Hello:
Daniel Shawul wrote:Hi Jesuz:
I understand your logarithmic fit but note that my data does not show significant diminishing returns so far. Maybe I need to test at bigger depths to show that but depth-15 is already too much. After a day and half run, here is new data with logarithmic interpolation
Code: Select all
Num. Name games score
0 scorpio6 10271 1567.5
1 scorpio7 13366 4279.5
2 scorpio9 16814 8437
3 scorpio11 14023 8185.5
4 scorpio13 9754 6111
5 scorpio8 16286 7323
6 scorpio10 15488 8442.5
7 scorpio12 11826 7154.5
8 scorpio14 6813 4703
9 scorpio15 4227 3230.5
Rank Name Elo + - games score oppo. draws
1 scorpio15 409 9 9 4227 76% 141 36%
2 scorpio14 338 8 7 6813 69% 149 38%
3 scorpio13 268 6 6 9754 63% 146 39%
4 scorpio12 184 6 6 11826 60% 87 37%
5 scorpio11 91 5 5 14023 58% 16 34%
6 scorpio10 -9 6 5 15488 55% -52 32%
7 scorpio9 -123 6 5 16814 50% -129 29%
8 scorpio8 -245 6 5 16286 45% -195 27%
9 scorpio7 -380 6 6 13366 32% -209 25%
10 scorpio6 -534 7 7 10271 15% -195 20%
As you can see the logarithmic fit of 1037ln(x) - 2397 (similar to yours) is very good and it doesn't show flattening out so quickly. Plot it and you will see you can reach astronomical elos > 20000 elo. That is why I think either more tests are required or there isn't significant dimnishing except at lower depths ...
Elo delta's b/n ply and ply-1 from top to bottom.
Daniel
I am glad to see that the logarithmic fit works fine! Well, regarding my model, this is a drawback... but more than 20000 Elo between depth 1 and depth d (in this case: Y(d) - Y(1) ~ 1037*ln(d/1) = 1037*ln(d)) would come with d > 2.37e+8, and I doubt that those depths can be reached. Please take a look to the first point of
this web:
The longest Chess game theoretically possible is 5,949 moves.
It means 11,897 or 11,898 plies; with the logarithmic fit and depending on each engine, the difference between depth 1 and this extreme depth would be around 9000, 10000, 12000... Elo, which is too much. But the game advances and the maximum possible depth also diminishes, so maybe differences are less than this upper bound. Differences between one ply and forty plies would be around 4000 Elo (depending on the engine; maybe a little more or a little less) with my cheap approach... I do not know if it is plausible or not, but my model suggests slow diminishing returns between consecutive depths.
Other question is the rightness (or not) of this number of maximum moves: 5949. I say that because if you take a look to the fourteenth and twentieth statements:
The number of possible ways of playing the first four moves per side in a game of Chess is 318,979,564,000.
According to the America’s Foundation for Chess, there are 169, 518, 829, 100, 544, 000, 000, 000, 000, 000 ways to play the first 10 moves of a game of Chess.
But we know that Perft(8) = 84,998,978,956 and it can be easily checked by JetChess 1.0.0.0 in my slow PC in just two minutes. This wrong number is an historical mistake that is explained at the bottom of this fantastic
web site by François Labelle. Regarding Perft(20), please read
this topic. It is also explained in Labelle's web.
So, we must take 5949 moves with a lot of care. Please read
this info trying to explain that the maximum number of moves should be above 5800; there are other interesting links in the comments of this web.
At least the fifth statement of the first web I link to seems good:
After each side has played three moves, the pieces could form any one of over nine million possible positions on the board.
The correct number is Positions(6) = 9,417,681 (easily checked with JetChess in nine seconds in my PC).
Regards from Spain.
Ajedrecista.