13 07 15 1

Page 1

Game Theory 13-7-2015 Surname: Name: Matricola: Exercise 1 (2+2+4 points ) Given the following bimatrix game: 0 @ (3, 2) (0 , 1) (3, 1) (1 , 3) (a, 0) ( 1, 2) 1 A 1. find the equilibria in pure strategies; 2. find for di ↵erent values of a the best reply of the first player to the strategy ( 1 4 , 3 4 ) of the second player; 3. find all the Nash equilibria if a = 2. Solution 1. The Nash equilibria in pure strategies are (1 , 3) for any value of a,( 3 , 2) if a  3. 2. If the second player plays ¯y = 1 4 , 3 4 , then we have to evaluate the expected payo ↵s for the first player from each row: EI ((1, 0, 0), ¯y)= 3 4 EI ((0, 1, 0), ¯y)= 6 4 EI ((0, 0, 1), ¯y)= a 3 4 If a< 9 the best reply is to play (0 , 1, 0); if a> 9 the best reply is (0 , 0, 1) and if a =9t h eb e s tr e p l y is (0 ,p , 1 p) for any value of p 2 [0, 1]. 3. If a = 2 the third row is dominated by the other two. So the game is reduced to ✓ (3, 2) (0 , 1) (3, 1) (1 , 3) ◆ There are the Nash equilibria in pure strategies: {(1, 0, 0), (1, 0)} and {(0, 1, 0), (0, 1)}.T o fi n d t h e equilibria in mixed strategies, the utility functions of the two players are: f (p, q)=( q 1)p +1+2 qg (p, q)=( 5 p 4)q +3 2p. The best reply functions are BRI = ( p 2 [0, 1] if q =1 p =0 i f q< 1 BRII = 8 >< >: q =1 i f p> 4 5 q 2 [0, 1] if p = 4 5 q =0 i f p< 4 5 The Nash equilibria in mixed strategies are {(p, 1 p), (1, 0)} with 4 5  p  1. 35 Exercise 2 (2+2+2+2 points ) Given the weighted majority game with four players: v =[ q; 40, 30, 15, 15] where q = 65 is the quota: 1. enumerate the winning coalitions; 2. find the core of the game; 3. find the Shapley value; 4. find the Banzhaf value. Solution 1. This is a weighted majority game with four players, then v(S)=1i ↵ P i2S xi 65. The winning coalitions are: {1, 2}, {1, 2, 3}, {1,…