Here it is.

Click Here To Show Diagram Code

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

At the risk of repeating Robert's points.

It isn't just that it is too complicated. It is relatively undefined. Does the region to the right on the bottom side belong to White? Does the region to the top on the left side belong to Black? Does White need to capture the two Black stones? Does Black need to keep the corner? Without knowing the answers, or without restricting the region of play, assuming that everything outside it is alive, we cannot calculate the size of a play.

Another problem, which is non unsurmountable, is that you do not know where to play.

Click Here To Show Diagram Code

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

Does Black play the atari or Uberdude's tesuji?

Click Here To Show Diagram Code

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

Where does White play?

----

Here is something that will help. Here are your main variations.

Click Here To Show Diagram Code

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

Click Here To Show Diagram Code

[go]$$Wc
$$ | . . . . . . . . . . .
$$ | . . . X X X X O O . .
$$ | . . . . X O O X O . .
$$ | . . . 2 1 . O X O . .
$$ | . . . . . . . . . . .
$$ -----------------------[/go]

It appears that you think that this position is double sente. If so, then make the play. The proverb should really be this: Play double sente right away.

But if you want to calculate the size of a play, forget double sente. A double sente is a free lunch. There is no free lunch (in theory ).

So we need to consider these variations.

Click Here To Show Diagram Code

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

elsewhere.

Click Here To Show Diagram Code

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

elsewhere.

You see how important it is how secure the stones framing the corner are.

----

Actually, we have evidence that this is Black's sente, especially early in the game. If it is not, then it appears that someone has made a mistake. Either Black should not have endangered the two Black stones, or White should not have put them into atari.

If this is Black's sente, then obviously White's reverse sente is large, even if Black controls the left side. The sente is probably urgent.

If this is Black's sente, then Uberdude's play has a lot to recommend it. Its threat is smaller than that of the atari, but as long as White replies it looks better.

Click Here To Show Diagram Code

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

Click Here To Show Diagram Code

[go]$$Bc Black follower
$$ | . . . . . . . . . . .
$$ | . . . X X X X O O . .
$$ | C C C C X O O . O . .
$$ | C C C C . X O . O . .
$$ | C C C C C 1 . O . . .
$$ -----------------------[/go]

Black gets 13 points in the corner.

Click Here To Show Diagram Code

[go]$$Wc White follower
$$ | . . . . . . . . . . .
$$ | . . . X X X X O O . .
$$ | C C C C X O O . O . .
$$ | C 0 6 2 1 B O . O . .
$$ | C 8 7 5 9 3 C O . . .
$$ -----------------------[/go]

elsewhere.

After - , White has a sente follower. The usual assumption is that Black cannot afford to play the ko. (Edit: Note that if Black does not control the left side White may play at 6 or the 3-3.)

Black gets 6 points and White gets 3, for a net of 3 points to Black. The average of the two gote followers in 8 points, which is how we evaluate the result of the Atari diagram.

Click Here To Show Diagram Code

[go]$$Bc Placement
$$ | . . . . . . . . . . .
$$ | . . . X X X X O O . .
$$ | C C C C X O O X O . .
$$ | C C C 3 2 4 O X O . .
$$ | C C C . 5 1 . 6 . . .
$$ -----------------------[/go]

is the orthodox response to . If is at 4, then Black can capture , but takes gote. Black can wait to play .

Black gets 10 points in the corner, 2 points better than with the atari.

Click Here To Show Diagram Code

[go]$$Bc Placement, var. 1
$$ | . . . . . . . . . . .
$$ | . . . X X X X O O . .
$$ | . . . . X O O X O . .
$$ | . . . . . . O X O . .
$$ | . . . . . 1 . 2 . . .
$$ -----------------------[/go]

Suppose that White replies with .

Click Here To Show Diagram Code

[go]$$Bc Black follower
$$ | . . . . . . . . . . .
$$ | . . . X X X X O O . .
$$ | C C C C X O O . O . .
$$ | C C C C . 1 O . O . .
$$ | C C C C C X . O . . .
$$ -----------------------[/go]

Black gets 13 points in the corner.

Click Here To Show Diagram Code

[go]$$Wc White follower
$$ | . . . . . . . . . . .
$$ | . . . X X X X O O . .
$$ | C C C C X O O . O . .
$$ | C C C 2 1 5 O . O . .
$$ | C C C 4 3 B C O . . .
$$ -----------------------[/go]

Because of , White cannot play at 4.

Black gets 10 points in the corner, White gets 3 points, for 7 points net to Black.

The average is still 10 points.

Bravo, Uberdude!

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

Thank you, Bill. This is kind of the type of analysis that I was looking for.

If the problem is not well-defined, then let's assume that the remainder of the board is empty. If this means that black needs to play elsewhere, I'm happy to hear that, too. I just want to know the valuation of doing so.

A couple of responses:


I do not know if the position is double sente. Maybe it is, but I am not calculating this in a precise way.

Let's say that the value of black playing first, locally, benefits black by A points vs. white playing first locally. Let's also say that if white plays first, locally, white benefits by B points. Let's say there's another place on the board. If black plays first there, he gets C. If white plays first there, he gets D.

