Cardinal Utility

Meaningfulness

A statement involving numerical scales is meaningful if its truth value remains unchanged if every scale is replaced by another scale. For $\langle X,\succ\rangle$, the set of scales are those related by composition with increasing transformations. So $u(x)>u(y)$ is preserved by increasing transformations, and so is meaningful. Utility differences $u(w)-u(x)>u(y)-u(z)$ are not preserved under increasing transformations, and so are not meaningful.

Meaningfulness is characterized by the set of transformations that preserve the scale properties. Thus we say that preferences are ordinal because utilities are invariant to increasing transformations.

Utility is cardinal if the set of transformations is restricted to those that are positive affine: $\phi(u)=a+bu$ with $b>0$. What kind of relational structure on a set $X$ of objects will have this property?