Representing qualitative capacities as families of possibility measures

Abstract

This paper studies the structure of qualitative capacities, that is, monotonic set-functions, when they range on a finite totally ordered scale equipped with an order-reversing map. These set-functions correspond to general representations of uncertainty, as well as importance levels of groups of criteria in multiple-criteria decision-making. We show that any capacity or fuzzy measure ranging on a qualitative scale can be viewed both as the lower bound of a set of possibility measures and the upper bound of a set of necessity measures (a situation somewhat similar to the one of quantitative capacities with respect to imprecise probability theory). We show that any capacity is characterized by a nonempty class of possibility measures having the structure of an upper semi-lattice. The lower bounds of this class are enough to reconstruct the capacity, and the number of them is characteristic of its complexity. An algorithm is provided to compute the minimal set of possibility measures dominating a given capacity. This algorithm relies on the representation of the capacity by means of its qualitative Möbius transform, and the use of selection functions of the corresponding focal sets. We provide the connection between Sugeno integrals and lower possibility measures. We introduce a sequence of axioms generalizing the maxitivity property of possibility measures, and related to the number of possibility measures needed for this reconstruction. In the Boolean case, capacities are closely related to non-regular modal logics and their neighborhood semantics can be described in terms of qualitative Möbius transforms.

Set Functions, Games and Capacities in Decision Making

“…Finding the minimal elements of the possibilistic core when the capacity is not strictly monotone is more difficult, and this problem was solved by Dubois et al [114]. Its solution is based on selectors, as in the definition of the selectope (Sect.…”

“…Dubois et al give in [114] an algorithm to find the minimal elements of the possibilistic core, which are possibility distributions induced by selectors. /; It should be noted that, contrarily to the classical selectope where marginal vectors correspond to selector values where the selector is consistent, possibility distributions generated by a permutation cannot always be generated by a selector: take X D fx 1 ; x 2 g and .X/ D .fx 2 g/ D 1, .fx 1 g/ D 0:5 and consider the identity permutation.…”

“…The proof of minimality and completeness is given in [114]. Put F D fS 2 2 X W m .S/ > 0g and order the sets in decreasing value of the ordinal Möbius transform; Step 1.…”

Generalized comonotonicity and new axiomatizations of Sugeno integrals on bounded distributive lattices

“…Let us mention that various types of integrals have many applications within the theory of aggregation functions. To illustrate that the study of Sugeno integrals from various points of view is still very active area of aggregation functions, we refer the reader to recent papers [3,4,8,9,14,17]. More specifically, in [8,9] a new promising approach via so-called clone theory to a study of aggregation functions on bounded lattices has been started.…”

“…We expect to apply these results for better understanding of properties of Sugeno integrals on distributive lattices. Moreover, following the spirit of papers [3,14,17] dealing with certain non-additive measures, we expect to introduce Sugeno integrals on non-distributive lattices.…”