Making Decision in a Qualitative Setting: from Decision under Uncertaintly to Case-based Decision

The paper improves a previously proposed axiomatic setting for qualitative decision under uncertainty in the von Neumann and Morgenstern' style, where only ordinal scales are required for assessing uncertainty and utility. A deeper analysis shows that only four axioms are needed, postulating i) that the choice is governed by a complete preorder, ii) that the decision maker is uncertainty-averse, iii) independence and iv) a (new) continuity assumption. Because uncertaintyaversion is assumed, the qualitative utility function enforced by these four postulates is pessimistic on a subset of highly plausible states. Substituting uncertainty-aversion by an uncertainty-prone postulate and suitably modifying the continuity postulate leads to a dual optimistic qualitative utility function (a decision is good as soon as it may have good consequences plausibly). This second function can help breaking ties between decisions having the same pessimistic evaluation. Besides, a case-based counterpart to expected utility has been recently proposed by Gilboa and Schmeidler, where similarity plays the role of probability. Elements for a qualitative case-based decision methodology are also proposed, with pessimistic and optimistic evaluations formally similar to the expressions which cope with uncertainty, up to modifying factors which cope with the lack of normalization of similarity.