Interaction transform for bi-capacities Fabien LANGE Université Paris I - Panthéon-Sorbonne Centre Marin Mersenne, 72 rue Regnault 75013 Paris, France email Fabien.Lange@univ-paris1.fr Michel GRABISCH Université Paris I - Panthéon-Sorbonne LIP6, 8 rue du Capitaine Scott 75015 Paris, France email Michel.Grabisch@lip6.fr 1 Introduction The concept of bi-capacity has recently been proposed by Grabisch and Labreuche [7, 5] as a generalization of capacities (or fuzzy measures) in the context of decision making. Specifically, let us consider a set X of alterna- tives in a multicriteria decision making prob- lem, where each alternative is described by a set of n real valued scores (a 1 ,...,a n ). Sup- pose one wants to compute a global score of this alternative by the Choquet integral w.r.t. a capacity µ, namely C µ (a 1 ,...,a n ). Then it is well known that the correspondence be- tween the capacity and the Choquet integral is µ(A)= C µ (1 A , 0 A c ), ∀A ⊆ N , where (1 A , 0 A c ) is an alternative having 1 as score on all cri- teria in A, and 0 otherwise. Such an alter- native is called a binary alternative, and the above result says that the capacity represents the overall score of all binary alternatives. However, in many practical situations, it is suitable to score alternatives on a bipolar scale, i.e. with a central value 0 having the meaning of a borderline between posi- tive scores, considered as good, and negative scores, considered as bad. It has been ob- served that most often human decision mak- ers have a different behaviour when faced with alternatives having positive and nega- tive scores, which means that a decision model based solely on the classical Choquet inte- gral, hence on binary alternatives, is no more sufficient. One should, in the general case, consider all ternary alternatives, i.e. alterna- tives of the form (1 A , −1 B , 0 (A∪B) c ). Clearly we need two arguments to denote the overall score of ternary alternatives, namely v(A, B), with A, B ⊆ N being disjoint. This defines bi-capacities, by analogy with capacities. An interesting question is to define for bi- capacities the concept of Möbius transform and interaction, since they are very useful in applications. This was done in [5], leading to a Möbius transform and an interaction index with two disjoint sets as arguments. Due to the complexity of bi-capacities (they require 3 n values to be defined), it is important to de- rive computations between the three possible representations of a bi-capacity, namely v it- self, its Möbius transform and its interaction index, as it was done for capacities [3]. The aim of this paper is precisely to provide a gen- eral framework for these computations, in the spirit of [3]. Throughout the paper, the cardinal of a set is denoted by the corresponding small letter, e.g. |N | = n. 2 Background We introduce necessary concepts for the se- quel. We consider a finite set N := {1,...,n} which can be thought as the set of crite- ria, states of nature, voters, etc. We denote P := P(N ). A capacity v : P → [0, 1] is a set func- tion satisfying v(∅) = 0 and A ⊆ B im- plies v(A)  v(B). If in addition v(N )=1, v is said normalized . Unanimity games u C , C ⊆ N , are particular capacities — except for C = ∅ — defined by: u C (A) :=  1 if A ⊇ C 0 otherwhise ,A ⊆ N. The Möbius transform of a capacity is