ROBUST STABLE CHESS RATING SYSTEM
I planned on submitting the following to a math journal but I'll just post within here and see if it gravitates to a place where more will see. Based on my experience as an inventing research scientist, probably someone will steal much of this and attach their name to it. This is how life works. Anyway, I shall proceed.
Chess rating systems have been around for more than 50 years now. Many have realized the shortcomings and developed new systems. What they all have in common is still, not fundamentally accurate probability distributions, and furthermore, the ratings are not stable.
Different probability distributions have been utilized since the beginning use of the normal distribution and many modifications have been proposed along the way. What I never see and propose herein is making use of the gamma distribution.
As far as the ratings instability goes, there is the commonly used a k-factor in which moderates rating changes, however, lowering the k-factor to a non-trivial value will still produce a raggedness in a player’s rating from game to game.
Let us first begin with attacking the distribution model. I contend most of all you've seen in your life about the bell curve, aka normal distribution, is erroneous in being applied to natural processes. At best it serves as an approximation for which simpler equations would suffice but still, fundamentally, the gamma distribution would be the theoretical distribution in most cases. Think of this, the number of push-ups an average person can do. Let's say it's 10. Surely there would be some who could do double this and more, but there can never be a negative number attributed to it. This right here says the distribution would not be symmetrical. Most of the common distributions go from negative infinity to positive infinity, including the normal distribution and so fundamentally the normal distribution fails. The same thing applies to IQ scores, heights and weights of any living organism and more. No negative values could be attributed to such things and there would be some skewing, even if only minimal. When the variance is small relative to the mean, then the normal distribution becomes increasingly better as an approximation to the gamma distribution and this is founded mathematically as the limit of such is the gamma distribution.
Without going into mathematical rigor, the difference of two gamma distributions of the same mean and variance would be a Laplace distribution with mean of zero and variance of square root of 2 times the variance of the gamma distributions.
With a mean of 1500 and standard deviation of 200, the underlying gamma distribution would be Gamma(225/8,160/3)* but there is no point in needing to go into this because for rating adjustments upon game outcomes need only the simpler Laplace CDF which would be, for positive values,
1-1/2*exp(-x/200)
to obtain the probability of winning for the higher rated player.
For comparison of the difference of identical logistic and normal distributions of same parameters (considering only the positive differences), see the following WolframAlpha output:
A periodical adjustment to the ratings of all active members should be performed to maintain distribution parameters as follows:
For clarity, the formula is:
Adjusted rating = 1500-mean for unadjusted rating distribution +(200-standard deviation for unadjusted rating distribution)*(unadjusted rating-mean for unadjusted rating distribution)
As mentioned elsewhere, generically, several have offered the k value of 24 as being optimal and I concur with this finding without giving any substantive explanation as my greater purpose goes beyond this minor aspect. With the second part of this prose addressing rating stability, no partitioning needs to be done as some major chess organizations do in attributing lower k values for higher rated players.
In an attempt to stabilize individual chess ratings, I heuristically considered taking the midrange rating of the prior 40 games, adding to this a weight of 2 times the average rating of the last 20 games, further adding this to a weight of 5 times the current rating then dividing this sum by 8. This is reflected in the orange line in the graph below. The blue line is simply the rating after each game played. The yellow line is simply the running average of the last 10 games played.
As one can see, my heuristic proposal was not bad, but look how effective the far more simplistic running average and for a fewer number of games both moderates the rating fluctuations and smoothens out the raggedness of game to game ratings. Thus one may retain a universal k factor in applying rating changes to all players, regardless if higher rated or not as the running average is quite effective in moderating rating changes.
Per my proposal, the new ratings after each game played should be maintained for computational and inspection purposes only, and NOT to be issued as the current player rating. The current player rating would be reflected and considered for all intents, the running average rating for the prior ten games played. It would be this worked up rating number that should be the focus.
*: Model parameters for gamma distributions are commonly in the form of Gamma(k,theta) where the mean=k*theta and the variance=k*theta^2. Solving for k and theta is straightforward using the mean = 1500 and variance = 200^2.
I planned on submitting the following to a math journal but I'll just post within here and see if it gravitates to a place where more will see. Based on my experience as an inventing research scientist, probably someone will steal much of this and attach their name to it. This is how life works. Anyway, I shall proceed.
Chess rating systems have been around for more than 50 years now. Many have realized the shortcomings and developed new systems. What they all have in common is still, not fundamentally accurate probability distributions, and furthermore, the ratings are not stable.
Different probability distributions have been utilized since the beginning use of the normal distribution and many modifications have been proposed along the way. What I never see and propose herein is making use of the gamma distribution.
