Please Illustrate a Mathematical Analysis of This Position

[Bill Spight]

Kirby, whilst it might be an interesting intellectual exercise to try to express the difference in size of follow-ups of the atari versus placement in a single number, it's unlikely to help you play good Go. In the vast majority of cases they will be both be sente at an easy to judge time of the game.

Indeed.

If the difference is relevant, it is likely that using the "calculate the miai value of a move and play them in decreasing order" strategy will not lead to optimum play, as it is based on there being a continuum of slightly decreasing sized miai plays available which is never quite true, and those cases where there is a difference between the atari and placement will likely be exactly those where the 'lumpiness' of the size of remaining moves is large resulting in tedomari etc being a very important factor. In such cases thinking of positions in terms of the tree of swing values and how these can interact for the various positions on the board is what you need to do to find optimum play.

P.S. I've not actually studied this rigourosly, so maybe Bill or someone else may correct me, but that's my understanding of it and how I try to play the best yose on OGS and it works well for me.

The strategy of making the largest play is surprisingly robust. When I started composing whole board yose problems I was very careful about constructing the background plays because I did not want any tedomari surprises. The results were unrealistic positions. Later on I found that if I just constructed a realistic background, ever with relatively few plays, the largest play was nearly always the best play.

[Kirby]

Contradiction example:

Code: Select all

                A
               / \
              B  -5
             / \
            20  0

It took me awhile to figure out what this notation meant, but from what I can gather, this represents a tree for a local position. The "A" represents the current board state. The "-5" means that, if white plays the best move he has in this local area, he gets 5 points, and the position is "finished". The "B" means that black can play his best move in the local area, but it won't finish the position - he can either play again or white may respond. The "20" means that if black plays twice in a row, starting at "A", the "final" position is 20 points for black. The "0" means that if black plays from A, then white responds, the "final" position is even for black and white.

Are these assumptions OK so far? This is what the rest of my thoughts are based on.

But we do. If Black makes play that gains 7.5 points but gives White the chance to reply and gain 10 points, then Black should not make that play. It gives White 2.5 points for free. If Black never makes a play at A, then it makes no sense to count A as 2.5 points.

I'm split on this. On one hand, I can agree with the evaluation of A to be zero - given the correct timing (when white doesn't have a play worth 10 or more), if black plays at position A, white will definitely respond. So we can ignore the intermediate branches, and simply say that, "Yes. Black can play here, and he'll get zero."

However, what confuses me is the argument that was given:

If Black never makes a play at A, then it makes no sense to count A as 2.5 points.

Black does play at A, at least provided the scenario that was given as an explanation to follow this quote. However, it seems to make sense that black should play at A only once there are few enough places on the board such that he wants to prevent white from getting 5 points.

Perhaps the valuation of a board position is a function of the moves remaining in the game...?

Suppose that there are plays elsewhere that gain between 10 and 15 points...

The argument makes sense if there are plays elsewhere that gain between 10 and 15 points. If there are no such points, is it fair to say that the value of the position at A is 0?

---

Both of the comments I made above are related to the timing of play. Is this perhaps simply because what can be defined as "sente" and "gote" fluctuate throughout the game? At the beginning of the game a local position may not be sente for black, but as the endgame approaches, it becomes sente...?

If this is the case, then perhaps local positions have associated counts, but these counts change as the game draws to a close.

Is this correct?

[Kirby]

OK, that's a start.

How do you assess the positions where Black completes his threat?

Do you mean the game trees? If so, maybe after we've confirmed that my assumptions from the previous post are correct, I'll give it a shot.

[Kirby]

Kirby, whilst it might be an interesting intellectual exercise to try to express the difference in size of follow-ups of the atari versus placement in a single number, it's unlikely to help you play good Go. ...

Perhaps, but I'm not a good go player in either case.

[Kirby]

...In such cases thinking of positions in terms of the tree of swing values and how these can interact for the various positions on the board is what you need to do to find optimum play..

What is a swing value?

[jts]

...In such cases thinking of positions in terms of the tree of swing values and how these can interact for the various positions on the board is what you need to do to find optimum play..

What is a swing value?

http://senseis.xmp.net/?BasicEndgameTheory#toc6

(By the way - does anyone know how the SL tags are supposed to work? I've never gotten them to work.)

[Dusk Eagle]

They don't seem to work with hashes in them, so I guess if you want to link partway down a page you just have to use the URL tags. They're supposed to work like this or like this: BasicEndgameTheory

[Kirby]

...In such cases thinking of positions in terms of the tree of swing values and how these can interact for the various positions on the board is what you need to do to find optimum play..