If white plays first in the first local position, and gets to play there a second time, the gain from the second play is worth E (for a net of B + E).

If black gets to play twice in the second local position, he gets F on the second play for a gain in that area of (C + F).

It seems that the following are possible:
1.) A > C and B > D. In this case, definitely, black should play in the first local position shown. Both black and white have this as the biggest spot on the board.
2.) A < C and B < D, but E > F . In this case, I'm not really sure. It's bigger for both black and white to play in the second spot, but maybe if (B + E) > (C + F), white gains more from the first local position than black gains from the second, even though both A and B individually were smaller...?

So the idea of "double sente" is hard for me to say that I should definitely play, because it's hard for me to define exactly what double sente is.

I dunno if that makes sense.

---



I added emphasis to the quote above. I appreciate the analysis of the average points that are gained, and it seems to support the quality of Uberdude's proposed move. But I feel the analysis should take as input something accounting for the fact that the threat is smaller. How is this expressed? It could be that I just don't understand the analysis completely, yet.

And again, let's assume the rest of the board is empty. Or if it makes it easier to consider the local position, feel free to setup the rest of the board however you would like.

This particular position is not important to me. I'm just curious how this analysis takes place.

Thank you.

_________________
be immersed

You have to distinguish between sente and gote. Not to mention kos. That is, you need to take the net number of plays into account.



Compare the threats yourself.

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

Hmm, ok.



It seems to me that sente and gote are used in calculating the value of a local position. In some ways it gives you a limit on the depth in which you need to calculate. You can say, "if black plays here, white will respond, because it is sente." And this information is used to determine the point value. Is this correct? This is my current understanding.

So if that assumption is correct, the determination of sente and gote is a prerequisite for determining local value.

But how do I determine sente and gote if I do not know what the value is of ignoring the play vs. playing the play?

I feel like I've stumbled upon this type of idea before and got confused then, too.

_________________
be immersed

Seriously. Show the diagrams.



If you can tell sente and gote, it simplifies your task.



It is not necessary.



You can start out assuming that a play is sente or gote, and if you are wrong, you will get a contradiction.

See http://senseis.xmp.net/?MiaiValuesList%2FDiscussion .

Practice helps.

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

Well, I think this originated from the comment:


which is something that I didn't come up with, but was asking about. My assumption on what you mean here is that

this:

Click Here To Show Diagram Code

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

threatens to capture the three white stones in a single move:

Click Here To Show Diagram Code

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

whereas this:

Click Here To Show Diagram Code

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

threatens to capture the same three stones, but leaves white the forcing move at the marked spot:

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]

But I don't think that this forcing move at the end is accounted for in the point average that was calculated earlier. I'd think that there is a way to quantify this numerically and include this in the "real" point average in order to determine the best move to play.



If this is the case, then sure - you can evaluate the local position since the knowledge of sente and gote are not necessary.

I assume this is to say that "guessing" if something is sente or gote is somewhat like a "hint" in determining the appropriate relative values.


It's not at all clear to me why this is the case, but I guess this is something I'll figure out on my own if I follow the advice provided in your next comment:



Thanks.

_________________
be immersed

Contradiction example:



Here is a simple game tree with Black moves to the left and White moves to the right. Bill Taylor used to argue for an evaluation system that was pure gote. Let's make the calculations.



We find that the count of A is 2.5 and each play at A gains 7.5 points. The count of B is 10 and each play at B gains 10 points. Bill Taylor sees no contradiction.

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.

OTOH, it makes no sense to wait until the end of play and let White play to -5. By playing first Black can guarantee at least 0. Therefore the count at A is at least 0. And White can prevent Black from getting more than 0; so that is the count at A.



Voila! This is a Black sente where the reverse sente gains 5 points.

----




Suppose that we believe that this is a Black sente.



Then the reverse sente gains 15 points, and a play at B gains 10 points.

Suppose that there are plays elsewhere that gain between 10 and 15 points. Then as a rule Black may play at A to prevent White from playing at A and gaining 15 points. But then as a rule White will make one of the other plays instead of playing at B, so Black's play will not be sente, but gote.

Another way of looking at it is to say that by not replying, White makes the count at A equal -2.5, which is better for White than 0.



This is the correct evaluation. A play at A is gote, gaining 12.5 points.

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

This is like



From a position worth -2.5, Black can increase by 12.5 to get 10. From a position worth -2.5, White can decrease by -12.5 to get -15. I.e., -2.5 is the average of the immediate followers.

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. 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.

OK, that's a start.

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

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

Indeed.



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.

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

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.




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:


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...?



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?

_________________
be immersed

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.

_________________
be immersed

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

_________________
be immersed

What is a swing value?

_________________
be immersed

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.)

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

_________________
We don't know who we are; we don't know where we are.
Each of us woke up one moment and here we were in the darkness.
We're nameless things with no memory; no knowledge of what went before,
No understanding of what is now, no knowledge of what will be.

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

_________________
be immersed

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.

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

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:

Click Here To Show Diagram Code

[go]$$Bc elsewhere
$$ | . . . . . . . . . . .
$$ | . . . X X X X O O . .
$$ | . . . . 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 -

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.

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 --

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.

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).