Komi on smaller boards

MarylandBill · Post by **MarylandBill** » Thu Apr 21, 2011 8:07 pm

Hi,

I am just curious, when you play on a smaller board, should the Komi be the same as it is on a full size board?

Thanks,
Bill

emeraldemon · Post by **emeraldemon** » Thu Apr 21, 2011 8:12 pm

Close enough. In 19x19, lots and lots of pro play confirms 7 to be about right. In smaller sizes there isn't nearly the volume of experience.

This might help:

http://senseis.xmp.net/?HandicapForSmallerBoardSizes

Harleqin · Post by **Harleqin** » Sat Apr 23, 2011 5:38 am

It depends on the size. Here are some values from the top of my head. The last two are educated guesses. The 7x7 value is mostly regarded as correct, although it is not completely proven yet, as far as I know.

2x2: 0
3x3: 8
4x4: 1
5x5: 24
7x7: 9
9x9: 6
13x13: 8

shapenaji · Post by **shapenaji** » Sat Apr 23, 2011 5:56 am

Harleqin wrote:It depends on the size. Here are some values from the top of my head. The last two are educated guesses. The 7x7 value is mostly regarded as correct, although it is not completely proven yet, as far as I know.

2x2: 0
3x3: 8
4x4: 1
5x5: 24
7x7: 9
9x9: 6
13x13: 8

Is there a theory out there somewhere suggesting convergence of komi to some value?

On an infinite board, what is the value of a move?

mw42 · Post by **mw42** » Sat Apr 23, 2011 6:02 am

shapenaji wrote:On an infinite board, what is the value of a move?

0.

gowan · Post by **gowan** » Sat Apr 23, 2011 6:06 am

shapenaji wrote:
Harleqin wrote:It depends on the size. Here are some values from the top of my head. The last two are educated guesses. The 7x7 value is mostly regarded as correct, although it is not completely proven yet, as far as I know.

2x2: 0
3x3: 8
4x4: 1
5x5: 24
7x7: 9
9x9: 6
13x13: 8

Is there a theory out there somewhere suggesting convergence of komi to some value?

On an infinite board, what is the value of a move?

I'd guess that the value of a move on an infinite "square" board is 0 since there is an infinite amount of potential territory and so whatever gain there might be from one move is 0 as a percentage. If there are also infinitely many stones of each color the game might never end. It's not clear that it makes sense to talk about go on an infinite board.

topazg · Post by **topazg** » Sat Apr 23, 2011 6:34 am

Harleqin wrote: 9x9: 6
13x13: 8

This is also what I heard (ok, actually 5.5 for 9x9, 7.5 for 13x13, and 6.5 for 19x19 - the 9x9 I know started from some professional opinion somewhere, but I can't remember who or when, and the 13x13 I have no idea about).

shapenaji · Post by **shapenaji** » Sat Apr 23, 2011 11:05 am

gowan wrote:
shapenaji wrote:
Harleqin wrote:It depends on the size. Here are some values from the top of my head. The last two are educated guesses. The 7x7 value is mostly regarded as correct, although it is not completely proven yet, as far as I know.

2x2: 0
3x3: 8
4x4: 1
5x5: 24
7x7: 9
9x9: 6
13x13: 8

Is there a theory out there somewhere suggesting convergence of komi to some value?

On an infinite board, what is the value of a move?

I'd guess that the value of a move on an infinite "square" board is 0 since there is an infinite amount of potential territory and so whatever gain there might be from one move is 0 as a percentage. If there are also infinitely many stones of each color the game might never end. It's not clear that it makes sense to talk about go on an infinite board.

Well, here's what I'm thinking, each move exerts a certain amount of influence around it, imagine each stone as a source of influence which drops off as a power-law or as an exponential. This influence reflects off sides of the board, leading to the strange nature of komi on small boards. (and in cases where the game is solved like on a 5x5, we have values of 24 for komi, which is silly, because this assumes perfect play, in which case it's not a game anyhow)

If we take theoretical komi to be the summed-up value of the influence around a stone, then komi is the value of the largest first move on the board.

If I assume influence is an inverse-squared law (Just a first approximation)

(and assuming a continuous board, for the time being (just so I don't have to do the discrete math))

then a stone in the corner, (where I can assume the greatest reflection) will effectively generate the influence of 4 stones, by the method of images.

the influence on the "real" portion of the board from these 4 stones, would then be:

integral(from 0->inf,integral(from 0->inf,
A/((x-xo)**2+(y-yo)**2)+A/((x-x1)**2+(y-y1)**2)+A/((x-x2)**2+(y-y2)**2)+A/((x-x3)**2+(y-y3)**2)

We then call this sum for the board involved, Komi

where A is a constant to be calibrated
xo, yo are the x,y coordinates of the actual stone
x1, y1 are the coordinates of image 1
x2, y2 are the coordinates of image 2
x3, y3 are the coordinates of image 3

(I'm currently a little busy, so I can't work this out just yet, but the logic here is that on a larger board, the number of places where you can get an advantage from reflection drops off.

As the board size gets larger and larger, the fraction of territory where reflection is an issue drops to 0, so if we can use the value of A from a 19x19 board, using the above equation. Then we would just need to find the integral of

A/(x**2+y**2) over all space, and that would be our convergent value of komi

shapenaji · Post by **shapenaji** » Sat Apr 23, 2011 11:08 am

An interesting caveat here is that under this, tengen gets better and better as the board gets smaller, (since, theoretically, there are 4 images contributing, (even more if the influence drops off slowly)

Also, I noticed a mistake above, a stone in the corner actually has 3 images, there's one on either side of the board, and then one behind the corner.

shapenaji · Post by **shapenaji** » Sat Apr 23, 2011 11:45 am

1/r**2 is wrong too, since that is clearly not integrable,

1/(1+r**2) is a possibility

Basically though, we want a quarter of the total potential of a quadrapole moment in the corner

xed_over · Post by **xed_over** » Sat Apr 23, 2011 7:31 pm

I don't think komi actually matters for amateurs.

When playing socially, I usually only count the score on the board (no komi).

daniel_the_smith · Post by **daniel_the_smith** » Sat Apr 23, 2011 7:38 pm

xed_over wrote:I don't think komi actually matters for amateurs.

When playing socially, I usually only count the score on the board (no komi).

Eh, komi definitely makes a difference to me. Especially on a small board.

Shaddy · Post by **Shaddy** » Sat Apr 23, 2011 7:41 pm

Komi has suddenly started mattering a lot more to me after 1-2d.

RobertJasiek · Post by **RobertJasiek** » Sat Apr 23, 2011 10:53 pm

emeraldemon wrote:In 19x19, lots and lots of pro play confirms 7 to be about right.

Why? Isn't Black still winning significantly more than 50%? So the komi must be 6.5 or greater!

RobertJasiek · Post by **RobertJasiek** » Sat Apr 23, 2011 11:01 pm

shapenaji wrote:If we take theoretical komi to be the summed-up value of the influence around a stone, then komi is the value of the largest first move on the board.

As I am about to show for 19x19 in Joseki / Volume 2 / Strategy, the value of the first stone's sum of territory and territorial value of influence is about 8.8 points. Details will be in the book.

Life In 19x19

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