1662537789 November 9 2021

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GAME THEORY 5 cfu November 9 2021 Surname: Name: Matricola: Only papers distributed with the text of the exercise must be returned for the evaluation of the exam. Solution of the Exercise below the exercise and on the back. All answers MUST be justified by reporting the main calculations. No copies with wide corrections will be graded. Points: 2 for every question Exercise 1 Given the following two player game in strategic form:   (2, 4) (5 ,a ) (2 , 2) (3, 3) (1 , 0) (0 , 4) (3, 3) (2 , 0) (2 , 4)   where a is a real parameter, 1. Find the best reaction of Player 2 to the strategy ( 1 3, 1 3, 1 3 ) of Player 1; 2. Find the mixed strategies (q1,q 2, 1 −q1 −q2) for which the best reaction of Player 1 is the first row; 3. Check, for every non negative real a, if there is a NEp with support first third row for Player 1 and first second column for Player 2; Answer of exercise 1 1. Since twice the expected utilities of the columns are respectively: 10,a, 10 the best reaction is (0, 1, 0) ifa> 10, (q, 0, 1 −q) fora< 10 and the whole simplex for a = 10; 2. The best reaction of Player 1 to the mixed strategies (q1,q 2, 1 −q1 −q2) of Player 2 is the set of non negative (q1,q 2) fulfilling the following system:    2q1 + 5q2 + 2(1 −q1 −q2) ≥ 3q1 + 1q2 + 0(1 −q1 −q2) 2q1 + 5q2 + 2(1 −q1 −q2) ≥ 3q1 + 2q2 + 2(1 −q1 −q2) q1 +q2 ≤ 1 which provides the conditions 3q2 ≥q1, q1 +q2 ≤ 1; 3. Setting p (q) the probability to play the first row (column) the conditions to check are: 2q + 5 − 5q = 3q + 2 − 2q ≥ 3q + 1 −q, 4p + 3 − 3p =ap ≥ 2p + 4 − 4p. This provides q = 3 4 and p ≥ 1 3, p = 3 a − 1, 4 ≤a ≤ 10; 1 GAME THEORY 5 cfu November 9 2021 Surname: Name: Matricola: Only papers distributed with the text of the exercise must be returned for the evaluation of the exam. Solution of the…