Reformulating decision theory using fuzzy set theory and Shafer's theory of evidence

Abstract

Utilities and probabilities in decision theory are usually assessed by asking individuals to indicate their preferences between various uncertain choices. In this paper, we argue that (1) The utility of a consequence can be assessed as the membership function of the consequence in the fuzzy set ‘ satisfactory’. (2) The probability of an event, instead of being directly assessed, should be inferred from the evidence associated with that event. The degree of evidence is quantified using Shaferian basic probability assignments. In addition, we use the Heisenberg Uncertainty Principle to argue for a change in one of the technical assumptions underlying decision theory. As a result of this change, some kinds of evidence will be observable in certain experiments but unobservable in others. Since probabilities are defined over the potential outcomes of an experiment, they will only be defined over some, but not all, the evidence. As a result, the probabilities associated with different experiments could be inconsistent. This formulation emphasizes the importance of new distinctions (and not just new information) in updating probabilities. We argue that this formulation addresses many of the observed empirical deviations between decision theory and experiment. It also addresses the anomalies of quantum physics. We close with a brief discussion of directions for further research.