Consider a table $w$ of win probabilities for N players well behaved such as
- No draws: $w(a,b) + w(b,a) = 1$, $w(a,a)=0.5$
- $w(a,b) > 0.5$ and $w(b,c) > 0.5$ imply that $w(a,c) > 0.5$
- if $w(a,b) > 0.5$ then for any other player $c$ we have $w(a,c) > w(b,c)$ and reciprocally.
It seems that it would be easy to build a compatible rating table $r(a)$ such that for any two players, $$r(a) > r(b) \iff w(a,b) > 0.5$$ In fact the third property is motivated to grant that a rating that is just the sum of wins, say $r(k)= (\sum_i w(k,i) - 0.5)/N$, is compatible: If $w(a,b) > 0.5$ then $\sum_i w(a,i) > \sum_i w(b,i)$ and so $r(a) > r(b)$. As a property, it is perhaps a bit overkill; not only the reciprocal is not needed, but also implies the second condition instead complementing it.
For the opposite direction, it seems also easy to provide some recipes to get a such well-behaved win table from a rating table.
But what about the complete route? I mean:
Under what conditions on $w()$ can be granted that a rating table $r()$ exists such that the win function can be exactly reconstructed from the rating?
I am aware that the ELO rating system claims to be able to reconstruct the win probabilities and to exist always; I do not know if it is proven for any arbitrary large number of players nor if it is the unique possible rating with reconstructability. I am aware also of other schemes to go from a rating table to a win table, as for instance the model $w(a,b)= r(a)/(r(a)+r(b)$), but I do not know if these schemes include a recipe to build the rating from the win function. Most importantly, besides particular models, I guess that there should be some general result on existence.
PS: comments about nomenclature are welcome too; I have not idea of the names of the properties and functions in the context of game theory. GPT 4 calls them symmetry, transitivity and dominance, but it does not look as stochastic dominance to me.