1662537864 February 2020

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GAME THEORY 5 cfu February 3 2020 Surname: Name: Matricola: Exercise 1 Given the following bimatrix game, where a, b are real parameters   (3, 2) (4 , 6) (4 , 4) (a, 3) (3 , 0) (10 , 0) (20, 0) (0 , 0) (4 , b)   1. find the Nash equilibria in pure strategies for different values of a, b∈ R; 2. find the best reaction of the first player to the strategy ( 1 3 , 1 3 , 1 3 ) of the second player; 3. for a = 25 say if there is a NE profile such that the first player plays the third row with positive probability; 4. find for which values of a and b there is a Nash equilibrium such that the first player plays only the third row. Answer of exercise 1 1. NE outcomes: (4, 6) for all a, b, (20, 0) if b≤ 0, a≤ 20, (a, 3) if a≥ 20 and all b; 2. Evaluating the utilities form the three rows we get    (0, 0, 1), if a < 11 (0, 1, 0), if a > 11 (0, p, 1− p) otherwise 3. Such a NE profile cannot exist since the last row is strictly dominated by the second one; 4. It must be b≤ 0; suppose player two plays (q, 1− q, 0). Then it must be 3q + 4− 4q≤ 20q, aq + 3− 3q≤ 20q. this implies a≤ 20, q ≥ max{ 4 21 , 3 23− a}. 1 Exercise 2 Let (N, v) be the following TU game: N ={1, . . . , n}, v(S) = ( s + 1)! (where S is a nonempty coalition and s =|S|). 1. Find the Shapley value for every n; 2. Find the core when n = 3. Answer of exercise 2 1. All players are symmetric thus the Shapley value is the vector ( (n+1)! n , . . . , (n+1)! n ); 2. The core of the game is co {(2, 4, 18), (4, 2, 18), (18, 2, 4), (18, 4, 2), (2, 18, 4), (4, 18, 2)}. Exercise 3 From a pile of cards the two players can take either 3 or 4 cards. The player that cannot move looses. 1. Tell who is the winner if there are 10 cards and how she wins; 2. Find all P and N positions. Answer of exercise 3 1. the first player, she takes 3…