Multichoice games, as well as many other recent attempts to generalize the notion of classical cooperative game, can be casted into the framework of lattices. We propose a general definition for games on lattices, together with an interpretation. Several definitions of the Shapley value of a multichoice games have already been given, among them the original one due to Hsiao and Raghavan, and the one given by Faigle and Kern. We propose a new approach together with its axiomatization, more in the spirit of the original axiomatization of Shapley, and avoiding a high computational complexity.
The Shapley value is a central notion defining a rational way to share the total worth of a cooperative game among players. We address a general framework leading to applications to games with communication graphs, where the feasible coalitions form a poset whose all maximal chains have the same length. Considering a new way to define the symmetry among players, we propose an axiomatization of the Shapley value of these games. Borrowing ideas from electric networks theory, we show that our symmetry axiom and the efficiency axiom correspond to the two Kirchhoff's laws in the circuit associated to the Hasse diagram of feasible coalitions. We thank both reviewers for their work and valuable remarks, as well as the Associate Editor. Below is the list of changes and argued rebuttals. Fabien Reviewer #2• Another definition of regular set systems is proposed: N is regular iff there is a set O of strict orders of N s.t. N contains precisely all initial segments of orderings in O. The "if part" is wrong: for instance, if N = {1, . . . , 4} and O := {(1, 2, 3, 4); (3, 4, 1, 2)}, the resulting N does not satisfy the definition ({3} ≺ {1, 2, 3} but |{1, 2, 3} \ {3}| = 1).• We quite disagree with the idea that the statement of the regularity axiom is too long, and the proposed restatement does not enable to understand its intuition. As specified, this axiom means that for an equidistributed game v, the sum of marginal contributions of players should not depend on the considered maximal chain. For such a game, the path taken from coalition A to B has no effect on the successive increasing worth v(C) (A ⊆ C ⊆ B), so that our statement of the axiom is defended.• It is true that the "Shapley-Kirchhoff value" is not explicitely presented.Actually, it appears that the coefficients are computable only for a given regular set sytem, since a linear system has to be solved, where equations depend on the structure of the system. By the way, it is false to say that this value is the arithmetic average of the order values of Weber over all orderings in O. This is actually a compatible-order value, and consequently an average of the order values, but the mean is not arithmetic (coefficients are not the same, depending on the ordering, and may be negative as a corollary of the counterexample of Annex B).• Theorem 12 does not show that the SK value is monotonic (which is false, according to the aforementionned counterexample), it shows its aggregate monotonicity. 1 * Response to ReviewersReviewer #1• About the fact that we do not give very much support to the claim that the class of regular set systems is natural: in addition of the example of communication situations, we added after Proposition 1 a paragraph with another approach in the process of building regular set systems : by considering permitted/forbidden permutations. The introduction is also amended (p. 2, paragraph 1).• Technical points p. 8 and p. 12: these are now corrected. Abstract. The Shapley value is a central notion defining a rational way to share the total wor...
International audienceThe paper proposes a general approach of interaction between players or attributes. It generalizes the notion of interaction defined for players modeled by games, by considering functions defined on distributive lattices. A general definition of the interaction transform is provided, as well as the construction of operators establishing transforms between games, their M¨obius transforms and their interaction indices
Bi-capacities are a natural generalization of capacities (or fuzzy measures) in a context of decision making where underlying scales are bipolar. They are able to capture a wide variety of decision behaviours. After a short presentation of the basis structure, we introduce the Shapley value and the interaction index for capacities. Afterwards, the case of bi-capacities is studied with new axiomatizations of the interaction index.
To cite this version:Fabien Lange, Michel Grabisch. Interaction transform for bi-set functions over a finite set. Information Sciences, Elsevier, 2006, 176 (16)
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