In decision theory, a multi-attribute utility function is used to represent the preferences of an agent over bundles of goods either under conditions of certainty about the results of any potential choice, or under conditions of uncertainty.
Under certain conditions, a person's preferences can be represented by a numeric function.
The article ordinal utility describes some properties of such functions and some ways by which they can be calculated.
Another consideration that might complicate the decision problem is uncertainty.
Although there are at least four sources of uncertainty - the attribute outcomes, and a decisionmaker's fuzziness about: a) the specific shapes of the individual attribute utility functions, b) the aggregating constants' values, and c) whether the attribute utility functions are additive, these terms being addressed presently - uncertainty henceforth means only randomness in attribute levels.
This uncertainty complication exists even when there is a single attribute, e.g.: money.
Again, under certain conditions the preferences can be represented by a numeric function.
The article Von Neumann–Morgenstern utility theorem describes some ways by which they can be calculated.
The most general situation is that there are both multiple attributes and uncertainty.
The preferences here can be represented by cardinal utility functions which take several variables (the attributes).
which represents the person's preferences on lotteries of bundles.
is higher under A than under B: If the number of possible bundles is finite, u can be constructed directly as explained by von Neumann and Morgenstern (VNM): order the bundles from least preferred to most preferred, assign utility 0 to the former and utility 1 to the latter, and assign to each bundle in between a utility equal to the probability of an equivalent lottery.
[1]: 222–223 If the number of bundles is infinite, one option is to start by ignoring the randomness, and assess an ordinal utility function
, such that: The problem with this approach is that it is not easy to assess the function r. When assessing a single-attribute cardinal utility function using VNM, we ask questions such as: "What probability to win $2 is equivalent to $1?".
The latter question is much harder to answer than the former, since it involves "value", which is an abstract quantity.
A possible solution is to calculate n one-dimensional cardinal utility functions - one for each attribute.
[1]: 221–222 Often, certain independence properties between attributes can be used to make the construction of a utility function easier.
Hence, if an agent has additive-independent utilities, he must be indifferent between these two lotteries.
[1]: 229–232 A fundamental result in utility theory is that, two attributes are additive-independent, if and only if their two-attribute utility function is additive and has the form:
If the function u is additive, then by the rules of expectation, for every lottery
: This expression depends only on the marginal probability distributions of
This result generalizes to any number of attributes: if preferences over lotteries on attributes 1,...,n depend only on their marginal probability distributions, then the n-attribute utility function is additive:[1]: 295 where
Much of the work in additive utility theory has been done by Peter C. Fishburn.
Note that utility independence (in contrast to additive independence) is not symmetric: it is possible that attribute 1 is utility-independent of attribute 2 and not vice versa.
Given attributes 1,...,n, if any subset of the attributes is utility-independent of its complement, then the n-attribute utility function is multi-linear and has one of the following forms: where: It is useful to compare three different concepts related to independence of attributes: Additive-independence (AI), Utility-independence (UI) and Preferential-independence (PI).
[1]: 344 AI and UI both concern preferences on lotteries and are explained above.
PI concerns preferences on sure outcomes and is explained in the article on ordinal utility.
Moreover, in that case there is a simple relation between the cardinal utility function
representing the preferences on lotteries, and the ordinal utility function
PROOF: It is sufficient to prove that u has constant absolute risk aversion with respect to the value v.