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Game Theory - Simulated Exam Solution 12-12-2014 Surname: Name: Matricola: Exercise 1 Let ( N, v) be the cooperative game N = {1,...,n }: v(S)= |S|X i=1 i 1. find and draw the core of the game if n = 3; 2. find the nucleolus of the game for every n; 3. find the Shapley value for every n. Solution 1. The core is the convex hull of the point {(1, 2, 3), (2, 1, 3), (3, 1, 2, ), (3, 2, 1), (2, 3, 1), (2, 3, 1), (1, 3, 2)} 2. The sum of the first n numbers is n(n+1) 2 . Using e ciency and symmetry we find that the nucleolus is n+1 2 ,..., n+1 2 ) 3. The Shapley value is equal to the nucleolus. 1 Exercise 2 Given the bimatrix game: 0 @ (a, 3) (2 , 2) (4, 1) (3 , 2) (0, 3) (4 ,b ) 1 A , 1. find the Nash equilibria in pure strategies for di↵erent values of a, b 2 R; 2. find the Nash equilibria in mixed strategies if a< 0,b = 1; 3. find the correlated equilibria of the form 0 @ xy z 0 00 1 A if a = 5 and b = 1. Solution 1. The equilibria in pure strategies are (4 ,b )i f b 3 and ( a, 3) if a 4. 2. If a< 0,b = 1 there is only one equilibrium in mixed strategies given by (0, 2 3 , 1 3 ), ( 1 5 , 4 5 ) . 3. If a = 5 and b = 1, there is only one correlated equilibrium of that form with x = 1 and y = z = 0. 2 Exercise 3 1. Which player has a winning strategy in the Nim game (10 , 2, 7)? Why? 2. Find the Nash equilibria of the two player game ( X, Y, u, v)w h e r eu(x, y)= x2 +2 xy +4 i s t h e u t i l i t y of player I and v(x, y)= y2 +4 xy 4y is the utility of player II, in the following cases (a) X = Y =[ 0, 10] (b) X =[ 0, 1] and Y =[ 0, 10]. 3. How many strategies do the players have in the following game? Which are the strategies? E F C D A C D B I II II I Solution 1. The first player has a winning strategy because the starting position is a N-position. A winning move is to take 5 from…