Author: Andrew Dados
Date: 12:37:05 04/29/04
Go up one level in this thread
On April 29, 2004 at 09:26:21, Tony Werten wrote: >On April 29, 2004 at 08:40:03, Anthony Cozzie wrote: > >>On April 29, 2004 at 03:13:07, Tony Werten wrote: >> >>>Hi all, >>> >>>a while ago we had some discussions about diminishing returns in search for >>>chess. >>> >>>My opinion was that you can't prove that with programs searching d vs d+1 ply >>>depth because the advantage of the d+1 program gets smaller. ie at d=1 it has a >>>100% depth advantage, at d=2 it's 50% etc. >>> >>>Some people claimed that you can't compare it that way because bla bla >>>exponential something bla :) >>> >>>Well, I found an easier way to explain it. >>> >>>A few assumption: >>> >>>The easiest way to win is when you see a trick, your opponent doesn't see. >>> >>>The depth that needs to be searched to see a trick is equally divided. ie there >>>are as many tricks hidden 1 ply away as there are tricks at 2 ply ( it doesn't >>>really matter but it's easier to visualize ) >>> >>>w is player d+1 >>>b is player d >>> >>>d=1: b sees tricks 1 ply away, w sees ply 1 and 2 => w sees 2.0x as many tricks >>>d=2: b:1,2 w:1,2,3 => w: 1.5x >>>d=3: b:1,2,3 w:1,2,3,4 => w: 1.3x >>>... >>>d=10: b: 1..10 w: 1..11 => w:1.1 x >>> >>> >>> >>>Conclusion: There may or may not be diminishing returns in chess, but d vs d+1 >>>are not going to prove it, because those matches by itself are a clear example >>>of diminishing returns regardless what game is played. >>> >>>disclamer: I know chess isn't only about tricks, but it is an advantage to see >>>more of them then your opponent. Clearly the win percentage is depending on >>>other (random) stuff as well. BUT When you see less more, the advantage becomes >>>less. >>> >>>Tony >> >>I'm not even sure I agree with "the tricks are equally divided". > >Me neither, but for the thinking it is easier. > >>It would be >>possible to get some sort of statistics for this, but my guess is trick % >>declines with depth :( >> >>Even so, just the extra positional help makes more depth worth it IMO. > >It's the same. Replace "depth to see tricks" with "depth needed to find the >correct positional move" and the same story holds. > >Don't misunderstand, I know an extra ply of depth gives more strength. But every >extra ply will give less extra strength. Not because of chess ( or checkers or >whatever ), it's just a logical consequence that comes from search. > >Maybe not even search alone. > >If you drive 10 km/h faster than me, it makes a big diffence if I'm driving 10 >km/h. You will get there twice as fast. When driving 50, the difference is a lot >smaller. Certain researchers should call that "diminishing returns in car >driving" if they are consequent. I have my private proof of dimishing returns. Since chess is finite game (huge number of positions, but finite) then for each position there is some maximum depth beyond which better then current move can't be found. So even if completely solving position after 1.d4 may require 400 plies, after say 100 plies program will find no improvement to its move whether it goes 200 or 400 plies (it will only find score changes). So practical 'solving depth' for this position is 100. Now there are position where 'solving depth' is 0 (easy recaptures), 1, 2 etc Each class of solving depth takes away from total possible positions leaving less and less for bigger depths. It is like finite series: SD(0)+SD(1)+SD(2).... And since series is limited (by number of chess positions) it can't have property that SD(n+1) >= SD(n) for most n -Andrew- > >Tony > >> >>anthony
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