Abe's method: since there seems to be more to it than triangles, it would be nice to hear about it. A triangle for the outside influence part is better than a triangle for everything would be. Understanding it as a rule of thumb appoximation of the global territory potential of the local excess influence is better than understanding it as a rule of thumb appoximation of the local extra territory due to the outside influence. According to Ishida (Go World 41), Takagawa would have imagined global sequences to judge about the territory potential of local thickness. Does Abe justify his triangle value by imagined global sequences, or does he offer other reasons why (here, in this simple shape case) the triangle's value is a good estimation of the outside influence's extra territory potential?

My method: It is approximative, because the values of T and I are not always exact. OTOH, the range of values 1.5 ~ 3.5 tolerates some imprecision. For purposes other than josekis, T and I can be calculated, but not related to each other as easily, because the josekis' implied axiom of reasonably fair construction for Black and White is missing.

mitsun, T and I are re-evaluated if the position changes, such as after the exchange White a for Black b.

BTW, this is not the only possible joseki follow-up. Other standard follow-ups are a) tenuki or b) White reinforces by extending to the center and building a box.

I wouldn't really know what old abe would claim but what I feel sure of is that he would expect common sense to kick in pretty quickly and he would not expect students to let the formula be sovereign. What I'd draw from that is a feeling that if application of the formula raises more questions than it answers, you probably need to be standing back to let CS have a go. In other words, treat it as you would any go proverb.

Are we supposed to know what these mean?

_________________
Occupy Babel!

Eventuelly you finish darts as long as you don't miss the board always completely.
But for not really strong players ("really" means really "really") applying the double-out-rule changes darts from a game of skill to a game of luck.
The same happens when playing 9-ball pool compared to 8-ball.

hyperpape, you choose some locale; due to this choice, you know what is 'local'.

The territory count is defined as the difference of Black's and White's territory. My joseki evaluation method uses a specific kind of territory, the 'current territory'.

The influence stone difference is defined by me as the difference of Black's and White's numbers of [significant] influence stones.

If the territory count favours a player, this is his excess. If the influence stone difference favours the same player or the opponent, this is either's excess.

In the example, Black's excess of current territory is 7 [points]; White's excess of influence stones is 5 [stones].

If Black has 1 apple and White has 6 apples, then, concerning the apple difference, White's excess is 5 apples.

The territory difference was calculated as 15 for B minus 8 for W. But the 8 points of W territory was just hand waving. After the a-b exchange, how much does W "current territory" change?

mitsun, not hand waving, but current territory remaining after imagined sente reduction sequences with pretty passive replies. (I lack time to repeat principles for how to construct them.) Here is just a hint for a relevant start of a reduction sequence:

Click Here To Show Diagram Code

[go]$$B
$$ +---------------------------------------+
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . X X O . . . . . . . . . . . . . . |
$$ | . . O X O . . . 2 O . . . . . . . . . |
$$ | . X X O . . . . 1 , . . . . . , . . . |
$$ | . . X O . . . . . . . . . . . . . . . |
$$ | . X O O . . . . . . . . . . . . . . . |
$$ | . . X O . . . . . . . . . . . . . . . |
$$ | . . X . . . . . . . . . . . . . . . . |
$$ | . . . , . . . . . , . . . . . , . . . |
$$ | . . . . . . . . . . . . . . . . . . . |[/go]

It could well be that that idea was around for a long time before Abe wrote about it. Many suggestions for evaluating walls have been proposed.



Yes, it does.



Of course, Abe relied upon imagined sequences to evaluate positions. That is standard practice.

Simple walls standing on the third line require an extension to make eye space. The standard extension is n + 1 spaces, where n is the number of stones in the wall. The following is speculation, but one could estimate the value of the wall plus extension by drawing sector lines. Doing so gives us a triangle from the third line up plus a rectangle on the first and second lines. The territory within the triangle is approximately n(n+1)/2 and the territory within the rectangle is 2(n+1). One could take the triangle to be the value of the wall and the rectangle as the value of the extension interacting with the wall.

