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

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

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

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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?

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0.

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.

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).

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

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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.

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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

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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.

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

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

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.

Since when is 7 no longer greater than 6.5?

Current statistics for professional play with 6.5 komi show a winning percentage of about 50.5%. Previously, at 5.5 komi, this was around 53%.

In which countries? I think in China and Taiwan with 7.5 komi (equivalent) they also have Black above 50%. Until dynamic komi and / or komi bidding are used, pro game statistics will remain too partial because playing style adapts to a given komi.

Given that I'm speaking of professional play at 6.5 komi, I think it is pretty obvious I'm talking about Japan and Korea.



In fact, according to SL's http://senseis.xmp.net/?Komi/Statistics, area scoring with a komi of 7.5 or 8 (Ing rules, so effectively 7.5) show a winning percentage of about 50.5% for White.

If anyone has statistics over a larger sample size, please post them.



At the moment, it is the best method we have, and the statistcis strongly hint that 7 komi is correct. We would only really have absolute certainty on the issue if we find perfect play, which we can pretty much rule out.

Rather they show that too few games are measured.

I wouldn't say that. Statistics on games in Japan back when komi was 4.5 indicated a median board score of 7. Ing based his komi of 7.5 on statistics many years ago. Many, many, many games have been used for komi statistics.

It is possible that, as pro play improves, komi will change. But the games which would indicate that have yet to be played.

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