What is a swing value?

http://senseis.xmp.net/?BasicEndgameTheory#toc6

(By the way - does anyone know how the SL tags are supposed to work? I've never gotten them to work.)

Thanks. Any ideas on the other questions I had, by the way?

[jts]

They don't seem to work with hashes in them, so I guess if you want to link partway down a page you just have to use the URL tags. They're supposed to work like this or like this: BasicEndgameTheory

I see - like an idiot, I was trying to put in the phrase "basic endgame theory", with spaces.

Kirby - the point bill is trying to make (I think) is that you can't measure the small differences between two similar endgame plays without using the most extreme endgame plays for each side as a point of comparison.

In other words, you need to know which points are at stake and which points aren't at stake in order to anchor your analysis. This is fairly easy to do on a board that has naturally reached yose, but weird to do on an empty board. I think bill is saying that if you make what you think is a good estimate on the anchor values, he can use that to help you do a consistent analysis of every branch of the yose tree, but he can't tell you where to start.

I'll switch to a different apple product to make diagrams.

[jts]

Okay, so presumably this is the most extreme position for Black:

$$Bc
$$ | . . . . . . . . . . .
$$ | . . . X X X X O O . .
$$ | . . . . X O O X O . .
$$ | . . . . . . O X O . .
$$ | . . . . . X . B C . .
$$ -----------------------

Click Here To Show Diagram Code

[go]$$Bc
$$ | . . . . . . . . . . .
$$ | . . . X X X X O O . .
$$ | . . . . X O O X O . .
$$ | . . . . . . O X O . .
$$ | . . . . . X . B C . .
$$ -----------------------[/go]

Now, I suppose that black could play this, say, as a ko threat, with the implied threat being:

$$Bc :w2: elsewhere
$$ | . . . . . . . . . . .
$$ | . . . X X X X O O . .
$$ | . . . . X O O X O . .
$$ | . . . . . . O X O 6 .
$$ | . . . . . X 5 1 3 4 .
$$ -----------------------

Click Here To Show Diagram Code

[go]$$Bc :w2: elsewhere
$$ | . . . . . . . . . . .
$$ | . . . X X X X O O . .
$$ | . . . . X O O X O . .
$$ | . . . . . . O X O 6 .
$$ | . . . . . X 5 1 3 4 .
$$ -----------------------[/go]

However, this seems profoundly unlikely, and I will assume that if is playable, forces -

$$Bc
$$ | . . . . . . . . . . .
$$ | . . . X X X X O O CC
$$ | . . . . X O O X O CC
$$ | . . . . . . O X O CC
$$ | . . . . . X 3 1 2 CC
$$ -----------------------

Click Here To Show Diagram Code

[go]$$Bc
$$ | . . . . . . . . . . .
$$ | . . . X X X X O O CC
$$ | . . . . X O O X O CC
$$ | . . . . . . O X O CC
$$ | . . . . . X 3 1 2 CC
$$ -----------------------[/go]

So this is the best possible local result for B, and this is the diagram from which we can calculate the swing in the score, treating this diagram as "W has nothing." Whether or not W gets points in the marked area is irrelevant for our purposes. (Except for the fact that saving the black stones is much bigger if W is weak or could become weak! Many yose problems hinge on threats to kill groups, or gote yose moves which are big because they weaken a group and turn other yose moves into sente threats. But we could do the problem on the assumption the the white group is absolutely unkillable, which I think is what you wanted - just so long as you understand we could also do it on the basis of the assumption that it will die if B captures, or on the assumption that this final diagram is sente and forces W to spend a move making an eye... each of the three assumptions leads to different mathematical analysis of the position.)

The best possible local result for W is a little bit trickier.

$$Wc
$$ | CCC . . . . . . . .
$$ | CCC X X X X O O . .
$$ | CCC . X O O X O . .
$$ | CCC 3 1 . O X O . .
$$ | CCC . . . . . . . .
$$ -----------------------

Click Here To Show Diagram Code

[go]$$Wc
$$ | CCC . . . . . . . .
$$ | CCC X X X X O O . .
$$ | CCC . X O O X O . .
$$ | CCC 3 1 . O X O . .
$$ | CCC . . . . . . . .
$$ -----------------------[/go]

At this point it would be much easier to figure out W's best possible local result if we knew which stones each side had around the marked area. For example, if this entire area could become black territory is quite big because is threatens something like --

$$Wc
$$ | . . . . . . . . . . .
$$ | . . b X X X X O O . .
$$ | . a 5 . X O O X O . .
$$ | . . . 3 1 . O X O . .
$$ | . . . . . . . . . . .
$$ -----------------------

