1662537926 June 2021

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GAME THEORY June 11, 2021 Surname: Name: Matricola: Exercise 1 Given the following bimatrix game, depending on the parameters a,b∈ R,   (12, 6) (4 , 4) (3 , 5) (3, 6) (8 , 5) (2 , 7) (a, 3) (6 , 7) (3 ,b )  , 1. find the pure Nash equilibria for every a,b ; 2. find the best reaction of the first player to ( 1 3, 1 3, 1 3 ); 3. find a,b such that the game is a potential game (if any); 4. find a,b such that there is a Nash equilibrium with second player playing ( 1 3, 1 3, 1 3 ); 5. show that for every a,b there is a Nash equilibrium with the first player playing a pure strategy and the second one a mixed strategy where exactly two pure strategies are played with strictly positive probability. Answer of exercise 1 1. The pure Nash equilibria outcomes are (12, 6) for every a,b such that a≤ 12 and (3,b ) if b≥ 7; 2. The expected utilities form the three rows (multiplied by 3) are respectively 19, 13, 9 + a. Thus the best reaction of the first player to ( 1 3, 1 3, 1 3 ) is (1, 0, 0) if a< 10, (0, 0, 1) if a> 10, and (p, 0, 1−p), 0≤p≤ 1 if a = 10. 3. A candidate potential (up to a constant) should look like   6 4 5 −3 8  . However, looking at the second player, with second row fixed for the first one, we see that 6− 5̸=−3− 8 thus for no a,b the game is a potential game; 4. For no pure strategy of the first player the best reaction of the second one is ( 1 3, 1 3, 1 3 ). Thus, if such a strategy exists, it must be a = 10 and such a strategy should be of the form (p, 0, 1−p). If player one plays (p, 0, 1−p), in order that the second mixes among the three columns, all of them must be optimal, thus providing the same expected value. Thus 6p + 3− 3p = 4p + 7− 7p = 5p +b−bp providingp = 2 3,b = 5; 5. the only possible case is when b = 7 and first player playing the last row,…