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After giving the matter much thought, they have narrowed the list of possible names to four: Abigail, Iris, Irene, and Olga.
A committee comprised of 15 members is called upon to choose the prettiest color: red, blue, or yellow.
Every resident ranks the candidates from most preferred to least preferred, and places this ranked list in a ballot box.
Since x* is the nucleolus, and since a1 is the greatest value of the excesses, it must be the case that e(S, x) = 0 for every coalition S ? D(a1, x*).
Find a weighted majority game that is not constant sum, and satisfies the property that if S is a winning coalition, then Sc is a losing coalition.
Prove that if [q; w] is a representation of a weighted majority game (N; v), then [q(w); w] is also a representation of that game.
There are five wives, who are owed 60, 120, 180, 240, and 300, respectively, out of an estate of 600.
Prove directly that the function f defined in Equation is continuous, its first coordinate is monotonically nondecreasing in d1 for every fixed d2.
Jeff owes Sam $140, and owes Harry $80. Jeff declares bankruptcy, because he has only $100, and the decision of how to divide his $100 between Harry.
Compute the prenucleolus of the three-player coalitional game with the following coalitional function.
Compute the prenucleolus of the three-player coalitional game with the following coalitional function: v(1) = v(2) = v(3) = v(2, 3) = 0, v(1, 2) = v(1, 3)
Let K be the triangle in R3 whose vertices are (1, 0, 0), (0, 1, 0), and (0, 0, 1), and let K0 be its boundary.
Compute the nucleolus and the prenucleolus of the three-player coalitional game (N; v) in which v(1, 2) = 1 and v(S) = 0 for every other coalition S.
Prove that the nucleolus is covariant under strategic equivalence: for every coalitional game (N; v), for every set K ? RN , for every a > 0.
Find a two-player coalitional game (N; v), and a V -shaped set K, i.e., a set that is the union of two line segments sharing an edge point.
My Aunt and I Auntie Betty can complete a certain job together with me or with any of my three brothers, with the payment for the work being $1,000.
Compute the core and the nucleolus of the following spanning tree game. v0 is the central point to which Players I, II, and III.
Deduce that the nucleolus is contained in the minimal core, and that if the core is nonempty, then the nucleolus is contained in the core.
Suppose that (N; v) is a coalitional game such that the set of imputations X(N; v) is nonempty, and such that the nucleolus x* differs from the prenucleolus.
Show that every null player is a dummy player, but there are dummy players that are not null players.
Find a game (N; v), a coalitional structure, and an imputation x such that: (a) 1 ?x 3, (b) 1 ~x 2, and (c) 2 ~x 3.
A simple coalitional game (N; v) is called a veto control game if there is a player i such that v(S) = 0.
Find a two-player coalitional game (N; v), and a bounded set K that is not closed, such that the nucleolus N (N; v; K) is the empty set.
Let (N; v) be a coalitional game satisfying the following property: there is an imputation x ? X(N; v) such that all the excesses at x are nonnegative.