Suppose we have a game with characteristic function v that satisfies the property v(S) + v(N − S) = v(N) for all coalitions S ⊂ N.
These are called constant sum games.
(a) Show that for a two-person constant sum game, the nucleolus and the Shapley value are the same.
(b) Show that the nucleolus and the Shapley value are the same for a three-person constant sum game. (c) Check the result of the preceding part if v(i) = i,i = 1, 2, 3, v(12) = 5, v(13) = 6, v(23) = 7 and v(123) = 8.
Suppose we have a game with characteristic function v that satisfies the property v(S) + v(N − S) = v(N) for all coalitions S ⊂ N.