That's not a logical derivation, OC, but people have been guessing at formulas for evaluating walls for over forty years.

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

Visualize whirled peas.

Everything with love. Stay safe.

What book did Abe present the formula in? He authored a lot of books. Did he claim originality? Did he offer a derivation of the formula? (My guess on the latter is not. If he did, please spell it out for us. )

I could speculate further on where the formula comes from, but I don't think that would be of much help As I said, there have been many guesses about how to evaluate walls for over forty years. None have been proven.



Since standing walls on the third line normally have extensions, territory is then made in the vicinity of the wall, and since the extensions are linear with regard to the stones in the wall, it is certainly plausible that the resulting territory will be 2 dimensional in the number of stones. (Well, I speculated, anyway. ) As for thickness in general, indeed, we do not expect territory to be made near it, unless it arose late enough in the game that territory has been relatively settled elsewhere, so that there is not much place else to make territory. But it would be desirable for influence to be conserved, given correct play, no matter where territory is eventually made. That is, the initial influence of a wall is local, but when much of the potential local territory is taken by the opponent, it is traded for territory elsewhere. In fact, if a wall has a definite value, but we do not know where it will be realized, then that is how things must work. (OC, a wall may not have a definite value, but it still should have an average, or expected value. )



Very good. But these walls should have different formulae associated with them, right?



Of course.



When you and I were learning go, professionals estimated the value of the corner 4-4 point as 10 points. That is consistent with the saying that ponnuki is worth 30 points, as there are 3 net stones in a ponnuki. It is also consistent with komi at the time, which was 4.5 or 5.5 points. Now, in 1975 I estimated its value at almost 14 pts., based upon results from pro-pro handicap go. At the time I predicted a komi of 6.5 by the turn of the century. (Close, but no cigar. ) A year or two later someone published the results of a statistical study of komi in the American Go Journal, claiming that correct komi was 7. That is what you would expect if the value of the 4-4 point is 14. By now it seem plain that correct komi is closer to 7 than 5, which indicates that the accuracy of professional opinion in the mid-twentieth century was off by some 40%! (BTW, I estimate the value of the diamond shape ("ponnuki") as about 42 pts., which would mean that the value of 30 pts. is also off by 40%.)

It is plain that standard pro evaluations using imagined kikashi sequences is inaccurate and biased towards the low side. But, since each player will have played the same number of stones, or Black will have played one more, these systematic biases tend to cancel out. Also, there is practical value to doing a kind of worst case analysis.

As you know, this is an area that I have researched, and I have a method which is accurate for single stones, except in the center. It evaluates the 4-4 stone at 14.5 points, for instance. I attribute that mainly to luck, however. I thought that it would overestimate the value of that point, because it would assume that the stone is stronger. (Since stones in a wall are strong, it should be fairly accurate for walls.) Much more research needs to be done.

The n(n+1)/2 formula seems to me to overestimate the value of a wall, as influence should drop off more rapidly. But since other standard estimates are on the low side, the combination may end up producing a fairly accurate positional evaluation.

Assessing thickness and influence is difficult, and many researchers have worked on the problem for decades. They are far from a consensus, and, since modern computer programs do not use positional evaluation, little work is now being done. Thomas Wolf and I are doing research, with quite different approaches. But I do not think that it is high priority for either of us.

Anyway, people are guessing, and so is Abe. I would not count on the formula to be accurate.

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

Visualize whirled peas.

Everything with love. Stay safe.

Bill

The earliest discussion in depth I have seen of the value of thickness [atsumi] is that by Yamazaki Masuo in Kido 1959. However, he does not discuss numerical values. Rather he is concerned with whether we can count it at all and what factors might go into it. I give a short excerpt below (NOT to be copied elsewhere) so that others know what we are talking about. Yamazaki was a pro but also an intellectual, hence the style.

