cardinal and ordinal utilities

February 04, 2026

A common mistake is to take one's formalism as metaphysics. This is especially true in domains tangentially related to the study of human behavior: "just because you can be described by a coherence theorem does not mean you are a coherence theorem."

I note that the difference between cardinal and ordinal utilities is not as deep as it may seem. Cardinalists use a utility function $u : X \to R$ to describe preferences, while ordinalists restrain themselves to only defining an ordering over $X .$

Under natural conditions, orderings over $X$ can be described as utility functions over $X .$ If the ordering is complete, transitive, continuous¹, and admits an order-dense subset², there exists a continuous function $u$ such that $u (x) \leq u (y)$ if and only if $x \leq y .$

As an example: if $X$ is any convex subset of $R^{n}$ and $\leq$ is continuous, then this holds and there exists a corresponding continuous utility function.

Give $X$ topological structure. If the upper and lower contour sets of an ordering $\leq$ are closed in $X$ for every $x \in X,$ then $\leq$ is continuous. ↩

There exists $Z \subseteq X$ such that for every pair $x, y \in X$ such that $x \leq y,$ there exists $z \in Z$ such that $x \leq z \leq y .$ ↩