No, wait, cancel that. That's an obvious way of doing it, but yours is more general, because it would (I think) cover any formula, whereas mine covers only linear ones.

Also, mine covers non-Archimedean orders. The IOC's ranking system, for instance, is a lexicographic order in which one gold beats any number of silvers; that can't be expressed at all as an additive scheme with a positive real score for each of gold, silver and bronze, because for any additive scheme there must be some number of silvers large enough to beat one gold.

Good point. I think I assumed x,y and z would be rational numbers (equivalent to integers), and ignored tie-breaking as some extra step that might or might not involve counting red-haired witch-doctors in each team :)

(Although, I could retrospectively pretend that I intended the domain to include infinite ordinals.)