Qualitative capacities: Basic notions and potential applications

Abstract

Qualitative capacities are set functions valued on a finite chain, increasing in the wide sense with respect to set inclusion. This paper exploits formal analogies between qualitative capacities and numerical capacities: we first recall qualitative counterparts to Möbius transforms, game-theoretic core, and conjugate set functions. However, in the qualitative setting, possibility measures play the same role as probability measures in the quantitative setting. Then possibility and necessity measures sometimes do not convey the same type of information. This situation creates difficulties to interpret qualitative capacities and related notions inspired from the quantitative setting. In particular, we propose three different ways of using qualitative capacities: either as bounds on ill-known possibility or necessity measures, or as tools to express the decision maker attitude in qualitative criteria under uncertainty, or yet qualitative counterparts of belief functions that handle both incompleteness and inconsistency of pieces of information stemming from several sources. In the latter framework possibility and necessity measures do not represent the same type of information. We define order relations between capacities with a view to compare them in terms of informational contents. We also study a counterpart of Dempster rule of combination in the qualitative setting. We compare several capacity combination rules in the framework of an information fusion problem. Finally, we address the problem of eliciting qualitative capacities based on human-originated information.