But if we accept the "stand on two, extend three" kind of rule two underlie most counts, especially those based on rectangles, then we can of course go back to the 13th century and beyond. In fact I think it was mentioned in the 6th century Dunhuang Classic but I'm too busy to check.

As to the point of previous researchers coming up with different estimates and maybe underestimates, I always had the impression (no more than that) that Takagawa evaluated thickness rather more generously than others, and what was distinctive about his approach was that he was concerned not with an "external" value but an "internal" one based on tewari.

QUOTE
The value of thickness, however, is extremely vague, and in the form in which it constitutes itself it does not have a value in the (absolute) sense that applies for territory. The value of thickness can be said to be governed by the situation of the surrounding stones. Depending on the surrounding circumstances, no matter how splendid the shape of the group of stones that constitute it may be, it may have absolutely no value. Also, the fact that groups of the same shape can have a higher or lower value to the extent that is hard to relate to anything else is no doubt something that any player who is aware of at least some go theory has constantly experienced. In other words, the value of thickness is not determined only by the stones of the side that constituted it, but is also governed by relationships with the opponent’s stones that are in proximity to it.

The fact that the value of thickness is thus governed by other elements means that it is totally different from territory, which is of itself and has an independent value, and an absolute value in the above sense would have no significant meaning in actual play, and so the problem becomes one of relative values only. Whether it be territory, whether it be thickness, the question: how much benefit have you given to the opponent in compensation?

However, there are no criteria to determine the evaluation of thickness as in the case of territory. It is impossible to state objectively that, if thickness is converted into territory after certain negotiations, it is worth so-and-so points. Naturally, therefore, such evaluations inevitably depend on experience. Consequently, it is only natural that such evaluations will unavoidably differ somewhat from person to person. But those differences are not enough to worry about. Of course, it is inescapable that large differences will be apparent if one player is extremely skilled and another is substantially weaker, but if there are two players in the same sort of range of skill, irrespective of differences in their personalities or go styles, on the whole they will normally agree. There is a certain universality about this.

Yet there is a criterion for evaluation and assessment of thickness. If we accept that it is not a number as in the case of territory, then what is it? It is intuition. In other words, do we feel happy or unhappy about it? After a given variation, if our thickness is greater than the territory given to the opponent, we probably feel a happy sensation, because it means we are that that much closer to victory. In the opposite case, we will have an unhappy feeling. Also, in the case of equality, this means that, as the tension mounts, we cannot say that a person confident about his thickness will feel any happier whereas he who lacks confidence will feel any less happy. We may also sense feelings of happiness or unhappiness as regards thickness during a game or when enjoying playing over a game record, and these too are not unrelated to our evaluation and assessment of thickness, even though at present they are intangible.

Naturally, territory is a solid thing and once it has been made there is no need to make any changes until the end of the game, but in contrast thickness of itself ultimately has no meaning, and so we must obtain a result whereby its effect is transmuted into some number of points of territory over the course of a game, be it by direct means or by indirect means. In other words, what it means when a player feels happy or unhappy as regards thickness is that he has a premonition that he is likely to make an amount that corresponds to at least (or at most) a certain size of territory, even though it is unclear exactly what course future events might take. It is as if the evaluation of thickness is the possibility of territory materialising. Next to the probability of the inevitability of this, this is the most important element. For whereas the evaluation of territory is fixed, and the player cannot possibly change that in any way, thickness puts in hock various possibilities as regards the course of the future, and so there is the possibility that we can control its value, at least to some degree. In other words, whereas territory has an independent, natural existence which cannot in any way be changed by the player’s volition, with thickness the player’s will can have repercussions. We have to do with Free Will – anti-Mechanism. It is as if thickness is the locus for attack and defence for both sides, and precisely in that we can see the throbbing life of a game of go.

