Symmetry in Decision Evaluation

Several models for handling vague and imprecise information in decision situations have been suggested. In those contexts, various interval methods have prevailed, i.e. methods based on interval estimates of probabilities and, in some cases, interval utility estimates. Even if these approaches in general are well founded, little has been done for demonstrating whether the approaches are comprehensible for a decision maker. In particular, it is far from always clear what is actually expressed by a set of intervals where linear dependencies do occur. Furthermore, it is difficult to find reasonable decision rules that select an alternative out of a set of alternatives and correspond to the intuition of a decision maker. In this article, we investigate some problems that are inherent in interval approaches and suggest how the choice of particular evaluation rules might compensate for this. Introduction A quite widespread opinion is that the principle of maximising the expected utility captures the concept of rationality. However, the shortcomings of this principle, and of utility theory in general are severe, and have to be compensated for (Ekenberg, et al, 2001). One of the main problems with the principle is that it requires too hard aggregation of background data. Nevertheless, there is a need for efficient evaluation principles. A number of models with representations allowing imprecise probability statements have been suggested. Some of them are based on ̄ capacities (of order 2), ̄ evidence theory and belief functions, ̄ various kinds of logic, ̄ upper and lower probabilities, or ̄ sets of probability measures. The common feature of the approaches is that they do not include the additivity axiom of probability theory and consequently do not require a decision maker to model and evaluate a decision situation using precise probability (and, in some cases, utility) estimates. The approaches are in general well founded, but quite little has been done in Copyright © 2001, A/~I. All riohls reserved. demonstrating whether they are understandable for a decision maker facing a real decision situation. In particular, it is not always clear what an agent expresses when providing, for instance, a set of intervals where linear dependencies do occur. The problem becomes of particular signifcance when evaluation models are considered. An advantage of approaches for upper and lower probabilities is that it is not necessary to take particular probability distributions into consideration. On the other hand, it is then difficult to fred reasonable decision rules that chose an alternative outof a set of alternatives, andthat rcally corresponds tothe intuition f a decision maker. This problem is emphasised bythe fact that no distributions over the intervals are taken into account toindicate values of most significance. The low-dimensional intuition of decision makers further adds to the problem. Effects of changes on input data arc not always simple to communicate. Wc have found two complementary views on the input data to be particularly helpful tothe decision maker symmetry and quadracy.