Click Here To Show Diagram Code

[go]$$Wc
$$ | . . . . . . . . . . .
$$ | . . b X X X X O O . .
$$ | . a 5 . X O O X O . .
$$ | . . . 3 1 . O X O . .
$$ | . . . . . . . . . . .
$$ -----------------------[/go]

(Would be better at a? b? Hard to say!) On the other hand, if there is not a lot territory at stake here because B has already surrounded and killed a group in the corner, is much smaller, the swing between the best local result for each play is smaller, and the values of the plays change accordingly.

$$Wc
$$ | . BB . . . . . . . .
$$ | B W B X X X X O O . .
$$ | . W B . X O O X O . .
$$ | W W B 3 1 . O X O . .
$$ | . W B . . . . . . . .
$$ -----------------------

Click Here To Show Diagram Code

[go]$$Wc
$$ | . BB . . . . . . . .
$$ | B W B X X X X O O . .
$$ | . W B . X O O X O . .
$$ | W W B 3 1 . O X O . .
$$ | . W B . . . . . . . .
$$ -----------------------[/go]

(And of course, everything I said about the strength of the white group affecting the count applies equally to the black group.)

So anyway - when you say "I just want a mathematical analysis for a board that is empty except for this stones," I'm sure you see the problem. In a normal endgame problem, or at least a problem that hinges on counting rather than spotting tesuji, you see a large part of the board, and you can figure out the value of the most extreme local results for B and W on the basis of the surrounding stones (including the question of whether any groups are dead, or are exposed to lethal threats). Does this make sense? I hope this is helpful (and also hope that it is correct! .. if it isn't, I'm I'll hear all about it soon enough).

[Kirby]

...

So anyway - when you say "I just want a mathematical analysis for a board that is empty except for this stones," I'm sure you see the problem. In a normal endgame problem, or at least a problem that hinges on counting rather than spotting tesuji, you see a large part of the board, and you can figure out the value of the most extreme local results for B and W on the basis of the surrounding stones (including the question of whether any groups are dead, or are exposed to lethal threats). Does this make sense?...

Yes, this makes sense. But the reason that I am asking this is not for an analysis of the posted position. It is to get some examples of analysis. After I asked for this, I was told that it is too difficult without giving the rest of the board position. So I replied to say that the board was empty just to do that. I don't care what the position is - I just wanted an analysis.

Bill illustrated some interesting techniques with some kind of game tree, which is really exactly what I was looking for - an example of how some of this is done. The senseis pages have some additional information, too, now that I look at it.

So already, the responses to this thread have been useful to me.

My last set of questions are inquiries on the explanation that Bill already gave - which was already a good explanation.

It's nice to have a bunch of knowledgeable folks around here regarding this type of stuff, which is why I am eager to ask more questions about the information that's been presented.

My original question was probably too open-ended, I know. If I knew more about this topic, perhaps I could have presented a more focused question to begin with.

[jts]

Sorry for answering the wrong questions! There are a lot of question marks in this thread, and a lot of them involve sentence fragments that refer to a dialogue between you and Bill which probably makes more sense to you two than to me. But to answer one question you seemed to have -

Sente is used in different senses. (Same for gote.) We use it to mean "B played there and W replied," and also "B played there and W's best move was to reply," and finally "if B's best move were to play there, W's best move would be to reply." The first sense doesn't imply anything at all about the whole-board situation or the global temperature. The second sense requires us to know the whole-board situation, so we can compare W's best move elsewhere to B's local follow-up. The third sense doesn't require us to know anything about the rest of the board at any given time - it just requires that assume that when B plays , there are no bigger moves elsewhere.

The third sense is generally the most useful for counting problems. When we say "A is B's sente," or "A will become B's sente in yose," or "B has the privilege of playing A", we mean that the follow-up is bigger than the original move, so if A1 is the biggest move on the board, A2 will definitely be the biggest move on the board. This gives B a window in which he can play A in sente, but W will have to take gote to play the same point. By assuming that everyone will get their own sente points, you simplify the board a lot. (B normally doesn't care if W takes gote to take away B's sente, since B can expect to get an equally large gote move elsewhere in exchange.)

This sense of sente ("B's sente," "W's sente", "gote for both", "double sente") does require you to know the size of the follow-ups for both sides in the local situation, but it doesn't require you to know anything about the global temperature. If B waits until the temperature is really low and suddenly W is able to play a move, in sente, which had been B's privilege, that's bad play by B, but it doesn't affect the analysis of the position.