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Game Theory (Solutions) 10-07-2017 Surname: Name: Matricola: SOL VE THE EXERCISES AND ANSWER THE QUESTIONS USING ONLY THESE PAPERS Exercise 1 ( 2+3+4 ) Given the folllowing bimatrix, with a, b 2 R: ✓ (a, b)( 1 , 2) (2, 3) (4 , 0) ◆ , 1. find all Nash equilibria in pure strategies 8a, b; 2. find all a, b such that {(1, 0), ( 1 2 , 1 2 )} is a Nash equilibrium; 3. prove that in such a case there are always infinite Nash equilibria. Solution 1. If a 2 and b 2 there is an equilibrium in pure strategies with outcome ( a, b). If a  2 there is an equilibrium in pure strategies with outcome (2 , 3). 2. The given strategies profile is a Nash equilibrium if and only if b = 2 and a 5. 3. If b = 2 and a 5 the equilibria are {(1, 0), (q, 1 q)} with q 3 a+1 . 1 Exercise 2 ( 2+2+4 ) Let ( N, v) be the a TU-game where N = {1, 2,...,n }, v(;) = 0 and v(S)= ( |S| 1i f |S| is odd |S| + 1 otherwise. 1. Find the Shapley value for all n; 2. find the nucleolus for all n; 3. for which n is the core of the game nonempty? Solution 1. All players are symmetric. Since the Shapley value satisfies symmetry and e ciency, for any i 2 N i(v)= v(N ) n ( 1 1 n if n is odd 1+ 1 n otherwise. 2. In this game the nucleolus coincides with the Shapley value. 3. If n = 2 the core is nonempty. If n 3 the core is empty, since the nucleolus does not belong to it. 2 Exercise 3 - 8 cfu ( 2+4+3) 1. Find all n, m such that (3 ,n , 4,m ) is a P-position in the Nim game. 2. Let A = {A, B, C, . . . Z } be the set of alternatives and let F be the social welfare function assigning to every preference profile the ranking Z % ··· % B % A. Does F fulfil unanimity and independence from irrelevant alternatives? Is F dictatorial? Is F manipulable? Explain your answer. 3. Find a matching with three men and three women with the…