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GAME THEORY 2017-2018 5 cfu 6 Cooperative games Exercise 73. Given the game (N, v) with N ={1, 2, 3} and v({i}) = 0 , v({1, 2}) = v({1, 3}) = 1 , v({2, 3}) = 0 , v(N ) = 2 , represent the core of the game. Exercise 74. Given the TU game: N ={1, 2, 3}, v({i}) = 0 , v({1, 2}) = a, v({1, 3}) = v({2, 3}) = 1 , v(N ) = 2 , find a such that the core of the game is a singleton. In such a case find the nucleolus. Exercise 75. Consider the graph in Figure ?? where O is a power plant that provides electricity to the houses in N ={1, 2, 3}. The numbers on the edges are the costs of connecting two different houses or one house with O. How can we represent the cost of each coalition of houses? Write the characteristic function and find the core of the cost game. Exercise 76. Given the game (N, v) with N ={1, 2, 3} and v({i}) = 0 , v({1, 2}) = 2 , v({1, 3}) = v({2, 3}) = 3 , v(N ) = a, 1. say for which values of a the core is not empty 2. find the Shapley and the Banzhaf values for a = 5 3. find the nucleolus for a = 6 Exercise 77. Given the TU game (N, v) with N ={1, 2, 3, 4} and v(A) = { 1 |A|≥ 3, {1, 2}⊂ A 0 otherwise 1. say how many non empty coalitions there are s.t. v(A) = 0 2. find the core 3. find the nucleolus 4. find the Shapley value Exercise 78. Exam 11 July 2016 Let (N, v) be the TU-game defined as follows: N ={1, . . . , n} and for S⊆ N: v(S) = {|S| if S∩{ 1, 2}̸ =∅ and |S|≤ 4 0 otherwise. 1 1. Are there any symmetric players? 2. Compute the Shapley value for n = 3. 3. Find the core for n = 3. 4. Say for which n the core of the game is empty. Exercise 79. Given the game (N, v) with N ={1, 2, 3} and v({1}) = 10 , v({2}) = 15 , v({3}) = 20 , v({1, 2}) = 40 , v({1, 3}) = 30 , v({2, 3}) = 40 , v({1, 2, 3}) = 60 , represent the core of the game. Exercise 80. Given the game with three…