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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