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.
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. PUBLICATION ABSTRACT
The 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 Möbius transforms and their interaction indices.
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.
A voting situation is given by a set of voters and the rules of legislation that determine minimal requirements for a group of voters to pass a motion. A priori measures of voting power, such as the ...Shapley—Shubik index and the Banzhaf value, show the influence of the individual players in a voting situation and are calculated by looking at marginal contributions in a simple game consisting of winning and losing coalitions derived from the legislative rules. We introduce a new way to calculate these measures directly from the set of minimal winning coalitions and derive explicit formulae for the Shapley—Shubik and Banzhaf values. This new approach logically appealing as it writes measures as functions of the rules of the legislation. For certain classes of games that arise naturally in applications the logical shortcut drastically simplifies the numerical calculations to obtain the indices. The technique generalises directly to all semivalues.
Set functions appear as a useful tool in many areas of decision making and operations research, and several linear invertible transformations have been introduced for set functions, such as the ...Möbius transform and the interaction transform. The present paper establish similar transforms and their relationships for bi-set functions, i.e. functions of two disjoint subsets. Bi-set functions have been recently introduced in decision making (bi-capacities) and game theory (bi-cooperative games), and appear to open new areas in these fields.
Les fonctions de treillis, apparaissent être des outils essentiels en recherche opérationnelle. Elles ouvrent en effet de nouveaux champs d'application en théorie des jeux coopératifs, et en aide à ...la décision (les jeux sont dans ce cas des capacités, ou mesures floues). Cette thèse a pour objet l'investigation de concepts de solutions pour les jeux définis sur des structures générales de coalitions. À cette fin, nous proposons plusieurs généralisations et axiomatisations de la valeur de Shapley pour les jeux multi-choix, les jeux à actions combinées, et les jeux réguliers. L'indice d'interaction quantifie la véritable contribution d'une coalition par rapport à toutes ses sous-coalitions. Mathématiquement, il s'agit d'un prolongement de la valeur de Shapley. Nous proposons des axiomatisations de l'indice d'interaction de Shapley pour les jeux bi-coopératifs, ainsi que des procédés calculatoires permettant de déterminer l'opérateur d'interaction et son inverse.
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