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The FIBS rating formula : why sqrt(n) ?

Posted By: Fabrice Liardet
Date: Thursday, 15 September 2011, at 6:49 p.m.

In Response To: The FIBS rating formula : why sqrt(n) ? (Fabrice Liardet)

Thanks for the numerous interesting answers, I still don't get it but am getting closer. Let me try to summarize what I have learned until now, which is a tough exercise since I don't understand all of the answers.

First of all I would like to separate clearly the three ingredients in the KG formula : 1) The standard Elo formula 2) The sqrt(n) adaptation for calculating the MWC. 3) The coefficient and the sqrt(n) adaptation of it.

1) The standard Elo formula, giving the expected result in function of the rating difference for one-game events. On http://en.wikipedia.org/wiki/Elo_rating it is said that it derives from the assumption that any player's results follow a normal distribution with the same variance. It has been said that it is a too strong assumption but until we use a different rating system in backgammon I think we have to content with it as an approximation.

2) The sqrt(n) adaptation for calculating the expected result in n-point matches. That is the crucial one and the one I was specifically asking about. It was based on what happened in no-cube, no-gammon matches. Maik and possibly Kevin seem to understand how the sqrt(n) comes from approximating the binomial distribution by the normal distribution, which in addition to the endorsement of Trice and Kaufman (even if they were less than clear about it) make me strongly believe that though it is an approximation it is no hack.

The question of how it works with real backgammon matches seemed more or less settled by statistics I read about a long time ago, probably on BKGM (I think they came from FIBS matches). As many expected, the sqrt(n) adaptation was a bit to steep and favoured the stronger player in the shorter matches. n=2 is of course a special case. That is interesting to know but I wouldn't investigate it further until I understand the basic formula.

3) The sqrt(n) in the coefficient for computing the new rating from the old one after a game. That is definitely not what I am interested in and I am ready to believe that it is a hack if one tells me so. Coefficients are hacks in most Elo-based rating systems anyway.

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