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Section 8 of the FIDE Rating Regulations provides the following table

p and dp

I understand that dp=800 and dp=-800 are arbitary caps at the top and bottom of the table.

However, in relation to the other scores, what's the formula for the conversion of p into dp, and vice-versa?

As noted in the comments, the regular Elo formula almost works but does not quite work. Does anyone know what the formula is for the adjustment made in the tails?

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It's basically just table 8.1a with the columns switched. E.g. the entry in table 8.1b with rating difference 92-98 and expected score (for the higher rated player) of .63, corresponds with the entry (.63, 95) in table 8.1a.

That table is based on the expected score formula mentioned here:

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)

but it's also possible to directly generate table 8.1b from this formula. For example, a rating difference of 92 points (in A's favor) leads to EA = 0.6294.

The inverse formula, calculating the rating difference which corresponds to a certain expected score, is RB - RA = 400 log10(1 / EA - 1).

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    I checked some calculations in MATLAB. I get that if I enter for example 1/(1 + 10^((300-392)/400)) it equals 0.6294 just like you said. However, if I enter 1/(1 + 10^((300-838)/400)) I get 0.9568, when the table seems to indicate that a difference of 538 rating points should accord to .97. The table seems to be slightly 'out' in relation to your formula as soon as we get into high probabilities. – user1205901 - Reinstate Monica Oct 31 '18 at 7:49
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    Yeah, maybe they have a slightly different formula for the tails. It could be that they don't want people to gain rating from playing a lot of games against much weaker opponents, and therefore using a slightly higher expected score. – Glorfindel Oct 31 '18 at 7:53
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As stated in 12.1 the formula above gives a close approximation to tables 8.1a and 8.1b. However it does not explain the precise construction of the table. Take fore example Rb-Ra = -736. According to the table 8.1b, the resulting expectation equals 100%. But according to the formula, the expectation is 98,57%.

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