UTILITY
(1) In economics, utility means the real or fancied ability of a good or service to
satisfy a human want. An associated term is WELFARE FUNCTION (synonym:
utility function--not to be confused with UTILITy FUNCTION in decision theory; see
below), which relates the utility derived by an individual or group to the goods and
services that it consumes. MARGINAL UTILITY is the change in utility due to a one
unit change in the quantity of a good or service consumed. (2) In decision
theory, utility is a measure of the desirability of consequences of courses of action that
applies to decision making under risk--that is, under uncertainty with known
probabilities.
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)
