Abstract
Decision-making under uncertainty has evolved into a mature field. However, in most parts of the existing decision theory, one assumes decision makers have complete decision-relevant information. The standard framework is not capable to deal with partial or fuzzy information, whereas, in reality, decision-relevant information about outcomes, probabilities, preferences etc is inherently imprecise and as such described in natural language (NL). Nowadays, there is no decision theory with second-order uncertainty in existence albeit real-world uncertainties fall into this category. This applies, in particular, to imprecise probabilities expressed by terms such as likely, unlikely, probable, usually etc. We call such imprecise evaluations second-order information granules.
In this study, we develop a decision theory with second-order information granules. The first direction we consider is decision making with fuzzy probabilities. The proposed theory differs from the existing ones as one that accumulates non-expected utility paradigm with NL-described decision-relevant information. Linguistic preference relations and fuzzy utility functions are used instead of their classical counterparts as forming a more adequate description of human preferences expressed under fuzzy probabilities. Fuzzy probability distribution is incorporated into the suggested fuzzy utility model by means of a fuzzy number-valued fuzzy measure instead of a real-valued non-additive probability. We provide representation theorems for a fuzzy utility function described by a fuzzy number-valued Choquet integral with a fuzzy number-valued integrand and a fuzzy number-valued fuzzy measure. The proposed theory is intended to solve decision problems when the environment of fuzzy states and fuzzy outcomes are characterized by fuzzy probabilities. As the second direction in this realm we consider hierarchical imprecise probability models. Such models allow us to take into account imprecision and imperfection of our knowledge, expressed by interval values of probabilities of states of nature and degrees of confidence associated with such values. A decision-making process analysis and a choice of the most preferable alternative subject to variation of intervals at the lower and upper levels of models and the types of distribution on the sets of random values of probabilities of states of nature is also of significant interest.
We apply the developed theories and methodologies to solving real-world economic decision-making problems. The obtained results show validity of the proposed approaches.