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The FIBS rating formula : why sqrt(n) ?

Posted By: Michael Petch
Date: Tuesday, 13 September 2011, at 10:38 p.m.

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

Trice also had this to say on R.G.B in the Mid 90's, regarding the sqrt(n) the last 2 paragraphs shed some other light on the choice:

The KG system was written up in Inside Backgammon, Volume 1, Number 5 (Sept.-Oct. 1991) by Larry Kaufman, who invented the system. Vol. 1 of Inside Backgammon may still be available from the Gammon Press -- ask Robertie.

The KG system, as far as I know, uses the same formulas that FIBS does, except that KG starts off 'intermediate' players at a rating lower than 1500 (1400 or 1300 I think.)

Note that sqrt(n) occurs twice in the formula, once as an exponent in determining the winning probability P and again as a coefficient of P in calculating the rating change. Kaufman's justification of the exponent sqrt(n) is as follows: "... an answer is provided by random walk theory, which says that the expected distance of a random walker from his starting point is proportional to the square root of the number of random steps he has taken."

Skill-differenced match equity tables have actually been around since 1977 and the sqrt(n) prediction seems to conform pretty well to more sophisticated analytical models. But it would be EXTREMELY interesting to see a large volume of empirical data based on FIBS matches.

The other sqrt(n) which multiplies 4*P*(experience factor) to get the rating gain or loss has no special justification. In his article Kaufman says "It is clear that a long match victory should count more than a short match..." but to me it ISN'T clear. Given that the system says player A wins 40% against player B in a match, why should an upset in a long match count more than an upset in a short match? (Note that match length was already factored into the 40%.) This is what Doug Roberts calls a "truth-and-justice function" as opposed to its having any mathematical significance.

Originally the KG ratings used n rather than 4*sqrt(n). One noticable result was that people whose 1st tournament was Monte Carlo sometimes got truly spectacular ratings from the combination of long matches and a high experience coefficient. Going to 4*sqrt(n) reduced the disparity between matches of different lengths but also INCREASED the rating changes for short matches. In my opinion the factor is much too high -- totally ridiculous rating swings are very common both in the KG listings and on FIBS.

-- Walter Trice

https://groups.google.com/group/rec.games.backgammon/msg/217cc56b4d2c4540?hl=en

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