Decision making theory with imprecise probabilities

There is an extensive literature on decision-making under uncertainty. Unfortunately, up to day there are no valid decision principles. The widely used principle of maximization of expected utility has serious shortcomings. The first approach in classical theory of choice was formulated axiomatically in the expected utility theory of von Neumann and Morgenstern and is the predominant economic theory for decision making under uncertainty. In another, more advanced approach of Savage subjective expected utility it is assumed that peoples assess the state of the economy and the effects of their behavior in a forward-looking way by processing available information according to standard statistical principles. Experimental evidence has repeatedly shown that people violate the axioms of von von Neumann-Morgenstern-Savage preferences in systematic ways. It resulted in initiation of numerous non-expected theories to take into account for this discrepancy, from weighted utility to rank-dependent utility. There are a lot of non-expected utility models. Each of them has its advantages and disadvantages. One of the most successful attempts so far at capturing experimental results has been Prospect theory. Utility function and non-additive measures used in non-expected utility models to model human preferences are mainly considered as real-valued functions despite of the fact that in reality human preferences are imprecise and therefore are described in natural language (NL). On the other hand, what is not available is a methodology for dealing with second-order uncertainty, that is, uncertainty about uncertainty, or uncertainty2. What is not widely recognized is that in wide sense, real-world uncertainties fall into this category. This applies, in particular, to imprecise probabilities expressed by terms such as likely, unlikely, probable, usually etc.