In
economics, an ordinal utility function is a function representing the
preferences of an agent on an
ordinal scale. Ordinal
utility theory
As a topic of economics, utility is used to model worth or value. Its usage has evolved significantly over time. The term was introduced initially as a measure of pleasure or happiness as part of the theory of utilitarianism by moral philosoph ...
claims that it is only meaningful to ask which option is better than the other, but it is meaningless to ask ''how much'' better it is or how good it is. All of the theory of
consumer decision-making under conditions of
certainty
Certainty (also known as epistemic certainty or objective certainty) is the epistemic property of beliefs which a person has no rational grounds for doubting. One standard way of defining epistemic certainty is that a belief is certain if and o ...
can be, and typically is, expressed in terms of ordinal utility.
For example, suppose George tells us that "I prefer A to B and B to C". George's preferences can be represented by a function ''u'' such that:
:
But critics of
cardinal utility claim the only meaningful message of this function is the order
; the actual numbers are meaningless. Hence, George's preferences can also be represented by the following function ''v'':
:
The functions ''u'' and ''v'' are ordinally equivalent – they represent George's preferences equally well.
Ordinal utility contrasts with
cardinal utility theory: the latter assumes that the differences between preferences are also important. In ''u'' the difference between A and B is much smaller than between B and C, while in ''v'' the opposite is true. Hence, ''u'' and ''v'' are ''not'' cardinally equivalent.
The ordinal utility concept was first introduced by
Pareto in 1906.
Notation
Suppose the set of all states of the world is
and an agent has a preference relation on
. It is common to mark the weak preference relation by
, so that
reads "the agent wants B at least as much as A".
The symbol
is used as a shorthand to the indifference relation:
, which reads "The agent is indifferent between B and A".
The symbol
is used as a shorthand to the strong preference relation:
, which reads "The agent strictly prefers B to A".
A function
is said to ''represent'' the relation
if:
:
Related concepts
Indifference curve mappings
Instead of defining a numeric function, an agent's preference relation can be represented graphically by indifference curves. This is especially useful when there are two kinds of goods, ''x'' and ''y''. Then, each indifference curve shows a set of points
such that, if
and
are on the same curve, then
.
An example indifference curve is shown below:
Each indifference curve is a set of points, each representing a combination of quantities of two goods or services, all of which combinations the consumer is equally satisfied with. The further a curve is from the origin, the greater is the level of utility.
The slope of the curve (the negative of the
marginal rate of substitution of X for Y) at any point shows the rate at which the individual is willing to trade off good X against good Y maintaining the same level of utility. The curve is convex to the origin as shown assuming the consumer has a diminishing marginal rate of substitution. It can be shown that consumer analysis with indifference curves (an ordinal approach) gives the same results as that based on
cardinal utility theory — i.e., consumers will consume at the point where the marginal rate of substitution between any two goods equals the ratio of the prices of those goods (the equi-marginal principle).
Revealed preference
Revealed preference theory addresses the problem of how to observe ordinal preference relations in the real world. The challenge of revealed preference theory lies in part in determining what goods bundles were foregone, on the basis of them being less liked, when individuals are observed choosing particular bundles of goods.
Necessary conditions for existence of ordinal utility function
Some conditions on
are necessary to guarantee the existence of a representing function:
*
Transitivity: if
and
then
.
* Completeness: for all bundles
: either
or
or both.
** Completeness also implies reflexivity: for every
:
.
When these conditions are met and the set
is finite, it is easy to create a function
which represents
by just assigning an appropriate number to each element of
, as exemplified in the opening paragraph. The same is true when X is
countably infinite
In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural number ...
. Moreover, it is possible to inductively construct a representing utility function whose values are in the range
.
[Ariel Rubinstein, Lecture Notes in Microeconomic Theory]
Lecture 2 – Utility
/ref>
When is infinite, these conditions are insufficient. For example, lexicographic preferences In economics, lexicographic preferences or lexicographic orderings describe comparative preferences where an agent prefers any amount of one good (X) to any amount of another (Y). Specifically, if offered several bundles of goods, the agent will ch ...
are transitive and complete, but they cannot be represented by any utility function.[ The additional condition required is continuity.
]
Continuity
A preference relation is called ''continuous'' if, whenever B is preferred to A, small deviations from B or A will not reverse the ordering between them. Formally, a preference relation on a set X is called continuous if it satisfies one of the following equivalent conditions:
# For every , the set is topologically closed in with the product topology (this definition requires to be a topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
).
# For every sequence , if for all ''i'' and and , then .
# For every such that , there exists a ball around and a ball around such that, for every in the ball around and every in the ball around , (this definition requires to be a metric space
In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general set ...
).
If a preference relation is represented by a continuous utility function, then it is clearly continuous. By the theorems of Debreu (1954), the opposite is also true:
::Every continuous complete preference relation can be represented by a continuous ordinal utility function.
Note that the lexicographic preferences In economics, lexicographic preferences or lexicographic orderings describe comparative preferences where an agent prefers any amount of one good (X) to any amount of another (Y). Specifically, if offered several bundles of goods, the agent will ch ...
are not continuous. For example, , but in every ball around (5,1) there are points with and these points are inferior to . This is in accordance with the fact, stated above, that these preferences cannot be represented by a utility function.
Uniqueness
For every utility function ''v'', there is a unique preference relation represented by ''v''. However, the opposite is not true: a preference relation may be represented by many different utility functions. The same preferences could be expressed as ''any'' utility function that is a monotonically increasing transformation of ''v''. E.g., if
:
where is ''any'' monotonically increasing function, then the functions ''v'' and ''v'' give rise to identical indifference curve mappings.
This equivalence is succinctly described in the following way:
::An ordinal utility function is ''unique up to increasing monotone transformation''.
In contrast, a cardinal utility function is unique up to increasing affine transformation
In Euclidean geometry, an affine transformation or affinity (from the Latin, ''affinis'', "connected with") is a geometric transformation that preserves lines and parallelism, but not necessarily Euclidean distances and angles.
More generally, ...
. Every affine transformation is monotone; hence, if two functions are cardinally equivalent they are also ordinally equivalent, but not vice versa.
Monotonicity
Suppose, from now on, that the set is the set of all non-negative real two-dimensional vectors. So an element of is a pair that represents the amounts consumed from two products, e.g., apples and bananas.
Then under certain circumstances a preference relation is represented by a utility function .
Suppose the preference relation is ''monotonically increasing'', which means that "more is always better":
: