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Game Theory 5 cfu 24-2-2015 Surname: Name: Matricola: Exercise 1 (1+2+4+3 points ) Given the bimatrix game: ✓ (1, 3) (2 , 1) (1 , 0) (0, 1) (4 , 2) ( a, b) ◆ , 1. find the Nash equilibria in pure strategies for di ↵erent values of a, b 2 R; 2. find the best reply of the first player to the strategy ¯ y = 1 2 , 0, 1 2 of the second one; 3. find a, b such that ¯y belongs to a Nash equilibrium profile; 4. find all the Nash equilibria for b< 2. Solution 1. The Nash equilibria in pure strategies are (1 , 3) for any value of a,( 4, 2) for any value of a and if b  2, (a, b)i f a 1 and b 2. 2. If the second player plays ¯ y = 1 2 , 0, 1 2 , then we have to evaluate the expected payo ↵s for the first player from each row: EI ((1, 0), ¯y)=1 EI ((0, 1), ¯y)= a 2 If a< 2 the best reply is to play (1 , 0); if a> 2 the best reply is (0 , 1) and if a =2t h eb e s tr e p l yi s (p, 1 p) for any value of p 2 [0, 1]. 3. To find the Nash equilibria we have to check if the strategy ¯ y is a best response for player II to the best responses of player I (that we found in the previous point). • If a< 2 and I plays (1 , 0) then there is not a Nash equilibrium because the second player gets more from the first column than from the third one. • If a> 2 and I plays (0 , 1) then there is not a Nash equilibrium because the second player gets more from the second column than from the second one. • If a = 2, then we can find p such that ¯y is a best response to the strategy ( p, 1 p). First of all we have to check that the second player gets the same from the columns he is playing with positive probability (the first and the last one), then we have to check that what he gets from the second row (played with null probability) is less than what he gets from the other two: 3p +1 p = b(1 p)= ) p = b 1 b +2 p +2…