The concept of utility applies to both SINGLE-ATTRIBUTE and MULTIATTRIBUTE consequences. The fundamental assumption in UTILITY THEORY is that the decision maker always chooses the alternative for which the expected value of the utility (EXPECTED utility) is maximum. If that assumption is accepted, utility theory can be used to predict or prescribe the choice that the decision maker will make, or should make, among the available alternatives. For that purpose, a utility has to be assigned to each of the possible (and mutually exclusive) consequences of every alternative. A UTILITY function is the rule by which this assignment is done and depends on the preferences of the individual decision maker. In utility theory, the utility measures u of the consequences are assumed to reflect a decision maker's preferences in the following sense: (i) the numerical order of utilities for consequences preserves the decision maker's preference order among the consequences; (ii) the numerical order of expected utilities of alternatives (referred to, in utility theory, as gambles or lotteries) preserves the decision maker's preference order among these alternatives (lotteries). For example if alternative A can have three mutually exclusive consequences, x,y,z, and the decision maker prefers z to y and x to z, the utilities Ul, U2, U3 assigned to x,y,z must be such that U3)U2)U1. If the probabilities of the consequences x,y,z are P1,P2,1-p1,-p2, respectively, the expected utility of alternative A is calculated as
E(u/P) = PlUl + P2U2 + (l-Pl-P2)U3
where P means the probability distribution, characteristic for the alternative (i.P1, P2, 1-P1-P2). (IIASA) If the decision maker prefers alternative B, which has probability distribution Q, to alternative A, the utility assignments in both alternatives must be such that
E(u/Q) 1/2 > E(u/P).
Utility theory provides a basis for the assignment of utilities to consequences by formulating necessary and sufficient conditions to satisfy (i) and (ii). A utility function is defined mathematically as a function u(.) from the set of consequences Y into the real numbers that provides for satisfaction of (i) and (ii). There exist various methods for constructing utility functions. The best-known method is based on indifference judgments of the decision maker about specially constructed alternatives(lotteries). Utility theory permits one to distinguish RISK-PRONE, RISK- NEUTRAL and RISK-AVERSE DECISION makers. For example, if the mutually exclusive payoffs xl,x2,x3 of an alternative A are all expressed in the same units (e.g., schillings), the decision maker is risk-prone if he prefers the alternative A (prefers the lottery) to receiving, with no risk, the expected value of the payoffs (calculated directly as
E(x/P) = plxl + p2x2 + (l-pl-p2)x3).
This preference can also be expressed as
E(u/P) > u(E(x/P))
i.e., the expected utility of the lottery to the risk-prone decision maker is larger than the utility of the expected value of the consequence. The risk-neutral and risk-averse decision makers are defined accordingly. The MULTIATTRIBUTE UTILITY FUNCTION is defined as a function u(.) from the set of multiattribute consequences into the real numbers. This means that it applies to cases where each of the mutually exclusive consequences has several attributes. Multiattribute utility functions, besides having properties (i) and (ii), also express the decision maker's TRADE OFFS among the attributes (compare MULTIATTRIBUTE VALUE FUNCTION). Several special forms of multiattribute utility functions have been developed, including the additive and the multiplicative forms. (IIASA)
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