Decision-Making under Ordinal Preferences and Comparative Uncertainty

This paper proposes a method that finds a preference relation on a set of acts from the knowledge of an ordering on events describing the decision-maker's uncertainty and an ordering of consequences of acts, describing the decision maker's preferences. However, contrary to classical approaches to decision theory, this method does not resort to any numerical representation of utility nor uncertainty and is purely ordinal. It is shown that although many axioms of Savage theory can be preserved and despite the intuitive appeal of the ordinal method, the approach is inconsistent with a probabilistic representation of uncertainty. It leads to the kind of uncertainty theory encountered in nonmonotonic reasoning (especially preferential and rational inference). Moreover the method turns out to be either very little decisive or to lead to very risky decisions, although its basic principles look sound. This paper raises the question of the very possibility of purely symbolic approaches to Savage-like decision-making under uncertainty and obtains preliminary negative results.