1662537891 July2021

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GAME THEORY 5 cfu July 14 Surname: Name: Matricola: These and only these papers must be returned for the evaluation of the exam. All answers MUST be justified by reporting the main calculations. No copies with corrections will be graded. Points: Ex. 1 : 2+3+3+3+3 Ex. 2 : 3+3+3 Ex. 3 : 2+2+3 Ex. 4 :2+3+3. SOLVE THE EXERCISES PROVIDING SHORT EXPLANATIONS USING ONLY THESE PAPERS Exercise 1 Given the following bimatrix game, where a, b are real parameters,   (3, 5) (2 , 2) (4 , 8) (2, 1) (4 , 1) (4 , 5) (3, b) ( a, 9) ( −1, 7)   , 1. Find the pure NE profiles for every a, b; 2. find a, b such that the game is a potential game; 3. find the best reaction of player II to the strategy ( 1 2 , 1 4 , 1 4 ) of Player I; 4. Say if there is a NE profile with Player I playing ( 1 2 , 1 4 , 1 4 ); 5. Say if there is a NE profile such that Player I plays (p, 0, 1− p) with 0 < p < 1. Answer of exercise 1 1. The pure NE outcomes are (4, 8) for every a, b, (3, b) if b≥ 9 and (a, 9) if a≥ 4 and b≤ 9; 2. A potential (up to a constant) could be 5 2 8 4 4 8 5 a 3 (this is built in the following way: at first write the first row by looking at the differences for the second player, thus build each column looking at the differences of the first player). In order to respect the differences of the second player in the last row, it must be a = 5, b = 9; 3. The expected values (multiplied by 4) of the columns when Player I plays ( 1 2 , 1 4 , 1 4 ) are, respectively: 11 + b, 14, 28. Thus the best reaction of Player II is: BRII ( 1 2 , 1 4 , 1 4 ) =    (1, 0, 0), if b > 17 (0, 0, 1), if b < 17 (q, 0, 1− q), if b = 17; 4. Clearly, no pure strategy of Player II can be part of a NE profile with Player I playing ( 1 2 , 1 4 , 1 4 ). Thus the only possible case is when b = 17 and thus Player II mixes between…