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Decision theory is a body of knowledge and related analytical techniques of different degrees of formality designed to help a decision maker choose among a set of alternatives in light of their possible consequences. Decision theory can apply to conditions of certainty, risk, or uncertainty. [decision UNDER certainty] means that each alternative leads to one and only one consequence, and a choice among alternatives is equivalent to a choice among consequences. In [DECISION UNDER risk] each alternative will have one of several possible consequences, and the probability of occurrence for each consequence is known. Therefore, each alternative is associated with a probability distribution, and a choice among probability distributions. When the probability distributions are unknown, one speaks about [DECISION UNDER uncertainty.] Decision theory recognizes that the ranking produced by using a criterion has to be consistent with the decision maker's objectives and preferences. The theory offers a rich collection of techniques and procedures to reveal preferences and to introduce them into models of decision. It is not concerned with defining objectives, designing the alternatives or assessing the consequences; it usually considers them as given from outside, or previously determined. Given a set of alternatives, a set of consequences, and a correspondence between those sets, decision theory offers conceptually simple procedures for choice. In a decision situation under certainty the decision maker's preferences are simulated by a single-attribute or MULTIATTRIBUTE VALUE FUNCTION that introduces ordering on the set of consequences and thus also ranks the alternatives. Decision theory for risk conditions is based on the concept of utility (see utility, sense 2). The decision maker's preferences for the mutually exclusive consequences of an alternative are described by a utility function that permits calculation of the EXPECTED UTILITY for each alternative. The alternative with the highest expected utility is considered the most preferable. For the case of uncertainty, decision theory offers two main approaches. The first exploits criteria of choice developed in a broader context by game theory, as for example the [MAX-MIN RULE,] where we choose the alternative such that the worst possible consequence of the chosen alternative is better than (or equal to) the best possible consequence of any other alternative. The second approach is to reduce the uncertainty case to the case of risk by using SUBJECTIVE PROBABILITIES, based on expert assessments or on analysis of previous decisions made in similar circumstances. See also: game theory, optimization, utility, value (IIASA)

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