Utility Theory

The branch of decision theory concerned with measurement and representation of preferences is called utility theory. Utility theorists focus on accounts of preferences in RATIONAL DECISION MAKING, where an individual's preferences cohere with associated beliefs and actions. Utility refers to the scale on which preference is measured.

Identification of preference measurement as an issue is usually credited to Daniel Bernoulli (1954/1738), who exhibited a prospect (probability distribution over outcomes) that had infinite expected monetary value, but apparently not infinite utility. Bernoulli resolved the "St. Petersburg paradox" by suggesting that utility be logarithmic in monetary amounts, which in this case would yield a finite expected utility.

That utility could apply to all sorts of outcomes, not merely monetary rewards, was first argued forcefully by Jeremy Bentham (1823/1789), who proposed a system for tabulating "pleasures" and "pains" (positive and negative utility factors), which he called the "hedonic calculus" (regrettably, as critics henceforth have confounded the idea of universal preference measurement with hedonism). Bentham further argued that these values could be aggregated across society ("greatest good for the greatest number"), which became the core of an ethical doctrine known as utilitarianism (Albee 1901).

Although modern economists are quite reluctant to aggregate preferences across individuals, the concept of individual utility plays a foundational role in the standard 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, J. 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, J → <var>R</var>, in the sense that U(y) ≤ U(x) if and only if y <∼ x. If U represents <∼, then so does j ° U, for any monotone function j 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

[F, p; F'](ω) ≡ p F(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


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),


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.

See also

Additional links

-- Michael P. Wellman


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