# A Unified Framework for Indeterminate Probabilities and Utilities

Sammanfattning: In classic decision theory, the different alternatives in a decision situation are merely objects of choice, and 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. However, in real-life situations, the ordering of alternatives from most to least preferred is often a delicate matter and an adequate mathematical representation is crucial. In attempting to address real-life problems, where uncertainty about data prevails, some kind of representation of imprecise information is important and several have been proposed. In particular, 1st order representations, such as sets of probability measures, upper and lower probabilities as well as interval probabilities and utilities of various kinds, have been suggested for enabling a better representation of the input sentences for a subsequent decision analysis. There are, however, problems with all the above approaches that are difficult to discuss in their own terminology.This thesis presents a family of methods and procedures for representing and analysing imprecise, vague, and incomplete probabilistic decision and risk problems. In particular, we suggest methods for stating and analysing probabilities and values (utilities) with belief distributions over them as well as optimization procedures for fast evaluation of decision rules with respect to such statements. We have extended the classical risk and decision evaluation process by the integration of procedures for handling qualitative, vague, and numerically imprecise probabilities and utilities. The shortcomings of the principle of maximising the expected utility, and of utility theory in general, have in part been compensated for by the introduction of various kinds of alternative decision rules. We have given a theoretical motivation for the methodology required for such evaluations involving vague and numerically imprecise data. There is also a need for a general discussion of interval decision methods, and such a discussion is included using an analytic method based on higher-order representations (e.g. a distribution over a probability distribution). A side effect is that the representation capability has increased by the inclusion of distributions over classes of probability and utility measures over sets of outcomes. Regarding this, we have investigated how local distributions, i.e. distributions over projections, can be derived from global distributions and investigated consistency measures expressing the extent into which user-asserted local distributions can be used for defining global distributions. The approach provides a decision-maker with the means for expressing varying degrees of imprecision in the input sentences.

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