It is a very natural process for the mind to order objects of a set. To achieve this, we intuitively assign values (they can be real values, qualitative values like "good", "bad" or more generally lattice values) that are easy to handle and to compare. The general theory for this is measurement theory that aims to give general conditions on the set X of objects that need to be compared, and on the binary relation, in order to have a function qb:X--- IR, such that: Vx, y X,x -y ¢ O(x) (y).
This paper presents an axiomatic framework for qualitative decision under uncertainty in a finite setting. The corresponding utility is expressed by a sup-min expression, called Sugeno (or fuzzy) integral. Technically speaking, Sugeno integral is a median, which is indeed a qualitative counterpart to the averaging operation underlying expected utility. The axiomatic justification of Sugeno integral-based utility is expressed in terms of preference between acts as in Savage decision theory. Pessimistic and optimistic qualitative utilities, based on necessity and possibility measures, previously introduced by two of the authors, can be retrieved in this setting by adding appropriate axioms.
The problem of assessing the value of a candidate is viewed here as a multiple combination problem. On the one hand a candidate can be evaluated according to different criteria, and on the other hand several experts are supposed to assess the value of candidates according to each criterion. Criteria are not equally important, experts are not equally competent or reliable. Moreover levels of satisfaction of criteria, or levels of confidence are only assumed to take their values in qualitative scales which are just linearly ordered. The problem is discussed within two frameworks, the transferable belief model and the qualitative possibility theory. They respectively offer a quantitative and a qualitative setting for handling the problem, providing thus a way to compare the nature of the underlying assumptions.
The multiplicity of modalities associated with the kinds of information that can appear in intelligent systems requires a wide spectrum of uncertainty representation calculi. Among those that are commonly used are probability theory, possibility theory, fuzzy sets, rough sets, Dempster-Shafer and random set theory. Close connections exist between some of these as is the case with Dempster-Shafer and random set theory as well as between fuzzy set theory and possibility theory. These calculi rather then being competing are needed to represent the different types of uncertainties such as randomness, lack of specificity and imprecision. As a result of this situation there arises a need for a unified framework in which to model the above mentioned and any other required type of uncertainty representations A promising unifying framework for this is a class of non-additive measures called fuzzy measures [1, 2] which have properties very suitable for the representation and management of uncertain information. In the following we shall shall briefly discuss our ideas in this direction.
Expected Utility: Algebraic Expected Utility In this paper, we provide two axiomatizations of algebraic expected utility, which is a particular generalized expected utility, in a von Neumann-Morgenstern setting, i.e. uncertainty representation is supposed to be given and here to be described by a plausibility measure valued on a semiring, which could be partially ordered. We show that axioms identical to those for expected utility entail that preferences are represented by an algebraic expected utility. This algebraic approach allows many previous propositions (expected utility, binary possibilistic utility,...) to be unified in a same general framework and proves that the obtained utility enjoys the same nice features as expected utility: linearity, dynamic consistency, autoduality of the underlying uncertainty measure, autoduality of the decision criterion and possibility of modeling decision maker's attitude toward uncertainty.