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Suppose a player with ELO rating X has improved (for simplicity, without playing in tournaments) and she is now actually playing at the level X+N. How many games will it typically take for her rating to reach the level X+N, depending on N? Is there a quantitative research into that?

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The way FIDE rating (and rating change) is calculated is described in FIDE Rating Regulations effective from 1 July 2017. For an already rated player the calculations for working out the rating change are given in section 8.5.

From here it can be seen that how fast a rating changes for a given performance depende on the "k factor". This is defined in 8.56:

8.56 K is the development coefficient.
K = 40 for a player new to the rating list until he has completed events with at least 30 games.
K = 20 as long as a player's rating remains under 2400.
K = 10 once a player's published rating has reached 2400 and remains at that level subsequently, even if the rating drops below 2400.
K = 40 for all players until their 18th birthday, as long as their rating remains under 2300.
If the number of games (n) for a player on any list for a rating period multiplied by K (as defined above) exceeds 700, then K shall be the largest whole number such that K x n does not exceed 700.

Here are the relevant sections of 8.5 describing the calculations:

8.5 Determining the rating change for a rated player

8.51 For each game played against a rated player, determine the difference in rating between the player and his opponent, D.

...

8.54 A difference in rating of more than 400 points shall be counted for rating purposes as though it were a difference of 400 points.

8.55 (a) Use table 8.1(b) to determine the player’s score probability PD
(b) ΔR = score – PD. For each game, the score is 1, 0.5 or 0.
(c) ΣΔR x K = the Rating Change for a given tournament, or Rating period.

As to your question:

How many games will it typically take for her rating to reach the level X+N, depending on N?

Table 8.1(b) is non-linear so as well as depending on the k factor the number of games also depends on N in a non linear fashion. So, you will have to perform separate calculations for different values of N.

Here is an example for N=100. How many games would a player require to draw against players on average 100 points higher to raise the rating by 100 points?

In that case, according to table 8.1(b), PD = 0.36. So, ΔR = 0.14 and the number of games required for a 100 point rating gain = 100 / 0.14 / k.

For k = 10 number of games = 72
For k = 20 number of games = 36
For k = 40 number of games = 18

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I considered an idealized scenario in this answer:

Let's say a FIDE player is rated 1800, and has k=20. Every month this player plays in one 4-game tournament, in which they play against opponents all rated exactly 2000, and scores exactly 50%. How fast does the rating converge?

After six months, they would be rated 1898. After one year, the player's rating would be 1949. After 18 months,the rating would be 1975, and after two years, it would be 1987. So it seems that the player is getting approximately halfway to their performance rating every six months or so in this scenario.

After 32 months, the player would be rated 1997, and after that the rating no longer moves because a rating difference of 3 has an expected score of 50% according to the table FIDE uses.

Of course, a "typical" player won't play people with exactly their target rating. They'll play some people above and some people below. This tends to slow down the speed at which the rating converges with the performance. For example, the difference in expected score of an 1800 player and a 2000 player when playing a 2000 player is about 0.26, but the difference in expected score when playing a 2200 player is only about 0.16. A smaller difference in score means that it will take more games for the rating system to realize that the player really did get stronger.

In my own experience in the USCF, after resuming tournament play after a multi-year break (while improving my play during that break by playing online and joining a local club), it took approximately 38 games over 9 tournaments for my rating to reach its new level, about 300 points above my old one. The USCF, however, has "bonus points" which allow ratings to adjust more quickly when they are significantly lower than a person's playing strength, and also typically has a higher K value.

I tried manually rating my USCF games using FIDE rating rules, and it would have taken about 108 games over 25 tournaments for my rating to catch up to my approximate strength.

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It depends. Ability changes constantly.
Ratings only reflect performance history not ability per se.

How accurately do you want it to reflect the true ability at this moment?

Are you improving rapidly like many kids do?
Do you play a lot of rated games?

If your strength was not changing and your opponents did not change andor play erratically then ten games would give you a fair approximation, and 100 would give you an answer within the margin of error of all the other factors influencing it.

In your example ten would give you a fair idea and 100 a good one. But it would depend on how big an improvement and what sort of tournaments you played in. Open tournaments against a wide range of opponents is best as in a national swiss open. Playing in small local tournaments only and you might never get a true answer.

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