Cardinal Utility

A set $X$ of objects is given, along with a quaternary relationship: $(w,x)\rhd(y,z)$ means, $w$ is preferred to $x$ more than $y$ is preferred to $z$. The more than relationship has the following properties.

1. $\rhd$ is asymmetric and negatively transitive.
2. If $(w,x)\rhd(y,z)$, then $(z,y)\rhd(x,w)$.
3. If $(a,b)\rhd(a',b')$ and $(b,c)\rhd(b',c')$, then $(a,c)\rhd(a',c')$.
4. An Archimedean axiom.
5. If $(w,x)\unrhd (y,z)\unrhd (w,w)$, then there are $a,b$ such that $(a,z)\equiv(y,z)$ and $(b,x)\equiv(y,z)$.

Theorem. If these five axioms are satisfied, then there is a function $u:X\to\R$ such that $(w,x)\rhd(y,z)$ iff $u(w)-u(x)>u(y)-u(z)$. If $v$ also represents $\rhd$, then $v(x)=a+bu(x)$ with $b>0$.