A main problem in nearly all contexts is that unguided decision making is tremendously difficult and can lead to inefficient decision processes and undesired consequences. Therefore, decision support systems (DSSs) are of prime concern to any organization and there have been numerous approaches to such from, e.g., computational, mathematical, financial, philosophical, psychological, and sociological angles. However, a key observation is that efficient decision making is not easily performed by using methods from one discipline only. The case is rather that if real world decision making is taken seriously, several aspects must be included. This article describes some efforts of the DECIDE research group for approaching decision making and developing DSSs in a cross-disciplinary environment.

In attempting to address real-life decision problems, where uncertainty about input data prevails, some kind of representation of imprecise information is important and several have been proposed over the years. In particular, first-order representations of imprecision, such as sets of probability measures, upper and lower probabilities, and interval probabilities and utilities of various kinds, have been suggested for enabling a better representation of the input sentences. A common problem is, however, that pure interval analyses in many cases cannot discriminate sufficiently between the various strategies under consideration, which, needless to say, is a substantial problem in real-life decision making in agents as well as decision support tools. This is one reason prohibiting a more wide-spread use. In this article we demonstrate that in many situations, the discrimination can be made much clearer by using information inherent in the decision structure. It is discussed using

second-order probabilities which, even when they are implicit, add information when handling aggregations of imprecise representations, as is the case in decision trees and probabilistic networks. The important conclusion is that since structure carries information, the structure of the decision problem influences evaluations of all interval representations and is quantifiable.

In real-life decision analysis, the probabilities and utilities of consequences are in general vague and imprecise. One way to model imprecise probabilities is to represent a probability with the interval between the lowest possible and the highest possible probability, respectively. However, there are disadvantages with this approach; one being that when an event has several possible outcomes, the distributions of belief in the different probabilities are heavily concentrated toward their centres of mass, meaning that much of the information of the original intervals are lost. Representing an imprecise probability with the distribution’s centre of mass therefore in practice gives much the same result as using an interval, but a single number instead of an interval is computationally easier and avoids problems such as overlapping intervals. We demonstrate why second-order calculations add information when handling imprecise representations, as is the case of decision trees or probabilistic networks. We suggest a measure of belief density for such intervals. We also discuss properties applicable to general distributions. The results herein apply also to approaches which do not explicitly deal with second-order distributions, instead using only first-order concepts such as upper and lower bounds.

Software agents and humans alike face severe difficulties in making decisions in uncertain contexts. One approach is to formalise the decision situation by means of decision theory, i.e. probabilities and utilities leading to the principle of maximising the expected utility. Expected utility is here considered as a stochastic variable; under the assumption that all utility values are equally likely, and that each vector of probability values is equally likely, the probability distribution of expected utility is calculated for two, three, and four possible outcomes. The effect of these probability distributions concentrating around the middle value is explored and its significance for making decisions.

In classic decision theory it is assumed that a decision-maker can assign precise numerical values corresponding to the true value of each consequence, as well as precise numerical probabilities for their occurrences. In attempting to address real-life problems, where uncertainty in the input data prevails, some kind of representation of imprecise information is important. Second-order distributions, probability distributions over probabilities, is one way to achieve such a representation. However, it is hard to intuitively understand statements in a multi-dimensional space and user statements must be provided more locally. But the information-theoretic interplay between joint and marginal distributions may give rise to unwanted effects on the global level. We consider this problem in a setting of second-order probability distributions and find a family of distributions that normalised over the probability simplex equals its own product of marginals. For such distributions, there is no flow of information between the joint distributions and the marginal distributions other than the trivial fact that the variables belong to the probability simplex. marginal distributions may give rise to unwanted effects on the global level.

We consider the problem of maximizing expected utility when utilities and probabilities are given by discrete probability dis- tributions so that expected utility is a discrete stochastic variable. As for discrete second-order distributions, that is probability distributions where the variables are themselves probabilities, the multinomial family is a reasonable choice at least if first-order probabilities are interpreted as relative frequencies. We suggest a decision rule that reflects the uncertainty present in distribution-based probabilities and utilities and we show an example of this rule in action with multinomial second-order distributions.