This two-part state-of-the-art overview on aggregation theory summarizes the essential information concerning aggregation issues. An overview of aggregation properties is given, including the basic classification on ag-gregation functions. In this first part, the stress is put on means, i.e., averaging aggregation functions, both with fixed arity (n-ary… (More)
The main advances regarding the use of the Choquet and Sugeno integrals in multi-criteria decision aid over the last decade are reviewed. They concern mainly a bipolar extension of both the Choquet integral and the Sugeno integral, interesting particular submodels, new learning techniques , a better interpretation of the models and a better use of the… (More)
Bi-cooperative games have been introduced by Bilbao et al as a generalization of TU cooperative games, where each player can participate positively to the game, negatively, or do not participate. In this paper, we propose a definition of a share of the wealth obtained by some players after they decided on their participation to the game. It turns out that… (More)
This paper introduces four alternative representations of a set function: the Möbius transformation, the co-Möbius transformation, and the interactions between elements of any subset of a given set as extensions of Shapley and Banzhaf values. The links between the five equivalent representations of a set function are emphasized in this presentation.
An axiomatization of the interaction between the players of any coalition is given. It is based on three axioms: linearity, dummy and symmetry. These interaction indices extend the Banzhaf and Shapley values when using in addition two equivalent recursive axioms. Lastly, we give an expression of the Banzhaf and Shapley interaction indices in terms of… (More)
This paper addresses the question of which models fit with information concerning the preferences of the decision maker over each attribute , and his preferences about aggregation of criteria (interacting criteria). We show that the conditions induced by these information plus some intuitive conditions lead to a unique possible aggregation operator: the… (More)
Considering a linearly ordered set, we introduce its symmetric version , and endow it with two operations extending supremum and in-fimum, so as to obtain an algebraic structure close to a commutative ring. We show that imposing symmetry necessarily entails non asso-ciativity, hence computing rules are defined in order to deal with non associativity. We… (More)