lightvector wrote:I'm still doubtful.
moha - if I understand correctly, you argue that the number of Elo points to perfect play can be very approximately bounded by 400 Elo * the number of points difference to perfect play, coming from the fact that 1 point is the minimum discrete unit that can be lost/gained.
This would seem to imply that there should not be more than 800 Elo points range in Chess. since there are only two units difference total in Chess. Define the "score difference" of a chess game as 1, 0, or -1 points depending on white wins / draw / black wins. But in actuality, we know that there are more than 800 Elo points range in Chess, even if we exclude very bad and very pathological players. Start with some decent 1400 Elo casual amateur player and we see there are still something like 1400 more Elo points to the top humans, and many many more Elo beyond that as you push into levels only achieved by bots. So the argument can't be correct.
Winning/losing the game may not be the smallest discrete error that can be made in chess. Errors may be smaller, such as losing a pawn (often considered a basic unit in chess), or losing the right to castle or exchanging a bishop for a knight, wich may be a only a fraction of this pawn unit. So I think chess is not really suitable for this approach of considering a smallest discrete error size.
But you could still use an error approach in chess and in go by ignoring the size of the error and only counting the number of mistakes that potentially affect the game result. A near perfect player (1 "class" below perfect play) would make 1 such mistake in 2 games and draw in the other game against a perfect player, so he would score 25% against a perfect player and maintain a rating 200 Elo below the perfect player. The size of the game losing error doesn't matter, nor if the error has some discrete fundamental unit. For this near perfect player, only the frequency matters.
lightvector wrote:I think this is because as players get better, even though any mistake objectively can cost you only a whole discrete unit or not at all, there are worlds of complexity and gradations hidden within the fight for that discrete unit, such that advantages and disadvantages and gains and losses from mistakes can happen at levels much finer than one whole unit for practical players, even if an omniscient objective view would not care until they finally "accumulated" to actually a whole unit of effect.
So the big flaw in the model is to treat mistakes as also being discretized to units, such as having a probability of losing a point on each move or not. In practice there exist "mistakes" that even if objectively they lose nothing, each successive one makes it increasingly "easier" to actually then make a move that does objectively lose later, unless the opponent's own mistakes cancels them out, and depending on the distributions of these "sub-unit" mistakes, this means you can get much more than a 75% winrate difference within the fight for that unit.
Now of course, there does seem to be some suggestiveness that Chess Elo range is finite - top level games have increasingly high draw percentages, particularly in the TCEC matches. But 400 Elo per "unit" is clearly too tight of a bound - exactly what the bound is should also depend on how much "subunit" gradation there is for practical real players and the mechanics of how it change or build up or not, which seems very hard to objectively pin down.
Using this error frequency approach, I think 400 Elo per unit is not too tight. Perhaps top chess engines are fairly close to making only 1 potentially game losing error per game (400 Elo below perfect play). Casual chess players probably make many potentially game losing errors per game. As a 1st order approximation, taking 4000 Elo as a top engine rating and 1200 Elo as a casual player rating, the casual player would be separated from the top engines by 14 "classes", so the frequency of potentially game losing errors might be 7 times higher than once per game (i.e. about 7 times per game).
lightvector wrote:
Maybe if players got within a point or two of optimal in Go it would turn out that there's no "sub-point" complexity in Go, but even only in the endgame you have things like fractional miai values, all those messy CGT problems whose only result is tedomari to round the score in your favor, and stuff like that - where one can make mistakes that are *much less* than one point - which makes me doubtful that this is true. And that's only the endgame.
So I would strongly suspect that the Elo range in Go is still finite (and not in some silly way like being exponentially large, but actually something not too unreasonable), but also that the last few points could quite easily be much more than 400 Elo per point.
Using the error frequency approach still leaves the question to determine what qualifies as a potentially game losing error. Perhaps in chess and go we could use this 1st approximation: count the number of times a player makes a mistake that potentially changes the game result (like failing to convert a decisive advantage to a win, failing to hold a draw or throwing away a game where you had a decisive advantage).
For go, there probably exists a correlation between the frequency of potentially game losing moves and average error size per move. This would allow an approximate conversion between the error size distribution (in units of komi points or 14 point ranks or handicap stones) and the error frequency (assuming 400 Elo points for each error frequency increment per game).
Such a conversion may also increase the accuracy of measuring the quality of play from a small number of games, because the signal to noise ratio of frequency would be small for low frequencies. Also, when the opponents are badly matched (even game with large skill gap), the frequency approach would be unreliable, because chances are that the stronger player quickly builds up such a big lead that none of the mistakes is large enough to potentially swing the result, resulting in underestimating the error frequency and thus overestimating the quality of play. Perhaps we should use a less conservative measure of what counts as a potentially game swinging error (e.g. also including errors with a size above some threshold, like a 14 point mistake in go or hanging a knight in chess, even when it doesn't affect the game result).
A correlation between error frequency (assuming this is represented by Elo rating) and error size (average centipawn loss per move) also seems to exist in chess. See average centipawn loss per move vs Elo rating:
https://chess-db.com/public/research/qualityofplay.html.
