This paper investigates a purely qualitative version of Savage's theory for decision making under uncertainty. Until now, most representation theorems for preference over acts rely on a numerical representation of utility and uncertainty where utility and uncertainty are commensurate. Disrupting the tradition, we relax this assumption and introduce a purely ordinal axiom requiring that the Decision Maker (DM) preference between two acts only depends on the relative position of their consequences for each state. Within this qualitative framework, we determine the only possible form of the decision rule and investigate some instances compatible with the transitivity of the strict preference. Finally we propose a mild relaxation of our ordinality axiom, leaving room for a new family of qualitative decision rules compatible with transitivity.
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).
For a long time, Artificial Intelligence had not been much concerned by decision issues. However, many reasoning tasks are more or less oriented towards decision or involve decision steps. In the last past five years, decision under uncertainty has become a topic of interest in AI. The application of classical expected utility theory to planning under uncertainty and the algorithmic igsues raised by its implementation have been specially investigated (e.g., , ) as well as a search for more qualitative approaches .
In this paper we analyze two recent axiomatic approaches proposed by Dubois et al and by Giang and Shenoy to qualitative decision making where uncertainty is described by possibility theory. Both axiomtizations are inspired by von Neumann and Morgenstern's system of axioms for the case of probability theory. We show that our approach naturally unifies two axiomatic systems that correspond respectively to pessimistic and optimistic decision criteria proposed by Dubois et al. The simplifying unification is achieved by (i) replacing axioms that are supposed to reflect two informational attitudes (uncertainty aversion and uncertainty attraction) by an axiom that imposes order on set of standard lotteries and (ii) using a binary utility scale in which each utility level is represented by a pair of numbers.
Classical Decision Theory provides a normative framework for representing and reasoning about complex preferences. Straightforward application of this theory to automate decision making is difficult due to high elicitation cost. In response to this problem, researchers have recently developed a number of qualitative, logic-oriented approaches for representing and reasoning about references. While effectively addressing some expressiveness issues, these logics have not proven powerful enough for building practical automated decision making systems. In this paper we present a hybrid approach to preference elicitation and decision making that is grounded in classical multi-attribute utility theory, but can make effective use of the expressive power of qualitative approaches. Specifically, assuming a partially specified multilinear utility function, we show how comparative statements about classes of decision alternatives can be used to further constrain the utility function and thus identify sup-optimal alternatives. This work demonstrates that quantitative and qualitative approaches can be synergistically integrated to provide effective and flexible decision support.