The only thing I don’t like so far, small things hang me up, is
If we now assume the log-odds can be written as a linear combination of the inputs (or more probabilistically, if we assume the conditional densities are Gaussian with a shared covariance matrix and equally likely classes), then…
I’ve absolutely no idea if that’s a reasonable assumption to make or not, is it a strong assumption a “normal”/natural assumption
That assumption probably contains a lot of the meat of the “why” I imagine, and it’s over and done with in a sentence.
I’m no mathematician, but that infinite sum converges (as I’m sure you knew).
I’m not going to attempt proving by calculation that an infinite number of wins and zero losses against an infinite pool of opponents with rating R will result in a divergent Elo rating (in theory, with infinite precision rating calculation and updates).
But maybe it is enough to consider this:
When a player (A) has a finite rating, there has to be a non-zero probability for them to lose against an anchor player (B) having some finite Elo rating. This non-zero probability determines the (finite) Elo difference between player A and the anchor player B.
But when a different player (C) wins an infinite number of games without ever losing to anchor player B, the probability of player B winning is evidently zero, so player C can’t have a finite rating. No matter how large (but finite) the Elo difference gets, the probability of losing never vanishes completely.
It seems tricky to systematically work through so many games in any practical rating system.
For the construction of such rating systems, are we allowing the possibility of a function that can select exactly one game out of each and every tournament that a player has participated in?
In case a professional player is reading this topic and looking for a player B who wouldn’t tire of losing against them, I volunteer. You can challenge me here on OGS any time!!