[math-fun] Chess ratings probability distribution function
I found this (somewhat old) picture of the distribution of chess ratings: http://zwim.free.fr/ics/rating_distribution.gif It isn't exactly Gaussian, but it seems to have Gaussian qualities. 1. Has anyone done research on the distribution of chess ratings? 2. What is the _interpretation_ of chess ratings? I.e., if A has chess rating Ar and B has chess rating Br, what is the probability that A beats B?
That question about chess rankings reminds me of a book published in 2006 that mentions how meaningless these single-number ratings are: Does Measurement Measure Up?: How Numbers Reveal and Conceal the Truth by John M. Henshaw There is an excellent review in Nature, 27 July 2006, page 357. I quote: When you try to reduce separate measurements to a single number, you can get any number you want by adjusting the weights. In evaluating universities, why give 25% weight to peer assessment and 10% to expenditure per student, etc.? Changing the weights would produce different rankings... Neil On Fri, Sep 5, 2014 at 8:23 AM, Henry Baker <hbaker1@pipeline.com> wrote:
I found this (somewhat old) picture of the distribution of chess ratings:
http://zwim.free.fr/ics/rating_distribution.gif
It isn't exactly Gaussian, but it seems to have Gaussian qualities.
1. Has anyone done research on the distribution of chess ratings?
2. What is the _interpretation_ of chess ratings? I.e., if A has chess rating Ar and B has chess rating Br, what is the probability that A beats B?
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-- Dear Friends, I have now retired from AT&T. New coordinates: Neil J. A. Sloane, President, OEIS Foundation 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com
Does anyone know of work done to validate the predicted win/loss ratios of the Elo ratings? Charles Greathouse Analyst/Programmer Case Western Reserve University On Fri, Sep 5, 2014 at 9:15 AM, Neil Sloane <njasloane@gmail.com> wrote:
That question about chess rankings reminds me of a book published in 2006 that mentions how meaningless these single-number ratings are:
Does Measurement Measure Up?: How Numbers Reveal and Conceal the Truth by John M. Henshaw
There is an excellent review in Nature, 27 July 2006, page 357.
I quote: When you try to reduce separate measurements to a single number, you can get any number you want by adjusting the weights. In evaluating universities, why give 25% weight to peer assessment and 10% to expenditure per student, etc.? Changing the weights would produce different rankings...
Neil
On Fri, Sep 5, 2014 at 8:23 AM, Henry Baker <hbaker1@pipeline.com> wrote:
I found this (somewhat old) picture of the distribution of chess ratings:
http://zwim.free.fr/ics/rating_distribution.gif
It isn't exactly Gaussian, but it seems to have Gaussian qualities.
1. Has anyone done research on the distribution of chess ratings?
2. What is the _interpretation_ of chess ratings? I.e., if A has chess rating Ar and B has chess rating Br, what is the probability that A beats B?
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-- Dear Friends, I have now retired from AT&T. New coordinates:
Neil J. A. Sloane, President, OEIS Foundation 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com
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A good place to start on the mathematics of chess ratings is Mark Glickman's 1995 "A Comprehensive Guide to Chess Ratings". Links to that and subsequent work may be found here: http://www.glicko.net/research.html . A useful overview is here: http://www.chess.com/blog/kurtgodden/elo-to-glicko-your-rating-explained .
