# Utility

In economics, utility is a measure of the happiness or satisfaction gained from a good or service.

The concept is applied by economists in such topics as the indifference curve, which measures the combination of a basket of commodities that an individual or a community requests at a given level(s) of satisfaction. The concept is also used in utility functions, social welfare functions, Pareto maximization, Edgeworth boxes and contract curves. It is a central concept of welfare economics.

The doctrine of utilitarianism saw the maximisation of utility as a moral criterion for the organisation of society. According to utilitarians, such as Jeremy Bentham (1748-1832) and John Stuart Mill (1806-1876), society should aim to maximise the total utility of individuals, aiming for 'the greatest happiness for the greatest number'.

Utility theory assumes that humankind is rational. That is, people maximize their utility wherever possible. For instance, one would request more of a good if it is available and if one has the ability to acquire that amount, if this is the rational thing to do in the circumstances.

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## Cardinal and ordinal utility

There are mainly two kinds of measurement of utility implemented by economists: cardinal utility and ordinal utility.

Utility was originally viewed as a measurable quantity, so that it would be possible to measure the utility of each individual in the society with respect to each good available in the society, and to add these together to yield the total utility of all people with respect to all goods in the society. Society could then aim to maximise the total utility of all people in society, or equivalently the average utility per person. This conception of utility as a measurable quantity that could be aggregated across individuals is called cardinal utility.

Cardinal utility quantitatively measures the preference of an individual towards a certain commodity. Numbers assigned to different goods or services can be compared. A utility of 100 units towards a cup of vodka is twice as desirable as a cup of coffee with a utility level of 50 units.

The concept of cardinal utility suffers from the absence of an objective measure of utility when comparing the utility gained from consumption of a particular good by one individual as opposed to another individual.

For this reason, neoclassical economics abandoned utility as a foundation for the analysis of economic behaviour, in favour of an analysis based upon preferences. This led to the development of tools such as indifference curves to explain economic behaviour.

In this analysis, an individual is observed to prefer one choice to another. Preferences can be ordered from most satisfying to least satisfying. Only the ordering is important: the magnitude of the numerical values are not important except in as much as they establish the order. A utility of 100 towards an ice-cream is not twice as desirable as a utility of 50 towards candy. All that can be said is that ice-cream is preferred to candy. There is no attempt to explain why one choice is preferred to another; hence no need for a quantitative concept of utility.

It is nonetheless possible, given a set of preferences which satisfy certain criteria of reasonableness, to find a utility function that will explain these preferences. Such a utility function takes on higher values for choices that the individual prefers. Utility functions are a useful and widely used tool in modern economics.

A utility function to describe an individual's set of preferences clearly is not unique. If the value of the utility function were to be, e.g., doubled, squared, or subjected to any other strictly monotonically increasing function, it would still describe the same preferences. With this approach to utility, known as ordinal utility it is not possible to compare utility between individuals, or find the total utility for society as the Utilitarians hoped to do.

## Utility functions

While preferences are the conventional foundation of microeconomics, it is convenient to represent preferences with a utility function and reason indirectly about preferences with utility functions. Let X be the consumption set, the set of all packages the consumer could conceivably consume. The consumer's utility function [itex]u : X \rightarrow \textbf R[itex] assigns a happiness score to each package in the consumption set. If u(x) > u(y), then the consumer strictly prefers x to y.

For example, suppose a consumer's consumption set is X = {nothing, 1 apple, 1 orange, 1 apple and 1 orange, 2 apples, 2 oranges}, and its utility function is u(nothing) = 0, u(1 apple) = 1, u(1 orange) = 2, u(1 apple and 1 orange) = 4, u(2 apples) = 2 and u(2 oranges) = 3. Then this consumer prefers 1 orange to 1 apple, but prefers one of each to 2 oranges.

In microeconomics models, there are usually a finite set of L commodities, and a consumer may consume an arbitrary amount of each commodity. This gives a consumption set of [itex]\textbf R^L_+[itex], and each package [itex]x \in \textbf R^L_+[itex] is a vector containing the amounts of each commodity. In the previous example, we might say there are two commodities: apples and oranges. If we say apples is the first commodity, and oranges the second, then the consumption set X = [itex]\textbf R^2_+[itex] and u(0, 0) = 0, u(1, 0) = 1, u(0, 1) = 2, u(1, 1) = 4, u(2, 0) = 2, u(0, 2) = 3 as before. Note that for u to be a utility function on X, it must be defined for every package in X.

A utility function [itex]u : X \rightarrow \textbf{R}[itex] rationalizes a preference relation <= on X if for every [itex]x, y \in X[itex], u(x) <= u(y) if and only if x <= y. If u rationalizes <=, then this implies <= is complete and transitive, and hence rational.

In order to simplify calculations, various assumptions have been made of utility functions.

## Utility in game theory

In game theory, utility is represented as a function representing the anticipated payoff of each player corresponding to their selected strategy. The domain of any utility function is defined below.

Consider a system ζ of entities u, υ, ω­

A player has an anticipation in ζ for any given move and their utility function has a natural operation defined as:

αu + (1 - α)υ

where 0 ≤ α ≤ 1 and the probability of u is α(u)
and the probability of υ is (1 - α)(υ).

The correspondence of utility and preference is denoted by:

u → α = V(u)

u being the utility and V(u) the value attached to it.

The following axioms are required:

1) u > υ implies V(u) > V(υ) and is a complete ordering of ζ
2) u and υ can exist only in three mutually exclusive orderings:
u > υ; u < υ; u = υ;
and all ζ are fully transitive of order
3) u > υ implies that u > αu + (1 - α)υ
4) u > υ > ω implies that their exists an α such that
αu + (1 - α)υ > ω
therefore α(ζ) is continuous
5) entities in ζ can be combined algebraically such that
αu + (1 - α)υ = (1 - α)υ + αu
and
α(βu + (1 - β)υ) + (1 - α)υ = γu + (1 - γ)υ
where γ = α(β)

Daniel Bernoulli has shown how the personal utility vary with the personal degree of risk aversion, itself linked to the initial wealth situation of the person.

## Discussion and criticism

The theory of consumer choice has come under criticism from different angles. One is behavioral economics, where different experiments show that consumers have a higher loss aversion than preference for a related gain. Also, estimations for small probabilities and their utility payoffs are difficult for laypersons to estimate (See Matthew Rabin's homepage (http://emlab.berkeley.edu/~rabin) for some puzzles connected with consumer theory).

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