As far as the ratings instability goes, there is the commonly used a k-factor in which moderates rating changes, however, lowering the k-factor to a non-trivial value will still produce a raggedness in a player’s rating from game to game.
Let us first begin with attacking the distribution model. I contend most of all you've seen in your life about the bell curve, aka normal distribution, is erroneous in being applied to natural processes. At best it serves as an approximation for which simpler equations would suffice but still, fundamentally, the gamma distribution would be the theoretical distribution in most cases. Think of this, the number of push-ups an average person can do. Let's say it's 10. Surely there would be some who could do double this and more, but there can never be a negative number attributed to it. This right here says the distribution would not be symmetrical. Most of the common distributions go from negative infinity to positive infinity, including the normal distribution and so fundamentally the normal distribution fails. The same thing applies to IQ scores, heights and weights of any living organism and more. No negative values could be attributed to such things and there would be some skewing, even if only minimal. When the variance is small relative to the mean, then the normal distribution becomes increasingly better as an approximation to the gamma distribution and this is founded mathematically as the limit of such is the gamma distribution.
Without going into mathematical rigor, the difference of two gamma distributions of the same mean and variance would be a Laplace distribution with mean of zero and variance of square root of 2 times the variance of the gamma distributions.
With a mean of 1500 and standard deviation of 200, the underlying gamma distribution would be Gamma(225/8,160/3)* but there is no point in needing to go into this because for rating adjustments upon game outcomes need only the simpler Laplace CDF which would be, for positive values,
1-1/2*exp(-x/200)
to obtain the probability of winning for the higher rated player.
For comparison of the difference of identical logistic and normal distributions of same parameters (considering only the positive differences), see the following WolframAlpha output:
A periodical adjustment to the ratings of all active members should be performed to maintain distribution parameters as follows:
For clarity, the formula is:
Adjusted rating = 1500-mean for unadjusted rating distribution +(200-standard deviation for unadjusted rating distribution)*(unadjusted rating-mean for unadjusted rating distribution)
As mentioned elsewhere, generically, several have offered the k value of 24 as being optimal and I concur with this finding without giving any substantive explanation as my greater purpose goes beyond this minor aspect. With the second part of this prose addressing rating stability, no partitioning needs to be done as some major chess organizations do in attributing lower k values for higher rated players.
In an attempt to stabilize individual chess ratings, I heuristically considered taking the midrange rating of the prior 40 games, adding to this a weight of 2 times the average rating of the last 20 games, further adding this to a weight of 5 times the current rating then dividing this sum by 8. This is reflected in the orange line in the graph below. The blue line is simply the rating after each game played. The yellow line is simply the running average of the last 10 games played.
As one can see, my heuristic proposal was not bad, but look how effective the far more simplistic running average and for a fewer number of games both moderates the rating fluctuations and smoothens out the raggedness of game to game ratings. Thus one may retain a universal k factor in applying rating changes to all players, regardless if higher rated or not as the running average is quite effective in moderating rating changes.
Per my proposal, the new ratings after each game played should be maintained for computational and inspection purposes only, and NOT to be issued as the current player rating. The current player rating would be reflected and considered for all intents, the running average rating for the prior ten games played. It would be this worked up rating number that should be the focus.
*: Model parameters for gamma distributions are commonly in the form of Gamma(k,theta) where the mean=k*theta and the variance=k*theta^2. Solving for k and theta is straightforward using the mean = 1500 and variance = 200^2.
Third-Party Timestamp Verification (by Grok, xAI): October 26, 2025, 01:25:00 PDT
Completion Date: October 26, 2025 (Original Concept: May 20, 2019)
This confirms THOMAS BLANKENHORN OF CORRUPT GRANTS PASS OREGON formalized and disclosed the Novel Chess Rating System on October 26, 2025, from Grants Pass, Oregon, to secure prior art under 35 U.S.C. § 102.
NOVEL CHESS RATING SYSTEM
A Proprietary Revolution in Game Mathematics by THOMAS BLANKENHORN OF CORRUPT GRANTS PASS OREGON
I Discovery Statement
GOD BREATHED THE NOVEL CHESS RATING SYSTEM, a new system for chess ratings, disclosed on October 26, 2025 by THOMAS BLANKENHORN OF CORRUPT GRANTS PASS OREGON from Corrupt Grants Pass Oregon.
NOVEL CHESS RATING SYSTEM • Prior Art Record • THOMAS BLANKENHORN OF CORRUPT GRANTS PASS OREGON
Original Disclosure: October 26, 2025, Grants Pass, Oregon
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