The ratings are designed to be self-adjusting. If my rating is x points greater than yours, then I gain f(x) points when I win and lose g(x) points when I lose, and you gain g(x) points when you win and lose f(x) points when you lose. f is decreasing and g is increasing. So for any fixed chance of winning, if we play each other a lot, our rating difference will converge to the appropriate number. For example, f(200) = g(200)/3. So suppose I win 3 games out of every 4 we play (pretend draws don't exist to simplify the discussion). So if my rating is more than 200 points above yours, playing will reduce my rating and increase yours, while if the rating difference is less than 200, playing will increase my rating and decrease yours, so if we play a lot, our rating difference will converge to 200. So far so good. The fact that f(200) = g(200)/3 encodes the fact that "a ratings difference of 200 represents a player who wins 3/4 of the time". But as soon as we add a third person to the system, things get trickier. In particular, the ratio of f(100) to g(100) encodes the probability that I win if my correct rating is 100 points more than yours. So imagine 3 players, with A rated 100 higher than B, who is rated 100 higher than C. Implicit in the functions f and g, evaluated at 100 and 200, is a statement of the form "if A beats B with probability p, and B beats C with probability p, then A beats C with probability q (= .75 in this case)". If this statement is false, there is no stable equilibrium, and ratings will converge to one equilibrium if A plays B and B plays C, and a different equilibrium if A plays C and A plays B. This is undesirable, because my rating should depend on how good I am, not on who I happen to play. The statement in the paragraph above is an empirically testable one, not a statement about probability and statistics, and I have no idea if anyone has done any empirical studies on it. Andy On Fri, Sep 5, 2014 at 9:35 AM, Charles Greathouse <charles.greathouse@case.edu> wrote:
Does anyone know of work done to validate the predicted win/loss ratios of the Elo ratings?
Charles Greathouse Analyst/Programmer Case Western Reserve University
On Fri, Sep 5, 2014 at 9:15 AM, Neil Sloane <njasloane@gmail.com> wrote:
That question about chess rankings reminds me of a book published in 2006 that mentions how meaningless these single-number ratings are:
Does Measurement Measure Up?: How Numbers Reveal and Conceal the Truth by John M. Henshaw
There is an excellent review in Nature, 27 July 2006, page 357.
I quote: When you try to reduce separate measurements to a single number, you can get any number you want by adjusting the weights. In evaluating universities, why give 25% weight to peer assessment and 10% to expenditure per student, etc.? Changing the weights would produce different rankings...
Neil
On Fri, Sep 5, 2014 at 8:23 AM, Henry Baker <hbaker1@pipeline.com> wrote:
I found this (somewhat old) picture of the distribution of chess ratings:
http://zwim.free.fr/ics/rating_distribution.gif
It isn't exactly Gaussian, but it seems to have Gaussian qualities.
1. Has anyone done research on the distribution of chess ratings?
2. What is the _interpretation_ of chess ratings? I.e., if A has chess rating Ar and B has chess rating Br, what is the probability that A beats B?
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-- Dear Friends, I have now retired from AT&T. New coordinates:
Neil J. A. Sloane, President, OEIS Foundation 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com
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-- Andy.Latto@pobox.com
The Elo rating system is derived from a simple model where each player's performance is normally distributed around some per-player score with some common-to-all-players standard deviation. Things are scaled so that a 200-point rating different corresponds to an 0.75 expected score -- that is, the stronger player's p_win + 1/2 p_draw = 0.75. http://en.wikipedia.org/wiki/Elo_rating_system I don't know anything about the distribution of ratings themselves. --Michael On Fri, Sep 5, 2014 at 8:23 AM, Henry Baker <hbaker1@pipeline.com> wrote:
I found this (somewhat old) picture of the distribution of chess ratings:
http://zwim.free.fr/ics/rating_distribution.gif
It isn't exactly Gaussian, but it seems to have Gaussian qualities.
1. Has anyone done research on the distribution of chess ratings?
2. What is the _interpretation_ of chess ratings? I.e., if A has chess rating Ar and B has chess rating Br, what is the probability that A beats B?
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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. 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 "Zermelos 1928 paper on measuring participants playing strengths in chess tournaments is a remarkable work in the history of paired comparison modeling." http://www.glicko.net/research/preface-z28.pdf At 05:23 AM 9/5/2014, Henry Baker wrote:
I found this (somewhat old) picture of the distribution of chess ratings:
http://zwim.free.fr/ics/rating_distribution.gif
It isn't exactly Gaussian, but it seems to have Gaussian qualities.
1. Has anyone done research on the distribution of chess ratings?
2. What is the _interpretation_ of chess ratings? I.e., if A has chess rating Ar and B has chess rating Br, what is the probability that A beats B?
participants (6)
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Andy Latto -
Charles Greathouse -
Hans Havermann -
Henry Baker -
Michael Kleber -
Neil Sloane