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T eoria Matematica dei Giochi - 5 CFU 7-2-2014 Cognome: Nome: Matricola: Exercise 1 Given the zero sum game 0 @ 264 413 22 a 1 A 1. find the conservative values (in pure strategy), with a 2 R; 2. find for which values of a 2 R there is an equilibrium in pure strategy; 3. find the optimal strategies of the players, if a< 3. Solution 1. For all a 2 R, vI =2e vII = 4; 2. there are not equilibria in pure strategies. 3. If a< 3, the third row is dominated by a convex combination of the first and the second row, the game reduces to ✓ 264 413 ◆ The third column is dominated. We can use the indi ↵erence principle with the matrix ✓ 26 41 ◆ We get p = 3 7 and q = 5 7 . The equilibrium of the game is given by 3 7 , 4 7 , 0 , 5 7 , 2 7 , 0 . 5 Exercise 2 Given the bimatrix: 0 @ (1, 2) (2 , 3) (4 , 4) (1, 0) (4 , 2) (4 , 1) (a, b)( 1 , 0) (2 , 1) 1 A 1. find Nash equilibria in pure strategy with a, b 2 R; 2. find the best reaction of the second player, if the first one plays the strategy (0 , 1 2 , 1 2 ); 3. is there any Nash equilibrium in which first player’s strategy is ( 1 3 , 1 3 , 1 3 )? Solution 1. 8a, b,( 4 , 2) and (4 , 4) are Nash equilibria. If a 1e b 1, there is another Nash equilibirum given by ( a, b). 2. If b> 2, second player’s best reaction is (1 , 0, 0); if b = 2 second player’s best reaction is any strategy of the form q1,q 2, 1 q1 q2 ;i f b< 2 it is a strategy of the form 0,q , 1 q 3. If a = 1 and b> 4, the Nash equilibrium is 1 3 , 1 3 , 1 3 , 1, 0, 0 .I f b = 4 and a 1, the Nash equilibrium is 1 3 , 1 3 , 1 3 , 2 a+1 , 0, a 1 a+1 . 6 Exercise 3 Let N = {1,...,n } and consider the game ( N, v) such that 8S ⇢ N ,w i t hs = |S|: v(S)= es 1. Show that the core is non-empty for all n. 2. Find the Shapley value and the nucleolus for all n. 3. Find the Banzhaf value for all…