Qualitative decision theory: from savage's axioms to nonmonotonic reasoning

This paper investigates to what extent a purely symbolic approach to decision making under uncertainty is possible, in the scope of artificial intelligence. Contrary to classical approaches to decision theory, we try to rank acts without resorting to any numerical representation of utility or uncertainty, and without using any scale on which both uncertainty and preference could be mapped. Our approach is a variant of Savage's where the setting is finite, and the strict preference on acts is a partial order. It is shown that although many axioms of Savage theory are preserved and despite the intuitive appeal of the ordinal method for constructing a preference over acts, the approach is inconsistent with a probabilistic representation of uncertainty. The latter leads to the kind of paradoxes encountered in the theory of voting. It is shown that the assumption of ordinal invariance enforces a qualitative decision procedure that presupposes a comparative possibility representation of uncertainty, originally due to Lewis, and usual in nonmonotonic reasoning. Our axiomatic investigation thus provides decision-theoretic foundations to the preferential inference of Lehmann and colleagues. However, the obtained decision rules are sometimes either not very decisive or may lead to overconfident decisions, although their basic principles look sound. This paper points out some limitations of purely ordinal approaches to Savage-like decision making under uncertainty, in perfect analogy with similar difficulties in voting theory.