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T eoria Matematica dei Giochi - 5 CFU 26-9-2014 Cognome: Nome: Matricola: Exercise 1 Given the game ( N, v)w i t hn = |N | and v(S)=2 s s2 8S : |S| = s 1 1. prove that if n = 3 the core is empty; 2. find the Shapley and the Banzhaf values if n = 3; 3. find the Shapley and the Banzhaf values for all n; Solution 1. If n = 3, the game is given by v(i)=1 8i 2 N , v(i, j)=0 8i, j 2 N and v(N )= 3. The core is given by {(x1,x 2,x 3)} such that xi 1 8i 2 N , xi + xj 0 8i, j 2 N and x1 + x2 + x3 = 3. So the core is empty. 2. Using symmetry and e ciency we get that the Shapley value is i(v)= 1 8i 2 N .Computing the Banzhaf value we find i(v)= 1. 3. Shapley value is i(v)=2 n and Banzhaf value is i(v)= 1 2n 1 P n 1 s=0 n 1 s (1 2s) 8i 2 N . 35 Exercise 2 Consider the following game in extensive form 1. solve this game using backward induction; 2. write the game in strategic form and find all Nash equilibria (in pure and mixed strategies). Solution 1. Second player plays D,fi r s tp l a y e rp l a y sB and the solution of the game is (2 , 2). 2. The strategic form of the game is given by the bimatrix ✓ (1, 1) (1 , 1) (0, 1) (2 , 2) ◆ The Nash equilibria in pure strategies are (1 , 1) and (2 , 2) corresponding to the strategies ( A, C) and (B, D). The Nash equilibria in mixed strategies are in the form (1, 0), (q, 1 q) with 1 2  q  1. 36 Exercise 3 Given the bimatrix: 0 @ (1, 1) (3 , 2) (1 , 0) (2, 2) (0 , 1) (0 ,a ) (0, 0) (1 , 2) (3 , 1) 1 A 1. is there any Nash equilibrium in mixed strategies, such that the second player plays a pure strategy? 2. if a< 1, is there any Nash equilibrium such that the second player plays a strategy like ( q, 1 q, 0)? 3. for which values of a is there a Nash equilibrium with strategies (p, 1 p, 0), (q, 1 q, 0) with 0 <p< 1 and 0 <q< 1? Solution 1.…