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Game Theory 15-12-2014 Surname: Name: Matricola: Exercise 1 (2+5+4p o i n t s ) Given the bimatrix game: 0 @ (a, 2) (2 , 1) (3 , 0) (1, 0) (5 , 0) (0 ,b ) (2, 0) (5 , 3) (0 , 5) 1 A , 1. find the Nash equilibria in pure strategies for di ↵erent values of a, b 2 R; 2. find a, b such that the strategy of the first player 1 4 , 1 4 , 1 2 belongs to a Nash equilibrium profile; 3. for every a, b, find the correlated equilibria of the following form: 0 @ xyz 000 000 1 A Solution 1. The equilibria in pure strategies are ( a, 2) if a 2 and (5 , 0) if b  0. 2. Suppose the first player plays x = 1 4 , 1 4 , 1 2 , then we have to evaluate the expected payo ↵s for the second player from each columns: EII (x, (1, 0, 0)) = 1 2 EII (x, (0, 1, 0)) = 7 4 EII (x, (0, 0, 1)) = 10 + b 4 The first column is dominated by the second one. We have to find for di ↵erent values of b the best reaction of player II. If 7 4 < 10+b 4 ,i . e . b> 3, then the best reaction of player II is (0 , 0, 1). These strategies are not a Nash equilibrium because player I will get more playing the first row than the second or the third. If b< 3 the best reaction of player II is (0 , 1, 0) but this is neither part of a Nash equilibrium. Instead if b = 3 the best reaction of player II is (0 ,q , 1 q)w i t hq 2 (0, 1). This strategy is part of a Nash equilibrium if the first player gets the same from all the rows played with positive probability. We can apply the indi ↵erence principle: 2q +3 3q =5 q =5 q =) q = 1 2 . So if b = 3, the Nash equilibrium is given by { 1 4 , 1 4 , 1 2 , 0, 1 2 , 1 2 } 5 3. To find the correlated equilibria we have to find the non-negative x, y, z such that: 8 >>>>>>>>>>>< >>>>>>>>>>>: ax +2 y +3 z x +5 y ax +2 y +3 z 2x +5 y 2x x y 2y 0z 2z 0z z x + y + z =1 These conditions imply y =0 ,z = 0 and…