Decision Under Risk and Uncertainty
This chapter opens the part of the book on applications of set functions in decision making. The foundations of decision making are mainly due to John von Neumann and Oskar Morgenstern, although the concepts of utility function and expected utility go back to Daniel Bernoulli and Blaise Pascal. The area of decision making which is addressed in this chapter is decision under risk and uncertainty. It deals with situations where the decision maker is faced with uncertainty: the consequences of his possible decisions depend on contingencies which are out of his control. The occurrence or the non-occurrence of these contingencies determine what is called the states of nature. If probabilistic information on the states of nature is available, one speaks of decision under risk. Otherwise, it is assumed that the decision maker has a personal, subjective probability measure on the states of nature in his mind, in which case one speaks of decision under uncertainty. While classical models solely rely on probability measures (expected utility), the observation of various paradoxes, unexplained by expected utility, has lead to considering capacities (viewed as nonadditive probabilities) and the Choquet integral in decision making. This chapter tries to show the emergence of these new models. It does not pretend to a full exposition of decision under risk and uncertainty, which would require a whole book. For this reason, and because already many textbooks exist on this subject, proofs of results are not given, except for some results which either are not so well known, or for which it is less easy to find comprehensive references. Recommended references for full exposition and details are Gilboa  and Wakker ; see also Wakker , Quiggin , Takemura  and a survey by Chateauneuf and Cohen . The chapter ends with a presentation of qualitative decision making and the use of the Sugeno integral, a topic which is generally absent from monographs on decision making.
KeywordsUtility Function Risk Aversion Prospect Theory Stochastic Dominance Standard Gamble
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