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Don Dailey

Joined: 29 Apr 2008
Posts: 4320

Post subject: Re: Elo Increase per Doubling    Posted: Wed May 09, 2012 5:19 pm

petero2 wrote:
Don wrote:
 petero2 wrote: Yes, the logarithmic formula goes to infinity, so not a good approximation for large x. Here is an exponential fit instead: Elo=1495+2996*(1-exp(-Name/12.8))

I've added 2 more levels to the same test and I'm going to get a larger sample of games. This is interesting, so even though I don't know if it's a very accurate way to estimate the highest rating it's a lot of fun to try.

I wrote a quicky program to find the 3 constants in this formula that minimizes the "least squares" error. With the new data points I get this:

Elo = 1488.20 + 2692.70 * (1 - exp( -level / 11.09 )

And the maximum achievable ELO is ... (drum roll please) is 4180! I don't have large samples at these high levels and I will continue to run the test for at least a couple more days or longer - and recheck the estimate.

It may be interesting to run the same test on a different program and see if the estimate is in the same ballpark. Preferably a program that has a solid fixed nodes testing level. This would make the "guess" more believable if they agreed.

Don

My parameters were also computed by minimizing the least squares error. I happened to have an old octave implementation of the Gauss-Newton method that I used. To see how much the estimate is affected by measurement errors, I added normally distributed noise with standard deviation 10 to the rating values and computed the corresponding maximum ELO. I repeated this 100000 times and made a histogram:

Average value: 4497
Standard deviation: 116

Although, I believe an even bigger error source is the fact that the true rating curve is most likely not an exponential function, so the large extrapolation is probably unsound.

ELO itself of course is not perfectly sound for this and I agree there is too much we don't know. However I'm pretty amazed at how will this exponential curve fits the data so far at these levels.

Also, I have believed for a long time that we have AT LEAST 1000 more ELO to achieve perfect play. Of course that is just my own intuition, a bigger source of error than any of these other things

I am playing a 5m+5s match with Houdini 1.5 64 right now and I notice that 40% of the games are draws. That means each program had about 30 chances out of 100 to convert a loss to a draw. It would take a LOT of ELO to do that.

But we are not even considering the lost opportunities to convert draws to a win that both program almost certainly missed. So it's very easy for me to believe that another 1000 ELO is not that much and it could be much higher.

 Code: Rank Name           Elo      +      -    games   score   oppo.   draws    1 4428.15      3006.4   43.4   43.4     228   50.9%  3000.0   39.5%    2 Houdini_1.5  3000.0   43.4   43.4     228   49.1%  3006.4   39.5%

Anyway, it's fun to speculate on this. I have some confidence in it because I don't believe any program is capable of playing perfect chess (not matter how much resources given) due to GHI and zugzwang issues. So any asymptote could be short of the mark. Maybe in 20 years we will look back and not be able to believe our progress and computers will be playing 1000 ELO stronger and still winning and losing games!
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Adam Hair Mon May 07, 2012 11:40 am
Julien MARCEL Mon May 07, 2012 12:04 pm
Adam Hair Mon May 07, 2012 4:48 pm
Julien MARCEL Mon May 07, 2012 4:54 pm
Adam Hair Mon May 07, 2012 8:50 pm
Julien MARCEL Mon May 07, 2012 9:00 pm
Mark Pearce Thu May 10, 2012 4:43 pm
H.G.Muller Mon May 07, 2012 12:13 pm
Michel Van den Bergh Mon May 07, 2012 2:01 pm
Adam Hair Mon May 07, 2012 5:12 pm
Don Dailey Mon May 07, 2012 6:40 pm
Robert Wagner Campos Tue May 08, 2012 12:38 am
Don Dailey Tue May 08, 2012 3:32 am
Adam Hair Tue May 08, 2012 10:02 am
Julien MARCEL Tue May 08, 2012 10:55 am
Adam Hair Tue May 08, 2012 8:08 pm
Don Dailey Tue May 08, 2012 11:29 am
Kai Laskos Tue May 08, 2012 11:59 am
Adam Hair Tue May 08, 2012 8:13 pm
Kai Laskos Tue May 08, 2012 10:21 pm
Don Dailey Tue May 08, 2012 10:52 pm
Adam Hair Tue May 08, 2012 11:14 pm
Kai Laskos Wed May 09, 2012 12:20 am
Adam Hair Wed May 09, 2012 2:44 am
H.G.Muller Wed May 09, 2012 8:05 am
Adam Hair Wed May 09, 2012 5:12 pm
Larry Kaufman Thu May 10, 2012 3:11 am
Don Dailey Thu May 10, 2012 5:03 am
Larry Kaufman Thu May 10, 2012 5:19 am
Don Dailey Thu May 10, 2012 12:43 pm
Peter Österlund Thu May 10, 2012 4:59 pm
Julien MARCEL Thu May 10, 2012 5:09 pm
Don Dailey Thu May 10, 2012 5:41 pm
Don Dailey Thu May 10, 2012 5:46 pm
Ed Schroder Tue May 08, 2012 12:09 pm
Don Dailey Tue May 08, 2012 12:10 pm
Ed Schroder Tue May 08, 2012 12:28 pm
Don Dailey Tue May 08, 2012 12:36 pm
Ed Schroder Tue May 08, 2012 1:15 pm
Don Dailey Tue May 08, 2012 2:37 pm
Julien MARCEL Tue May 08, 2012 3:15 pm
Don Dailey Tue May 08, 2012 6:15 pm
Peter Österlund Tue May 08, 2012 8:06 pm
Don Dailey Tue May 08, 2012 8:25 pm
Peter Österlund Tue May 08, 2012 9:00 pm
Julien MARCEL Tue May 08, 2012 9:02 pm
Don Dailey Tue May 08, 2012 9:06 pm
Don Dailey Wed May 09, 2012 4:26 pm
Peter Österlund Wed May 09, 2012 4:52 pm
Re: Elo Increase per Doubling Don Dailey Wed May 09, 2012 5:19 pm
Don Dailey Thu May 10, 2012 1:41 pm
Don Dailey Thu May 10, 2012 4:52 pm
Adam Hair Thu May 10, 2012 9:07 pm
Don Dailey Thu May 10, 2012 10:55 pm
Adam Hair Fri May 11, 2012 12:46 am
Adam Hair Fri May 11, 2012 1:06 am
Don Dailey Wed May 09, 2012 5:00 pm
Adam Hair Tue May 08, 2012 9:05 pm
Don Dailey Tue May 08, 2012 9:18 pm
Adam Hair Tue May 08, 2012 11:26 pm
Don Dailey Wed May 09, 2012 2:50 am
Adam Hair Tue May 15, 2012 3:17 am
Don Dailey Tue May 08, 2012 6:23 pm
Ed Schroder Tue May 08, 2012 12:18 pm
Don Dailey Tue May 08, 2012 12:25 pm
Don Dailey Tue May 08, 2012 12:27 pm
Robert Hyatt Tue May 08, 2012 3:36 pm
Don Dailey Tue May 08, 2012 5:32 pm
Daniel Shawul Fri May 11, 2012 5:48 pm
Don Dailey Fri May 11, 2012 6:30 pm
Julien MARCEL Fri May 11, 2012 6:52 pm
Don Dailey Fri May 11, 2012 7:05 pm
Daniel Shawul Fri May 11, 2012 7:08 pm

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