1662537962 November 2019

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GAME THEORY 5 cfu November 5 Surname: Name: Matricola: Exercise 1 Given the following bimatrix game, where a,b are real parameters   (3, 2) (4 , 6) (4 , 10) (a, 4) (2 ,b ) (1 , 1) (4, 3) (1 , 0) (2 , 0)  , 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 4, 0, 3 4 ) of the second player 3. find for which values of a and b there is a Nash equilibrium with support first and third strategy for both players; 4. prove that for a = 5 there is no NE profile with full support. Answer of exercise 1 1. (4, 10) for every a,b , (a, 4) if a≥ 4,b≤ 4, (4, 3) if a≤ 4. 2. utilities (multiplied by 4) from the rows: 15, a + 3, 10 . Thus    (1, 0, 0), if a< 12 (0, 1, 0), if a> 12 (p, 1−p, 0), otherwise. 3. Setting respectively (p, 0, 1−p) and (q, 0, 1−q) for the strategies of the two players, the inequalities/equalities to be checked are: 3q + 4− 4q = 4q + 2− 2q >aq + 1−q for the first player and 2p + 3− 3p = 10p> 6p providing the conditions p = 3 11, q = 2 3, a< 9 2 4. for a = 5 a convex combination of the first and second rows strictly dominates the third one. This implies that no NE can exist with full support for the first player 1 Exercise 2 Two players have three cards: the first has one 1 one 2 and one 3, the second has one 1 one 3 and one 4. They must select a card. The first player wins if the sum of the two cards is odd, otherwise the second wins. Setting 1 the utility of the victory, write the matrix of the (zero sum) game. Answer of exercise 2   −1 −1 1 1 1 −1 −1 −1 1  . Exercise 3 Consider the following cooperative game (N,v ) in which N ={1, 2, 3} and v(S) = ∑ i∈S i2, write the characteristic function of the game, find the Shapley value and the core. Answer of exercise 3 v(1) =…