09 02 15

Page 1

Game Theory 5 cfu 9-2-2015 Surname: Name: Matricola: Exercise 1 (2+2+3 points ) Given the zero sum game: ✓ 24 a 622 ◆ , 1. find the conservative values in pure strategies of the two players; 2. find for which a there is any equilibrium in pure strategies and which are the optimal pure strategies of the players; 3. find the optimal strategies of the players if a = 3. Solution 1. The conservative values for the players are: vI =2 8av II = 8 >< >: 2i f a  2 a if 2 <a< 4 4i f a 4 2. There is an equilibrium in pure strategies i ↵ vI = vII , this is possible only if a  2. If a< 2 the optimal pure strategies at the equilibrium are (0, 1), (0, 0, 1) ,i f a = 2 there are two equilibria: (0, 1), (0, 0, 1) and (1, 0), (0, 0, 1) . 3. If a = 3 the second column is weakly dominated by the third one, but if we are looking for all the equilibria this is not enough to reduce the game. If we suppose that player I plays ( p, 1 p) and draw his expected payo ↵, we see that the equilibrium of the game is given by a convex combination of the first and the third columns. So we can reduce the game to ✓ 23 62 ◆ . From the indi ↵erence principle we get 2p +6 6p =3 p +2 2p =) p = 4 5 and 2q +3 3q =6 q +2 2q =) q = 1 5 . So the optimal strategies are ( 4 5 , 1 5 ), ( 1 5 , 0, 4 5 ) . 13 Exercise 2 (2+1+2+1 points ) Let ( N, v) be the cooperative game with |N | = n 3 such that v(S)= ( 1i f S \ {1, 3} 6= ;and S \ {2} = ; 0 otherwise 1. Prove that the core is empty for any value of n. 2. Write the value of the game for each coalition if n = 3. 3. Find the Shapley value of the game if n = 3. 4. Find the Banzhaf value of the game if n = 3. 5. (Optional) Find the Shapley value of the game for any value n. Solution 1. For any value of n we have v(N ) = 0, v({1})= v({3}) = 1 and v({i}) = 0 for all i 6=1 ,…