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Game Theory (5 CFU) 01-02-2017 Surname: Name: Matricola: SOL VE THE EXERCISES AND ANSWER THE QUESTIONS USING ONLY THESE PAPERS Exercise 1 ( 1+2+3+3 ) Given the folllowing bimatrix, with a, b 2 R: 0 @ (2, 1) (1 , 3) (4 ,a ) (1, 0) (2 , 1) (0 , 2) (b, 3) (0 , 0) (3 , 2) 1 A , 1. find the Nash equilibria in pure strategies for di ↵erent values of a, b 2 R; 2. find the best reply of the second player to the strategy x =( 1 4 , 3 4 , 0) of the first player; 3. find for which values of a, b 2 R, x is part of a Nash equilibrium; 4. find all the Nash equilibria for a = 2 and b = 1. Solution 1. The Nash equilibria in pure strategies are (4 ,a )i f a 3 and ( b, 3) if b 2. 2. If Player I plays x, then for each of her pure strategies the expected outcome of II are: EII [x, (1, 0, 0)] = 1 4 EII [x, (0, 1, 0)] = 6 4 EII [x, (0, 0, 1)] = a +6 4 Thus, BRII = 8 >< >: (0, 1, 0) if a< 0 (0,q , 1 q)i f a =0 (0, 0, 1) if a> 0 3. If a< 0, x is not part of a Nash equilibrium since BRII (x)=( 0 , 1, 0) but BRI (0, 1, 0) = (0 , 1, 0). If a> 0, x is not part of a Nash equilibrium since BRII (x)=( 0 , 0, 1) but BRI (0, 0, 1) = (1 , 0, 0). If a = 0, x is part of a Nash equilibrium if and only if BRI (0,q , 1 q)= x. So the first player should be indi ↵erent among the first and the second row, and he should get more from these rows than from the third one: q + 4(1 q)=2 q 3(1 q)= ) q = 4 5 x is part of a Nash equilibrium for any value of b and if a = 0, the second player plays (0 , 4 5 , 1 5 ). 4. If a = 2 and b = 1 the third row is dominated by the first one, then the first column is dominated by the second. Thus, the game is reduced to ✓ (1, 3) (4 ,a ) (2, 1) (0 , 2) ◆ , and there are not equilibria in pure strategies. We can apply the indi ↵erence principle: 3 p +1 p = 2 q + 4(1 q)=2 q and get the Nash…