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Game Theory 14-9-2015 Surname: Name: Matricola: Exercise 1 (8p o i n t s) Given the following zero sum game: ✓ 24 a 31 a ◆ , 1. find the conservative values of the two players; 2. find for which a there is an equilibrium in pure strategies and which are the optimal pure strategies of the players; 3. (5 cfu) find the optimal strategies of the players if a =3 (8 cfu) find the optimal strategies of the players for di ↵erent values of a> 2. Solution 1. The conservative values of the two players are, respectively, vI = ( a if a  2 2i f a> 2 vII = ( a if a  3 3i f a> 3 2. There is an equilibrium in pure strategies if and only if vI = vII , i.e. for a  2. The optimal strategies for the players if a  2 are (1, 0), (0, 0, 1) and if a  1 there is also (0, 1), (0, 0, 1) . 3. (5 cfu) if a = 3, the third column is dominated by a combination of the first and second column and the optimal strategies of the players are given by ( 1 2 , 1 2 ), ( 3 4 , 1 4 , 0) . (8 cfu) if 2 <a< 5 2 , the optimal strategies of the players are given by (p, 1 p), (0, 0, 1) ,w h e r e a 1 3  p  3 a.I f a = 5 2 , the optimal strategy of the first player is given by ( 1 2 , 1 2 ) and the optimal strategy of the second player is a combination of all the three columns ( q1,q 2, 1 q1 q2) such that 0  q1 < 3 4 , q2 = 1 3 q1.I f a> 5 2 , the third column is dominated by a combination of the first and second column and the optimal strategies of the players are given by ( 1 2 , 1 2 ), ( 3 4 , 1 4 , 0) . 39 Exercise 2 (8p o i n t s) Let ( N, v) be the cooperative game N = {1, 2, 3} such that v({1})= v({2})=0 v({3})=2 v(N )= a v({1, 2})=2 v({1, 3})=0 v({2, 3})=1 1. For which values of a is the core empty? 2. Find and draw the core if a = 4; 3. find the Shapley value of this game 8a; 4. find the Banzhaf value of this…