the thirs row is dominated by a convex combination of first and second rows. The matrix reduces to ✓ 84 a 48 a ◆ First player never plays the third row, so is optimal strategy is ( p, 1 p, 0). 2. If a> 6, third column is dominated. From the indi ↵erence principle we get the optimal strategies ( 1 2 , 1 2 , 0), ( 1 2 , 1 2 , 0) . 3. If a = 6, the optimal strategies are ( 1 2 , 1 2 , 0), (q, q, 1 2q) with 0  q  1. 27 Exercise 2 Let a, b > 0.Given the bimatrix: 0 @ (1, 0) (0 , 0) ( a, 0) (0, 1) (2 ,b )( 1 , 1) (1, 1) (0 , 0) (2 , 1) 1 A 1. find all Nash equilibria in pure strategy; 2. find correlated equilibria in the form 0 @ x 00 000 y 0 z 1 A with a =1e b> 0; 3. Optional If a> 2 find if there is any value of b such that the strategy ( 1 3 , 1 3 , 1 3 ) is an equilibrium for the second player. Solution 1. (1 , 0) e (1 , 1) are Nash equilibria 8a, b 2 R. Other Nash equilibria are: ( a, 0) if a 2, (2 , 1) if a  2 and (2 ,b )i f b 1. 2. Correlalated equilibria are all the ones such that x, y, z 0 and x + y + z = 1. 3. If a> 2, the strategy (1, 0, 0), ( 1 3 , 1 3 , 1 3 ) is an equilibrium 8b. 28 Exercise 3 Let N = {1,...,n } and let ( N, v) tale che H ` e il sottoinsieme dei giochi ( N, v) tali che 8 S, T ⇢ N , S \ T = ; v(S [ T )= v(S)+ v(T ). 1. show that H is a subspace of the space of all games. What is the dimension of H? 2. show that for each fixed v 2 H, the core, the nucleolus, the Shapley and the Banzhaf values are…