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T eoria Matematica dei Giochi - 5 CFU 3-3-2014 Cognome: Nome: Matricola: Exercise 1 Given the following game in extensive form: 1. solve it using backward induction; 2. write the game in strategic form and find Nash equilibria (in pure and mixed strategies); Solution 1. Using backward induction we find that player I playes B,w h i l ep l a y e rI Ip l a y sT2B2 and the outcome of the game is (12 , 10). 2. Player I’s strategies are B, T . Player II’s strategies are T1B1 ,T1B2 ,T2B1, T2B2. The strategic form of the game is given by the ✓ (10, 4) (10 , 4) (8 , 8) (8 , 8) (15, 5) (12 , 10) (15 , 5) (12 , 10) ◆ , All Nash equilbria are (0, 1), (0,q , 0, 1 q) ,w i t h0  q  1. 16 Exercise 2 Given the bimatrix: 0 @ (a, b)( 2 , 3) (0 , 0) (4, 1) (3 , 0) (1 , 2) (2, 4) (3 , 3) (5 , 7) 1 A 1. find Nash equilibria in pure strategy with a, b 2 R; 2. find Nash equilibria in mixed strategy with b 2 R and a  2; 3. is there any Nash equilibrium where the first player’s strategy is ( 1 4 , 1 2 , 1 4 ) for some values of a, b 2 R? Solution 1. (5,7) is an equilibrium 8a, b, (a,b) is an equilibrium if a 4 and b 3. 2. If a  2, (5,7) is the only Nash equilibrium (we can find it using dominated strategy). 3. If b = 5 and a = 9 2 , a Nash equilibrium is given by ( 1 4 , 1 2 , 1 4 ), ( 2 3 , 0, 1 3 ) . 17 Exercise 3 Let N = {1,...,n } and consider the game ( N, v) such that 8S ⇢ N ,w i t hs = |S|: v(S)= s2 s 1. find the nucleolus and the Shapley value for all n; 2. show that the core is non-empty for all n; 3. find the core if n = 3; 4. find the Banzhaf value for all n (OPTIONAL). Solution 1. The Shapley value and the nucleolus are the same vector: n 1, ··· ,n 1 . 2. The core is non-empty, since the Shapley value is in it. 3. C(v)= co{(4, 2, 0), (2, 4, 0), (0, 4, 2), (0, 2, 4), (2, 0, 4), (4, 0,…