Ordinal utility

In economics, an ordinal utility function is a function representing the preferences of an agent on an ordinal scale. The ordinal utility theory 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 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 the only meaningful message of this function is the order ${\displaystyle u(A)>u(B)>u(C)}$; 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.

Suppose the set of all states of the world is ${\displaystyle X}$ and an agent has a preference relation on ${\displaystyle X}$. It is common to mark the weak preference relation by ${\displaystyle \preceq }$, so that ${\displaystyle A\preceq B}$ reads "the agent wants B at least as much as A".

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