MIT Encyclopedia of the Cognitive Sciences - Cryptome

860 Utility Theory
neoclassical theory. Recognition of this role was the result
of the so-called marginal utility revolution of the 1870s, in
which Carl Menger, W. Stanley Jevons, Francis Ysidro
Edgeworth, Léon Walras, and other leading “marginalists”
demonstrated that values/prices could be founded on utility.
The standard theory of utility starts with a preference
order, < ~ , typically taken to be a complete preorder over the
outcome space, ϑ. y <
~ x means that x is (weakly) preferred
to y, and if in addition ¬ (x < ~ y), x is strictly preferred.
Granting certain topological assumptions, the preference
order can be represented by a real-valued utility function, U:
ϑ → R, in the sense that U (y) ≤ U (x) if and only if y < ~ x. If
U represents < ~ , then so does ϕ ° U, for any monotone func-
tion ϕ on the real numbers. Thus, utility is an ordinal scale
(Krantz et al. 1971).
Under UNCERTAINTY, the relevant preference comparison
is over prospects rather than outcomes. One can extend
the utility-function representation to prospects by taking
expectations with respect to the utility for constituent outcomes.
Write [F,p; F'] to denote the prospect formed by
combining prospects F and F' with probabilities p and 1 – p,
respectively. If F(ω) denotes the probability of outcome ω
in prospect F, then
[ Fp; , F′ ] ( ω)
≡ pF( ω)
+ ( 1 – p)F′
( ω).
The independence axiom of utility theory states that if
F' < ~ F'', then for any prospect F and probability p,
[ F′ , p ; F]
<
~ [ F″ , p ; F].
In other words, preference is decomposable according to the
prospect’s exclusive possibilities.
Given the properties of an order relation, the independence
axiom, and an innocuous continuity condition, preference
for a prospect can be reduced to the expected value of
the outcomes in its probability distribution. The expected
utility Û of a prospect F is defined by
Û( F)
≡ E
F
[ U]
= U( ω)F(
ω)
ω∈ϑ For the continuous case, replace the sum by an appropriate
integral and interpret F as a probability density function.
Because expectation is generally not invariant with respect
to monotone transformations, the measure of utility for the
uncertain case must be cardinal rather than ordinal. As with
preferences over outcomes, the utility function representation
is not unique. If U is a utility function representation of
< ~ and ϕ a positive linear (affine) function on the reals, then
ϕ ° u also represents <
~ .
∑
Frank Plumpton Ramsey (1964/1926) was the first to
derive expected utility from axioms on preferences and
belief. The concept achieved prominence in the 1940s,
when John VON NEUMANN and Oskar Morgenstern presented
an axiomatization in their seminal volume on GAME
THEORY (von Neumann and Morgenstern 1953). (Indeed,
many still refer to “vN-M utility.”) Savage (1972) presented
what is now considered the definitive mathematical argument
for expected utility from the Bayesian perspective.
Although it stands as the cornerstone of accepted decision
theory, the doctrine is not without its critics. Allais
(1953) presented a compelling early example in which most
individuals would make choices violating the expectation
principle. Some have accounted for this by expanding the
outcome description to include determinants of regret (see
Bell 1982), whereas others (particularly researchers in
behavioral DECISION MAKING) have constructed alternate
preference theories (Kahneman and TVERSKY’s 1979 prospect
theory) to account for this as well as other phenomena.
Among those tracing the observed deviations to the premises,
the independence axiom has been the greatest source
of controversy. Although the dispute centers primarily
around its descriptive validity, some also question its normative
status. See Machina (1987, 1989) for a review of
alternate approaches and discussion of descriptive and normative
issues.
Behavioral models typically posit more about preferences
than that they obey the expected utility axioms. One
of the most important qualitative properties is risk aversion,
the tendency to prefer the expected value of a prospect to
the prospect itself. For scalar outcomes, the risk aversion
function (Pratt 1964),
– U″ ( x)
rx ( ) =
----------------- ,
U′ ( x)
is the standard measure of this tendency. Properties of the
risk aversion measure (e.g., is constant, proportional, or
decreasing) correspond to analytical forms for utility functions
(Keeney and Raiffa 1976), or stochastic dominance
tests for decision making (Fishburn and Vickson 1978).
When outcomes are multiattribute (nonscalar), the outcome
space is typically too large to consider specifying
preferences without imposing some structure on the utility
function. Independence concepts for preferences (Bacchus
and Grove 1996; Gorman 1968; Keeney and Raiffa 1976)—
analogous to those for probability—define conditions under
which preferences for some attributes are invariant with
respect to others. Such conditions lead to separability of the
multiattribute utility function into a combination of subutility
functions of lower dimensionality.
Modeling risk aversion, attribute independence, and other
utility properties is part of the domain of decision analysis
(Raiffa 1968; Watson and Buede 1987), the methodology of
applied decision theory. Decision analysts typically construct
preference models by asking decision makers to make
hypothetical choices (presumably easier than the original
decision), and combining these with analytical assumptions
to constrain the form of a utility function.
Designers of artificial agents must also specify preferences
for their artifacts. Until relatively recently, Artificial
Intelligence PLANNING techniques have generally been limited
to goal predicates, binary indicators of an outcome
state’s acceptability. Recently, however, decision-theoretic
methods have become increasingly popular, and many
developers encode utility functions in their systems. Some
researchers have attempted to combine concepts from utility
theory and KNOWLEDGE REPRESENTATION to develop flexible
preference models suitable for artificial agents (Bacchus
and Grove 1996; Haddawy and Hanks 1992; Wellman and
Doyle 1991), but this work is still at an early stage of development.