Editing Ordinal utility (section)

== 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

:<math>v(A) \equiv f(v(A))</math>

where <math>f: \mathbb{R}\to \mathbb{R}</math> 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]]. Every affine transformation is monotone; hence, if two functions are cardinally equivalent they are also ordinally equivalent, but not vice versa.