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Game Theory (5 CFU) 21-02-2017 Surname: Name: Matricola: SOL VE THE EXERCISES AND ANSWER THE QUESTIONS USING ONLY THESE PAPERS Exercise 1 ( 2+3+3 ) Given the following zero sum game: ✓ 5 a 2 b 53 ◆ , 1. for b = 4, prove that player II has a unique optimal strategy for every a 0; 2. find the optimal strategies of the players for a 0, b = 1; 3. find the optimal strategies of the players for a 0, b = 3. Solution 1. If b = 4 the conservative values of the two players coincide (they are both equal to 3) and the game has a unique equilibrium, in which player II plays the third column for every value of a 0. 2. If b = 1 the second column is not played at the equilibrium, applying the indi ↵erence principle we find the equilibrium in mixed strategies {( 2 5 , 3 5 ), ( 4 5 , 0, 1 5 )}, the value of the game is 13 5 . 3. if b = 3 the second column is still not played at the equilibrium, but the first player plays the pure strategy (0 , 1). This means that if the second player plays ( q, 0, 1 q), player I gets more from the second row than from the first: 5q + 2(1 q)  3= ) q  1 3 . The value of the game is 3 and all the equilibria of the game are given by {(0, 1)(q, 0, 1 q)} with q 2 [0, 1 3 ]. 1 p v a 0b =1 b =3 b =4 a 0 b =1 b =3 b =4 b =1 b =3 b =4 1 Exercise 2 ( 2+2+2+2 ) Let k be a real number and ( N, v) be the TU-game defined as follows: N = {1,...,n }, v(;) = 0 and for S ✓ N ,( S 6= ;): v(S)= ⇢ 2i f |S|  2 k otherwise. 1. If k = 4, for which values of n is the core empty? 2. Find and draw the core of the game for n = 3 and k = 7; 3. find the Banzhaf value of the game for n = 3 and every k; 4. find the Shapley value and the nucleolus of the game for every n and every k. Solution 1. If k = 4 the core is empty for any value of n 2. Actually if n = 2 the conditions for a vector (…