How can an expert system help in choosing the optimal decision?

Abstract

We introduce a rationality principle for a preference relation ~< on an arbitrary set of lotteries. Such a principle is a necessary and sufficient condition for the existence of an expected utility agreeing with ~<. The same principle also guarantees a rational extension of the preference relation to any larger set of lotteries. When the extended relation is unique with respect to the alternatives under consideration, the decision maker does not need a numerical evaluation in order to make a choice. Such a rationality condition needs little information in order to be applied, and its verification amounts to solving a linear system.

Rationality principles for preferences on belief functions

“…Two rationality principles (g-R) and (g-CR) for preference relations among random quantities equipped with a belief function (g-lotteries) are introduced, following the line of the rationality principle given in [5]. Such principles allow to handle representability of preference relations when the set of random quantities is arbitrary (not necessarily closed under convex combinations) and possibly finite.…”

“…In [5], referring to the EU model, a different approach (based on a "rationality principle") is proposed: it does not need all these non-natural comparisons but, instead, it can work by considering only the (few) lotteries and comparisons of interest. The mentioned "rationality principle" can be summarized as follows: it is not possible to obtain the same lottery by combining in the same way two groups of lotteries, if every lottery of the first group is not preferred to the corresponding one of the second group, and at least a preference is strict.…”

“…Our aim is to propose an approach similar to that in [5] for both the linear utility model and the Choquet expected utility model.…”

“…Since our main interest is to manage a finite set L of g-lotteries with a possibly partial preference relation ( , ≺), in analogy with [5], we search for a necessary and sufficient condition for the existence of either a linear utility function on L representing ( , ≺) or a utility function u : X → R such that its Choquet expected value represents ( , ≺).…”

“…The first axiom of rationality we introduce is formally equal to the one given in [5]: it requires that it is not possible to obtain the same g-lottery, by combining in the same way two groups of g-lotteries, if each g-lottery in the first group is not preferred to the corresponding one in the second group, and at least a preference is strict. Definition 3.1.…”