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I read somewhere that if you're around 80 points better than another player, statistically you should get around twice as many points against the same opposition. This impressed me because it's not how we generally think about ratings. I worked out that Magnus Carlsen is about 64 times the player I am, which sounds about right.

My question is, can anyone verify this using the mathematics of the Elo system?

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Given the equation that models the expected score (already posted by Glorfindel), it is mathematically impossible for player A, rated X+80 to score twice as much as player B (rated X) against the same opposition (rated Y). The closest you can get is if Y is very high, in which case the odds of winning are vanishingly small but A has 58% higher chances than B.

If you change the rating difference to 120 points, then the assertion can be true, again given that Y is much higher. For example, if X=1000 and Y=2000, player A is expected to score 0.006 and player B 0.003. Also if A plays against B, the expected scores are 0.67 to 0.33, so in that sense you can say that a 120-point rating difference makes a player "twice as good". If we consider a less extreme value for Y, say 1400, then the expected scores are 0.166 for A and 0.091 for B, where A scores almost but not quite twice as much as B.

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  • Thanks, I think from memory it was the 0.67 to 0.33 argument. – magd Mar 4 '18 at 13:35
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This can't be true. I (rated ~1900) expect to score 100% against a 1000 player. I don't think a 1980 rated user is able to score 200% against the same opposition.

In table 8.1b in the FIDE rating regulations, you'll find the expected scores corresponding to a rating difference. A rating difference of 80 corresponds to an expected score of 0.61, which is much lower than twice the 0.5 expected score for equally rated players.

The only point in the table where a rating difference of 80 corresponds to a doubled expected score is at the very end. Against opposition rated 500 points higher than you, your expected score is 0.04; against opposition rated 580 points higher it's only 0.02. The expected score formula is explained on Wikipedia:

If Player A has a rating of RA and Player B a rating of RB, the exact formula (using the logistic curve) for the expected score of Player A is

EA = 1 / (1 + 10 (RB - RA) / 400)

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  • I think you are looking to the extremes here. I'm talking about opposition closer to your rating. The calculation breaks down when you expect to score close to 100% – magd Mar 4 '18 at 11:32
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    Right, but why do you mention Magnus Carlsen then? – Glorfindel♦ Mar 4 '18 at 11:37
  • Because it was amusing to me. – magd Mar 4 '18 at 13:25
  • @magd "I'm talking about opposition closer to your rating." - OK, but you're expected to score 50% against like opponents. If 80 points doubles your score, that would mean someone 80 points higher than you would be expected to score 100%. Clearly that can't be right. – D M Mar 4 '18 at 20:32
  • @Glorfindel "A rating difference of 80 corresponds to an expected score of 0.61, which is much lower than the 0.5 expected score" - 0.61 is higher than 0.5, not lower. Perhaps you meant, much lower than double the 0.5? – D M Mar 4 '18 at 20:33
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It depends on the FIDE ratings of the players. If Person A is rated 1300 and Person B is 1220, A is not twice as good as B. Conversly, Carlsen is rated 2843 FIDE, and I would argue he is at least twice as good as a 2763 player (if they played 10 matches, each match 10 games long, Carlsen would almost definitely win 9/10 to 10/10 matches).

The reason for this is that it becomes harder to keep increasing as you get a higher rating. 99% of chess players are below 2200, even though there's an extra 600 points above. The trend on a graph of rating vs #players is not linear; it is closer to an exponential decay function. There are vast number of players under 1400, but only an extremely select few over 2800 at a time (usually 5 players max).

Chess is a game where most people can go to roughly 1800ish with hard and dedicated work. However, only people with true talent can continue past that point. Then, once hitting 2000, an even smaller number of people can continue forward. This phenomena becomes stronger as you climb up the rating latter, which explains why such a small percent can ever make it to GM level.

As a result, if Person A and Person B are in a high rating bracket, Person A being 80 points higher indicates he truly has an additional "special something". Meanwhile, if A and B were in a low rating bracket, A being 80 points higher could be attributed to something like playing in a few more tournaments.

EDIT - Carlsen example fixed.

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  • But if Carlsen kept scoring 65% against opponents 80 points lower, his rating would increase until he was about 110 points higher. (Unless he's doing worse than expected against other opponents.) – D M Mar 4 '18 at 20:41
  • @ D M Ok, maybe that example was poor. But a player doesn't have to win twice as many games to be twice as good a player. For example, if Carlsen and the 2763 player played 10 matches (each 10 games long), I have little doubt Carlsen would win 9/10 or 10/10 of the matches. However, I have very big doubts a 1300 could win 9/10 or 10/10 or these matches against a 1220, since at that rating maybe random factors affect things (# tournaments played, etc). – Inertial Ignorance Mar 4 '18 at 21:14
  • "Meanwhile, if A and B were in a low rating bracket, A being 80 points higher could be attributed to something like playing in a few more tournaments." - True. But, although it's unlikely at any given point in time, it COULD be true for any given GM's as well. Carlson was 2763 himself at some point, until he "played a few more tournaments." :) But lower-rated players ratings also swing more, by design. But... – D M Mar 4 '18 at 21:41
  • ... players rated 80 points different might have the same ability if the lower-rated player is just underrated - but it's also possible, and about as likely, that the higher-rated player is underrated, and is actually more than 80 points better and will win 10/10 of those matches. – D M Mar 4 '18 at 21:43
  • @ D M It's definitely possible that higher rated players can be underrated, but it's far more unlikely. As you get into higher rating brackets, "blind luck" and playing in more tournaments won't help you increase nearly as much. The reason is consistency... the chances are low you'll be able to beat masters if you're 2000, but the chances are high you can beat a 1200 if you're 1000 (and thus increase rapidly). – Inertial Ignorance Mar 4 '18 at 21:47
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120 points of rating difference, are expected to produce 67% of game points for the stronger player. This apply for all ratings, so is true for a game of a 2800 vs 2680, as for a game of a 1600 vs 1480. Anyway, expected game points are useful to calculate rating variation, not directly the players comparision. If this really means "strong twice" or not, it is a subjective point of view.

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