Weakening Commensurability Hypothesis in Possibilistic Qualitative Decision Theory

Abstract

Models for Qualitative Decision under Uncertainty assuming that uncertainty is of possibilistic nature have been proposed in 1990's. In them, as in the classical approach of Expected Utility Theory, two alternatives emerge: a la Von Neumann and Morgenstern (VNM) or a la Savage. In the first one, uncertainty and preferences on consequences are measured on finite scales. Decisions are ranked in terms of the ranking induced, on their associated possibility distributions on consequences, by qualitative utility functions. To define these utilities, an hypothesis of commensurability, i.e. the existence of an onto order-preserving mapping h linking both scales of uncertainty and preference, is assumed. This hypothesis forces us to restrict to decision problems where the cardinality of the uncertainty values set is greater than or equal to the cardinality of the preference set. This point has been attacked in the possibilistic a la Savage approach, but as far as we know, it is an open question in the qualitative VNM approach. As a first step to weaken the commensurability hypothesis in this model, the preference orderings resulting of applying the qualitative criteria without requiring h to be onto are characterized. These orderings may be not continuous, but they satisfy a relaxed continuity axiom.