Hybrid probabilistic-possibilistic mixtures and utility functions

The paper was written during the stay of the second author as visiting professor at University Paul Sabatier, in Toulouse, in June, 1999.International audienceA basic building block in the standard mathematics of decision under uncertainty is the notion of probabilistic mixture. In order to generalize decision theory to non probabilistic uncertainty, one approach is to generalize mixture sets. In the recent past it has been proved that generalized mixtures can be non trivially defined, and they have been instrumental in the development of possibilistic utility theory. This paper characterizes the families of operations involved in generalized mixtures, due to a previous result on the characterization of the pairs of continuous t-norm and t-conorm such that the former is conditionally distributive over the latter. What is obtained is a family of mixtures that combine probabilistic and possibilistic mixtures via a threshold. It is based on a restricted family of t-conorm/ t-norm pairs which are very special ordinal sums. Any practically useful theory of pseudo-additive measures must use such special pairs of operations in order to extend the additivity property, and the notion of probabilistic independence