Interestingly, the winning chances decline the higher the ratings go - even if the difference in Elo remains exactly the same.
eg. The chance of a 1400 beating a 1200 (200 pts difference) is 70% vs 18% (remaining 12% for draws)
but the chance of a 2200 beating a 2000 is 64% vs 12% with an almost doubled 23% chance for draws.
Maybe you should look at it as 1500-1750 is half as good as a 1750-2000. A 1500 could then be argued as 1/4 of a GM, with the understanding the drawing chances increases. So, now you have another variable, the ability to draw. This is why a 1500 can spot mates and tactics when a GM is in a winning position but doesn't have a clue what to do in the middlegame.
The problem I see with isolated rating comparisons (1500 vs. 1600) is that we don't know if that 1500 is actually a 2000 or if 1600 is a 2000. Why did Carlsen become chess champion, but Jon Ludvig Hammer is shown as ranked 199?
It shows Magnus was 1650 at age 10 and jumped to 2127 at age 11. Was Hammer also around 1650 but only jumped to about 1800 (which would still sound pretty good)?
I see the info was Norwegian to FIDE, so let's compare different ages instead with the same question.
Magnus at 11 was 2175 (FIDE) and at age 13 2450 (FIDE). The question is the same, did Hammer make the same increase? And this highlights yet another variable, the pool of players you are comparing. Apparently, in Norway they ranked harsher? Magnus was competing in a Hogwart's league of players, and then when he joined FIDE he played weaker players?
I don't think twice as good is well defined in the question here.
If by twice as good you simply mean twice as likely to win against a certain opponent then you can directly calculate your answer from the elo formula.
Now a more interesting set of questions would be at what rating differences would you expect a player to know twice as many tactical patterns, or twice as many branches of lines in the openings they choose to play, or the ability to see a line with half the chance of a tactical oversight, or the ability to see twice as deep. These questions aren't really something that could be answered without extensive research, however one would intuit that in all cases except for opening theory, these would increase exponentially from beginner all the way up to about 1500, at which point the growth would slow, significantly more so by 1800, and again by 2000, and again by each level of master. Opening theory would likely see a slow linear uptick all the way up to about 2k, where the linear slope would increase drastically until the top level of play, when the linear slope would again increase drastically, only slowing at the very very top level of play.
My intuition about the number of tactical patterns know would be that a 1200 knows about 4 times as many tactical patterns as a 900. A 1500 4 times more than a 1200. And so on up until about 2000 when growth slows significantly.
My intuition about the opening lines that a 1200 would know vs a 900 would again be about 4 times the number of moves. About 1.5 times as much up to 1500. About 1.25 times as much up to 1800, about 1.25 times as much up to 2000, before increasing sharply again.
Accuracy of calculation I would intuit would be about twice as good in a 1500 as a 1200. About twice as good again in an 1800, and about 1.25 times as good in a 2000, and then maybe 1.1 times as good at a 2000, and again at master.
But these are all extremely hazy suggestions that couldn't be verified without an extremely difficult to run set of tests.
Twice as good could mean a lot of different things, most of them almost impossible to measure.
Again twice as strong meaning twice as likely to win against a certain opponent is exactly what the elo formula measures, and is satisfying enough of an answer for me, although a lot less interesting.
At what level of rating improvement, would you say you’re now “twice as good” as you previously were?
Can such a statement be made?
Looking at Elo scales, if you are 1500 and climb to 1620, that’s an increase of 120 Elo points. If you plug in those two Elo ranges into an Elo win probability calculator, it’ll say that the 1620 player has a 56% chance of winning, the 1500 has a 27% chance of winning and a draw is 17%. This would mean the 1620 player is 2.07x more likely to win.
But does this mean you’re “twice as good”?
An Elo difference of 200 points is loosely speaking - a 68% chance of winning vs the lower rated player having a 17% chance of winning, and the remaining 15% going to drawing chances. So having 200 points more means you’re almost 4 times more likely to win…
It’d be a misconception to think improving from 800 to 1600 would mean you’re twice as good, as in reality - the 1600 would score a win, 99.6% of the time! Or phrased differently, would be almost 200x more likely to win than the 800!
So simply doubling rating wouldn’t mean twice as good. Can you define being twice as likely to win as being twice as good, or twice as skilled?