14 09 16

Pagina 1

Game Theory - 14/09/2016 Tot = Surname: Name: Matricola: Exercise 1 (2+3+3+1) Given the following zero-sum game: ✓ a 33 324 ◆ , 1. find the conservative values of the players and say if there exists a such that there are equilibria in pure strategies; 2. solve the game for a< 3; 3. solve the game for a> 3; 4. say how many strategies does the first player have if a = 3. Exercise 2 (4+3+2) Let (N, v) be the a TU-game where N = {1, 2, 3}, v({1})= v({2})= v({3}) = 0, v({1, 2})= a, v({1, 3})= b, v({2, 3}) = 2, v(N ) = 4, where a, b are positive real numbers. 1. Find a, b such that the core of v contains, as extreme points, (2 , 0, 2) (0 , 2, 2); for these a, b find the core and draw a picture of it; 2. Find the Shapley value when a = b. 3. Find the nucleolus for a = b = 2. 19 Excercise 3 (3+3+3) 1. Find the Nash equilibria in pure and mixed strategies of the following game ✓ (2, 0) (1 , 2) (2 , 1) (1, 3) (2 , 2) (0 , 0) ◆ 2. Give an example of a game in extensive form with perfect information in which player I has 2 strategies {A, B} and player II has 4 strategies {X, Y, W, Z}. Write the pay-o ↵s of the game in order to have has a unique optimal strategies A, Y . Can you find another example of such a game? 3. (5 cfu ) Consider the Nim game with starting position (12 , 4, 6, 7). Which player has a winning strategy? How many winning moves does he have when he plays for the first time? 3. (8 cfu ) Consider the following matching problem with W = {Catwoman, W onderwoman} and M = {Superman, Batman, F lash}. Suppose that Catwoman prefers Batman to Superman and Superman to Flash, while Wonderwoman prefers Superman to Flash and Flash to Batman. (a) If the women are visiting men, what is the stable set? (b) Find a preference profile for the men such that there is another stable set. (c)…