[math-fun] Chess ratings
Henry Baker: My results from several hours of Googling:
1. Huge amount of research on chess ratings. scholar.google.com "chess" "Elo"
2. Elo model was invented by Zermelo 1928 and Bradley&Terry 1952, rediscovered by Ford 1957.
Bradley, Ralph A. and Terry, Milton E. (1952). "The rank analysis of incomplete block designs. I. The method of paired comparisons." Biometrika, 39, 324-45.
Elo, Arpad E. (1978). The rating of chessplayers, past and present. Arco Publishing: New York.
https://en.wikipedia.org/wiki/Elo_rating_system
Ford, Lester R. Jr. (1957). "Solution of a ranking problem from binary comparisons." American Mathematical Monthly, 64(8), 28-33.
As best I can tell, the Elo model is based on the "logistic" distribution rather than the Gaussian distribution. They look very similar, but the logistic has slightly fatter tails.
--not so. Elo's system in his book (which I read prior to writing my 1984 BS thesis) based on Gaussian, the others based on logistic. But these functions employed in the rating formulas, not as probability distributions of rating distribution. My 1984 bachelor's thesis at MIT, which my paper #20 was based on, shows Elo was wrong to do what he did, and logistic is superior to Gaussian where he used it, at least in the sense I proved "minimatch self-consistency" results for logistic (my "WM^n model") while Gaussian has no such. If Glickman claims otherwise it is perhaps because he never read Elo's book, or for some other reason, I have no idea. However, it appears in the meantime that the world in general has somehow realized that Gaussians suck (presumably due to experiments, since few members of the world showed any signs of reading my BS thesis), and most (all?) rating agencies have switched away from Elo's original system and to a logistic-base system.
sech((x-mu)/(2s))^2/(4s)
Just remember "sech-mate" ;-)
Its cdf is the hyperbolic tangent function!
https://en.wikipedia.org/wiki/Logistic_distribution
The best discussion of ZermElo I was able to find is:
"Introductory note to 1928" Mark E. Glickman 10 pages
"Zermelo's 1928 paper on measuring participants' playing strengths in chess tournaments is a remarkable work in the history of paired comparison modeling."
And as far as the distribution of chessplayer ratings is concerned, that is neither Gaussian nor logistic, it instead is more like a Gompertz extreme value distribution. My paper #20 explains how to derive such from a model, and my 1984 BS thesis performed a fit to the actual US Chess Federation rating distribution. The histogram of internet chess ratings Baker linked to, is a different data set, which also looks to my eye like it would be fit well by Gompertz, anyhow obviously better than any Gaussian or logistic could fit it. If Baker has the energy he might want to try such fits himself. In 1984 I had great difficulty doing the fits because this was 1984 and I was a math undergrad at MIT, which meant my access to computers was total dogshit. Some physicist who had a computer with commercial data fitting software agreed to help me in a halfassed way which just barely enabled me to do the job. Nowadays this sort of job should be far easier. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)
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Warren D Smith