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Game Theory 5 cfu - 5/02/2016 Tot = Surname: Name: Matricola: Exercise 1 (2+2+2+3) Given the bimatrix game: ( (1, 2) (4 , 0) (3 , b) (1, 0) (2 , 0) ( a, 1) ) , 1. find the Nash equilibria in pure strategies for different values of a, b∈ R; 2. for b = 0, find the best reply of the second player to the strategy ¯ x = (1 5 , 4 5 ) of the first one; 3. for b = 0, find a such that ¯x belongs to a Nash equilibrium profile; 4. find all the Nash equilibria for a = 2, b = 1. Solution 1. The Nash equilibria in pure strategies are {(1, 0), (1, 0, 0)} with outcome (1 , 2) if b≤ 2∀a {(0, 1), (0, 0, 1)} with outcome ( a, 1) if a≥ 3∀b {(1, 0), (0, 0, 1)} with outcome (3 , b) if a≤ 3 and b≥ 2 2. If player I playes ¯x, player II expected outcome is 2 5 from the first column, 0 from the second column and 4 5 from the last one. Player II best response is to play the third column: (0 , 0, 1). 3. ¯x is part of a Nash equilibrium if it is a best response to the strategy (0 , 0, 1) of the second player, so if the first player is indifferent between the two rows, i.e. if a = 3. 4. If b = 1 the second column is dominated by the last one. The Nash equilibria are {(p, 1− p), (1, 0, 0)} with p≥ 1 2, as can be derived by analyzing the best reply functions: BRI = { 1 ∀q < 1 [0, 1] if q = 1 BRII =    0 if p < 1 2 [0, 1] if p = 1 2 1 if p > 1 2 q p1 2 1 1 1 Exercise 2 (2+2+3) Let ( N, v) be the a TU-game where N = {1, 2, 3}, v({1}) = v({2}) = 1, v({3}) = 0, v({1, 2}) = a, v({1, 3}) = v({2, 3}) = b, v(N) = 4. 1. Find and draw the core for a = 2 and b = 3 2. For a = 4, find for which values of b the Shapley value of player 3 is zero. 3. Find the nucleolus for a = b = 4. Solution 1. The conditions of the core are:    x1, x2≥ 1 x3≥ 0 x1 + x2≥ 2 x1 + x3≥ 3 x2 + x3≥ 3 x1 + x2 + x3 = 4 =⇒   …