How to Estimate your ELO for a game, using ACPL, and what it realistically means?

ELO Rating estimation for Humans = 3100e^(-0.01Average Centipawn Loss).

So, this means Carlsen's most accurate chess game was 6.62 ACPL, which using this - estimates his peak rating at just above 2900. Of course, most games aren't going to be this accurate.

And, then, we'd expect this to be the case, below:
So, a typical GM game should have better than 21.5 ACPL. (ELO > 2500).
So, a typical Master game should have better than 34 ACPL. (ELO > 2200).
So, a typical Expert game should have better than 54 ACPL. (ELO > 1800).
So, a typical Average game should have better than 95 ACPL. (ELO > 1200).
So, a typical Beginner game should have an ACPL > 100 ACPL. (ELO < 1140).

This is interesting. You should test this approximation by calculating AACPL(player), i.e., average ACPL of a player over several games, or AACPL(rating range), i.e., the average ACPL of a collection of games between players in a given rating range. For example, you could calculate AACPL(2500-2550) and see how close is it to ~20. Unfortunately, this will require a lot of computation.

I just checked caissabase has about 1.5M games in which both players are rated 2400+. At 10s/game, this would be 15M seconds or more than 4000 hours of computation. But maybe such a large sample is not required. One could just use GM games in one year and do that analysis at fewer seconds per game. It is likely that the AACPL(GM games) has been decreasing over the years.

@kajalmaya said in #2:

This is interesting. You should test this approximation by calculating AACPL(player), i.e., average ACPL of a player over several games, or AACPL(rating range), i.e., the average ACPL of a collection of games between players in a given rating range. For example, you could calculate AACPL(2500-2550) and see how close is it to ~20. Unfortunately, this will require a lot of computation.

I just checked caissabase has about 1.5M games in which both players are rated 2400+. At 10s/game, this would be 15M seconds or more than 4000 hours of computation. But maybe such a large sample is not required. One could just use GM games in one year and do that analysis at fewer seconds per game. It is likely that the AACPL(GM games) has been decreasing over the years.

That correlation would be interesting to see! If someone wants to do that ans share the actual ACPL and compare it to this generalization they are welcome to.

Also, i found this online, seeming to confirm this approximate calculation, in table form.

https://m.facebook.com/photo.php?fbid=10219916749660312&set=p.10219916749660312&type=3&cft%5B0%5D=AZVpBHqRbnso6a4aBq2jqKXyTpZlxi5OuZZzdz8-vWLYjmYo7rYVwINAHTokn2o_x8QOlhIb-RA5SJoDP2SZLi-jdCHT6g0_s6tCYXxElysvqcJrcnpaTNJXFgB01V6dyA65h1nBQ4RhsKnCJmPacjGHZHfU06j-AcGgErxDm9MlG53cMhvjzxROaqZEttJ2C0q0SZxwVqgCfzRdeXbJk7Sd&tn=R%5D-R

According to Patrick Coulombe, "There is not a strong relationship between aCPL and rating:" https://web.chessdigits.com/articles/predicting-rating-from-centipawn-loss

@CaseyReese said in #4:

According to Patrick Coulombe, "There is not a strong relationship between aCPL and rating:" web.chessdigits.com/articles/predicting-rating-from-centipawn-loss

Correlation isn't causation - I agree. The relationship isn't a strong correlation because the ACPL method is imperfect; but having said that, Stockfish plays at a higher playing strength not throwing this method of calculation out the window, so ACPL apparently has validity at its level. After all, can you prove where ACPL method went obviously wrong, and situations where it would be better just to throw it out the window?

Completely wrong, I play regularly at chess.com where I am 900, 1000 elo, and many of my opponents play at under 40 centipawn loss even in 1+1 bullet matches, after I analyse our games with my Chessbase.

I just had this game here

My opponent played 35 centipawn loss. Surely a 1600 lichess player is not FIDE Master level strength, right?

acpl is relative to how your opponent plays, not just on yourself is what I find so that's just my ten pence worth ,it's not really relative to elo otherwise ratings would fluctuate more, wouldn't they? Sorry if I'm showing my ignorance xxx

There is correlation one just have reject moves that are in lost or win positions. also moves that are part of opening knowledge
for master moves up 12 were rejected. for average people bit less is enough
https://web.archive.org/web/20161211211029/http://chess-db.com/public/research/qualityofplay.html
from there you will find link to original academic paper.

@i_use_engine_to_win said in #6:

Completely wrong, I play regularly at chess.com where I am 900, 1000 elo, and many of my opponents play at under 40 centipawn loss even in 1+1 bullet matches, after I analyse our games with my Chessbase.

I just had this game here

Typically, I'd expect a player to have more ELO potiental (ability to play better against a standard bot) having long time controls than playing really fast, bullet time controls. I'm not saying that there is Master playing potiental from the same players at that level; especially, if that bullet time control is the time control a player is used to, but the ability to actually think about a position in depth vs whip out moves that are good is worth something. And, memorizing good opening lines, coupled by a quick resignation could account for this low ACPL, after making the key losing mistake too. Would ACPL have dropped more if the game continued? That game was short (23 moves!) Also, odds are this was an above average game you shared in the distribution of games you played. Hence, the correlation isn't truly inverse (or opposite of what I'd expect), on a macro basis given the total distribution of games. I'm not convinced by 1 counter-example, but I do find that game interesting in the distribution of total games played. Thank you for sharing.

Also, I'd expect there to be some leeway too, in measuring approximate ELO to ACPL. How much leeway (volatility)? About up to 61.8% ACPL upside/downside, at most, I think, maybe! Therefore 40 centipawn loss = 2078 Rating using the above calculation, but given leeway that is (40golden ratio) = 1623 rating, (40(golden ratio minus 1) = 2421. So, that would bound the ELO between ~1623 and ~2421. (if that is a lowest level ACPL game or a highest level ACPL game that they played). I think saying your opponent is between ~1623 and ~2421 is probably accurate for the majority of all games, where its actually at the lower end, given their true playing strength in your example.

@petri999 said in #8:

There is correlation one just have reject moves that are in lost or win positions. also moves that are part of opening knowledge
for master moves up 12 were rejected. for average people bit less is enough
web.archive.org/web/20161211211029/http://chess-db.com/public/research/qualityofplay.html
from there you will find link to original academic paper.

My reading of the First Research Reference would be that lost positions are disregarded.

The difference between the First and Second Research Reference with one saying No and the other Yes to correlation is very odd, giving the very similar underlying methods used. My guess is that the cause is in the raw data used, the First uses Lichess data of roughly 10million where the later uses 100,000 games. Differences in number here are unlikely to be significant - both samples are very large. But what I would consider significant is the nature of the game datasets under consideration.

My guess is that the lichess dataset was hugely biased towards Blitz/Bullet games, whereas the Second Research Reference wasn't. If so, then the conclusion would be that ACPL is not a good predictor of outcomes for short duration formats but is for longer duration formats.

No surprise, really.