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Game Theory (Solutions) 07-20-2017 Surname: Name: Matricola: SOL VE THE EXERCISES AND ANSWER THE QUESTIONS USING ONLY THESE PAPERS Exercise 1 ( 5+4 ) Given the folllowing bimatrix, with a, b 2 R: ✓ (a, b)( 1 , 2) (2, 3) (4 , 0) ◆ , 1. find all Nash equilibria for a 6= 2, b 6= 2; 2. find all correlated equilibria for a 6= 2, b 6=2 Solution We have three di ↵erent cases: • If a< 2 thanks to the elimination of dominated strategies, the outcome is (2 , 3). Thus, there is a unique equilibrium in pure strategies: {(0, 1), (1, 0)} and a unique correlated equilibrium ✓ 00 10 ◆ . • If a> 2 and b> 2 thanks to the elimination of dominated strategies, the outcome is ( a, b). Thus, there is a unique equilibrium in pure strategies: {(1, 0), (1, 0)} and a unique correlated equilibrium ✓ 10 00 ◆ . • If a> 2 and b< 2 there is not an equilibrium in pure strategies. Using the indi ↵erence principle, we find the unique equilibrium in mixed strategies: {( 3 5 b , 2 b 5 b ), ( 3 a+1 , a 2 a+1 )} Since there is only one equilibrium and it is in mixed strategies, there is only one correlated equilibrium and it is the one associated to the equilibrium in mixed: ✓ pq p (1 q) (1 p)q (1 p)(1 q) ◆ with p = 3 5 b and q = 3 a+1 . 1 Exercise 2 ( 2+3+2+2 ) Let ( N, v) be the a TU-game where N = {1, 2, 3}, v(;) = 0 and v(i)=1 8i, v(1, 2) = v(1, 3) = 2 ,v (2, 3) = 4 ,v (N )= a. 1. find a such that the game is superadditive; 2. find a such that the core of the game is nonempty and find the core for a = 5; 3. find the the nucleolus for a = 5; 4. find the Shapley value for any a. Solution 1. The game is superadditive if for any S, T 2 2N such that S \ T = ; it holds v(S [ T ) v(S)+ v(T ). If we take S = {2, 3} and T = {1},w efi n d a 5. For these values of a the other inequalities are satisfied, too. 2. The core of…