On Wed, 2008-09-24 at 19:48 +0200, Erik van der Werf wrote: > I'm quite confident that 4.0 is the correct komi for 6x6.
I am quite confident that it is 2.0 :-)
I admit there is some room for error on my part, but I have just done a
fairly significant study of 6x6 Go using Leela. My primary room for
doubt is if there is some kind of end of game issue (in programs like
Leela) such as seki that causes a gross and systematic error.
Here is why I think komi should be 2.0 and if you can prove me wrong, I
think we will probably both have learned something interesting - I hope
you can.
Here is my analysis:
I did an analysis of 784 6x6 leela games. These games were played at
2.5 komi and white tends to win most of the games.
After converting all the game to a canonical representation, I
discovered that Leela always plays the first 4 moves the equivalent of
this:
C3 D4 C4 D3 - all 784 games started like this or the equivalent
After this, BLACK varied significantly:
BLACK WINS/GAMES PERCENTAGE
----------------- -------------
C3 D4 C4 D3 C2 0 out of 6 = 0.000 percent
C3 D4 C4 D3 C5 1 out of 4 = 25.000 percent
C3 D4 C4 D3 D2 103 out of 428 = 24.065 percent
C3 D4 C4 D3 D5 82 out of 339 = 24.189 percent
C3 D4 C4 D3 E3 0 out of 4 = 0.000 percent
C3 D4 C4 D3 E4 0 out of 2 = 0.000 percent
C3 D4 C4 D3 E5 0 out of 1 = 0.000 percent
For the rest of this discussion, the Wins and Percentage are ALWAYS
from BLACK'S point of view.
In this sequence, the only statistically interesting moves are D2 and
D5 for black, because these 2 choice constitute the vast majority of
the games. (It might be slightly more interesting if other moves showed
black doing well, but in the minor lines of play black is losing.) The
2 "good" moves appear to be approximately equal in value ... however,
let's check that out.
Let's consider D2 first. If black plays D2 white plays either E2 or
C5. C5 appears to be a mistake. When white plays C5 black wins 70%
of the games. However, Leela only played that move 20 times. 408
times it played E2 doing quite well - holding black to about 22%
BLACK WINS/GAMES PERCENTAGE
----------------- -------------
C3 D4 C4 D3 D2 E2 89 out of 408 = 21.814 percent
C3 D4 C4 D3 D2 C5 14 out of 20 = 70.000 percent
Let us assume that C5 is a blunder and with correct play
black aways wins (all of this assume 2.5 komi.)
AT this point black plays 3 different moves, all of them
losing:
C3 D4 C4 D3 D2 E2 C2 0 out of 7 = 0.000 percent
C3 D4 C4 D3 D2 E2 E4 1 out of 42 = 2.381 percent
C3 D4 C4 D3 D2 E2 D5 88 out of 359 = 24.513 percent
Again, the only interesting move here is D5 for black.
White responds with E5 or C2. C2 appears to be a blunder and Leela
played it 15 times, losing every time as white. However, E5 keeps
black down to 21.221 percent:
C3 D4 C4 D3 D2 E2 D5 E5 73 out of 344 = 21.221 percent
C3 D4 C4 D3 D2 E2 D5 C2 15 out of 15 = 100.000 percent
At this point we seem to be well into the 6x6 game and none of blacks
responses seem to be game winning.
Carrying this out 1 pair of moves farther (which I am not including
here), I see more of the same - there is nothing that indicates
that black has a surprise game winning move that white cannot avoid.
Ok. So let's go back to our other early black choice, 5. D5
C3 D4 C4 D3 D5 82 out of 339 = 24.189 percent
Leela played 2 moves here, again, one of them may be a blunder because
it allows black to win 81% of the games:
C3 D4 C4 D3 D5 C2 13 out of 16 = 81.250 percent
C3 D4 C4 D3 D5 E5 69 out of 323 = 21.362 percent
So let's see if E5 is interesting. If it is, we might be able to
furnish empirical proof that black can win a 2.5 komi:
C3 D4 C4 D3 D5 E5 D2 69 out of 284 = 24.296 percent
C3 D4 C4 D3 D5 E5 F5 0 out of 1 = 0.000 percent
C3 D4 C4 D3 D5 E5 C5 0 out of 4 = 0.000 percent
C3 D4 C4 D3 D5 E5 E3 0 out of 34 = 0.000 percent
Apparently, only 1 black choice here is reasonable, D2.
After Black plays D2, Leela probably loses after white plays C5, so E2
is the move of choice for white here:
C3 D4 C4 D3 D5 E5 D2 E2 62 out of 276 = 22.464 percent
C3 D4 C4 D3 D5 E5 D2 C5 7 out of 8 = 87.500 percent
The 3 lines Leela plays as black are:
C3 D4 C4 D3 D5 E5 D2 E2 E3 30 out of 128 = 23.438 percent
C3 D4 C4 D3 D5 E5 D2 E2 E4 32 out of 143 = 22.378 percent
C3 D4 C4 D3 D5 E5 D2 E2 E1 0 out of 5 = 0.000 percent
So now we have 2 lines to chase down, E3 and E4.
To make a long story shorter, skipping ahead to all the variations
and ignoring E1 which loses in all 5 lines we get:
C3 D4 C4 D3 D5 E5 D2 E2 E3 E4 E1 30 out of 128 = 23.438 percent
C3 D4 C4 D3 D5 E5 D2 E2 E4 E3 E6 32 out of 143 = 22.378 percent
^
And it looks like there is no salvation for black, that I can detect
in the data.
So I tried to find an example of a move for black that wins and that is
also out of Whites control. There are of course lines of play that
work for black but they depend on white playing badly (apparently.)
Of course it's entirely possible Leela missed a great winning line
somewhere, and I'm not a go player so I didn't look at the moves
myself, I am relying entirely on Leela's game play and statistics.
Some interesting (and tentative) conclusions:
1. Komi for 6x6 under these rules should be 2.0
2. The very strong and mighty leela still plays far from perfect at
this level on 6x6.
3. It would be easy to construct a book for Leela which would
correct the errors I found. White made errors that allowed
black to win and black made errors that at least might have made
it a little harder for white to win.
4. I believe Leela, at a higher level and with a "correction" book
would play perfect or very close to perfect on 6x6. This may
depend on seki issues however, it may not be possible for Leela
(or other Go programs) to play perfectly without some minor
adjustments to handle the weird corner cases.
- Don
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