multiple-button go

Basically I've just put forward an idea on this page. It's not written particularly well and it's currently lacking in detail too. Also, potentially there are blatant mistakes. Hopefully you will find it interesting nonetheless. -- The Count

Karl Knechtel: this seems to be the same idea as environmental go but applied specifically to the endgame.

The Count: Yes, Apparently it is. It was pointed out by Bill Spight too (on the forums?). I didn't mean for it to be applied specifically to the endgame though by the way... it's just the only place you can do calculations.

Okay, I'll take this page down as I don't think it's interesting if we already have environmental go stuff. I might put some /discussion in environmental go and add links across SL to tie together some related ideas which I didn't come across by myself.

Bill: Well, the environmental go material is not developed on SL to the extent you have taken it. It's up to you, but I think what you have here is valuable. You also relate it to your view of button go, which does not appear elsewhere. I would say, keep it. :-)

The concept of button go can be extended by the use of multiple buttons. This idea has already been considered, and is called environmental go. I believe, however, that the main concern of environmental go as already discussed is assessing the value of board plays, and causing the biggest play (by miai counting) to also be the best play. Here we will consider the effect on the granularity or sharpness of the score. Granularity here will be used to describe the closest possible difference in score. For instance, with area scoring, the score (without komi) will usually be an odd number. That is, Black could win by 1 point or by 3 points, but not by 2 or 4 points. It is not always the case but if there is no seki then this is true. So, we can say the area score is coarser-grained than the territory score. For the purpose of this page, granularity will refer to the number of points that the score can differ by, so that area scoring has a granularity of 2 and territory has a granularity of 1.

It has already been shown that by using a button, the granularity of area scoring can be modified so that it is equivalent to territory scoring. Multiple-button go shows that the granularity can be made more fine without limit, and so it can be argued that the particular granularity of territory scoring is of no special significance.

The simple modification to button go is to use many buttons, rather than just one button. Consider that we use n buttons with values of i.b where i ranges from 1 to n and b is the value of the smallest button. For example, we could use six buttons worth 1/4, 2/4, 3/4, 4/4, 5/4 and 6/4 respectively.

(I ought to show here that the granularity is twice the value of the smallest button, given you have enough buttons. It's quite easy to see anyway. Since an edit, this is actually the most important part of this page.)

The following is really environmental go material... Sorry to leave it here, I really must go.

example  

We will use these six buttons in a simple example. What is the miai value of a play at a? Using area scoring, it is 3/2. Now, what is the difference in score between when Black goes first and when White does.

Black first  

takes the button worth 6/4 points. takes 5/4 button. takes 4/4 button. takes 3/4 button. takes 2/4 button. takes 1/4 button.

The score is +4-5 + (-6+5+4-3+2-1)/4 = -1/4 if we count just the area score for the nine points we can see on the board.

White first  

takes the button worth 6/4 points. takes 5/4 button. takes 4/4 button. takes 3/4 button. takes 2/4 button. takes 1/4 button.

The score is +3-6 +(6-5+4-3+2-1)/4 = -7/4

So the difference between Black first and White first is -1/4 + 7/4 = 3/2, the miai value of the first local play. This example doesn't prove anything, but you can work through examples to see that the value of sente is always the miai value as long as a few condition are met.

These are (A) that the highest value button is equal to the miai value of the best play and (B) that the size of the smallest button is small enough so that it can accurately account for the fractional part of the miai value. I think the exact size is given by having the smallest button be 1/d where d is twice the denominator of the miai value (and if there are lots of miai values, take the lowest common multiple of all of the denominators).

Concening (A), when it actually becomes time to make the board play of a certain miai value, the actual difference in the score between making that play and passing is the miai value. If there are higher value buttons still to take, you would of course take them first.

-- The Count

I'll need this bit later. Don't read.

It is easier to imagine we have an infinite number of buttons. First, realise that if Black and White both take buttons alternately in a sequence without making any board plays, if the first button is worth n and the button just smaller than the last button taken is worth m, then the player with sente originally has gained (n-m)/2 points. For example, if Black takes buttons worth 12/4, 10/4, 8/4, 6/4 and White takes buttons worth 11/4, 9/4, 7/4, 5/4, then the net difference is (12/4 - 4/4) / 2 = 1 to Black. Exactly how many buttons are in the range of 3 (12/4) points to 1 (4/4) point does not matter. The net gain for the player who takes the first button is always 1. So, if we imagine we have an infinite number of buttons in the range, the gain is still 1...

Infinite button go

Tas: You could take the step completely and go to an infinite number of infinitesimally close buttons. The stack would start at a certain number of points. Then if one player decides to take buttons he gets the number of points corresponding to the value of the stack. If the other player also takes a button, the players bid the number value at which they would stop, whoever bid the highest then makes a play and his bid becomes the new stack value, and the other player gets a number of points equal to half the difference between the former and later value. If the initial value is set large enough the perfect komi will be half this value. In effect this will make the komi decided by the pie rule, since whoever bids highest gets to play first.

In this way all plays will have a value exactly equal to their miai value, fully removing the effects of tedomari. Note that if the two players happened to bid exactly the same, it won't matter who gets to play, since they must have spotted a play of the same value - it could be chosen randomly.