The reason I think that go is a fusion of inevitability and possibilities is that I see inevitability as Mechanism and possibilities as Free Will.
UNQUOTE

I think a wall can have many different applications depending on the overall game position

- Creating territory nearby via an extension stone
- Harrassing opponent's stones and thereby creating territory elsewhere
- Killing opponent's stones (by denying them a big fraction of their angular escaping space)
- Breaking ladders
- Enabling invasions into opponent's moyo
- Connecting two own weak groups thereby making them strong (this saves time and thus enables big plays elsewhere)
- I guess that there are many more possible applications which are too subtle for me to recognize

The approximation formula for the "value" of a wall (or: influence stones) discussed here seem only relevant for the first application I mentioned. Thus I think the value of a wall depends on what actually can be achieved with it in the actual game position. If a wall just results from being chased out of an opponent's framework, it can be pretty useless. A numerical formula like n(n+1)/2 may give an approximate idea for the nearby territory potential (first mentioned application) but should be IMHO strongly misleading in all other cases.

schawipp, the influence stone difference is not always sufficient, but can be used, e.g., for these purposes:

- joseki evaluation if stones in the neighbourhood do not dominate the local situation
- center control if the influence stones have similar center access and are similarly stable (the center will be pretty neutral if the influence stone difference is about 0; the (sufficiently big) center will be valuable for the player have a great favourable influence stone difference)
- domination of a fighting region (the attacker has a great influence stone difference)

For other considerations, influence stone difference might be insufficient. Then other aspects of influence might have to be judged for representative intersections in or near a sphere of influence: degrees of connection, life and territory.

Robert, this definition of local excess territory makes sense, though I share mitsun's worry.

_________________
Occupy Babel!

Many thanks, John! This is great. Yamazaki is very interesting.



I think we can say that Fujisawa Hideyuki also evaluated thickness highly. Both he and Takagawa were considered masters of fuseki.

I may have more to say about Yamazaki's thoughts, but let me respond to his opening statement.



There is also a vagueness in the estimation or assessment of territory, which should also be addressed. For instance, suppose that a joseki leaves behind a simple ko in a corner which has a swing of 27 points between winning and losing the ko. Everyone who understands evaluation agrees on the estimate for the corner territory. However, they also know that at the end of the game the actual corner territory will differ from the estimate by 9 points or 18 points, depending on who wins the ko.

This possible difference of 18 points in the corner territory does not bother them, because they also know that, when the ko is fought, there will almost certainly be many plays of approximately the same size as the ko on the board, so that whoever loses the ko will come out even, or almost even, by taking one or two of those plays. There will be a tradeoff between potential territory in the corner and territory elsewhere on the board.

That is not so different from the tradeoff between the influence of a wall and territory elsewhere, although the mechanisms of the different tradeoffs differ. But there is a different kind of uncertainty associated with walls. Nobody knows how to evaluate them in the first place. (Whereas, we have known for more than 200 years how to evaluate simple kos. )

That is not to say that it is impossible to come up with good evaluation methods for influence, even though no consensus method has emerged yet. But once we do, I think that the ordinary uncertainty associated with thickness will be less than the +/- 9 pts. of the example ko, perhaps on the order of +/- 5 points, perhaps less. Even now, my guess is that in Robert's initial diagram (assuming an otherwise empty board) White has an advantage of 5 - 10 points, though those are not hard limits.

_________________
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 very much for this excerpt John, very nice. I really like the "happy" or "unhappy" feeling.

I'm to weak so that my feeling could be trust, but it is really how i think in a game: cool, i'm strong and have free hands; damn, he's to thick and i have to play small otherwise i get in trouble; wow, no way you get my solid 25 points with this stupid wall; argh, i'll miss points...

Than a good part of trying to improve, in my case, is "correcting" these intuitions with comments from stronger players and review of pros games. Somehow i cannot imagine myself using some mathematical formulas during a game to evaluate things (maybe it is a weakness of mine?)

Anyway, it was interesting to read, thank you

gasana, most formulas applied in a game are simple sums, differences, multiplications, divisions or comparisons of two values. Iterative application can make this more complicated, e.g., during endgame value calculation. Some formulas combine two basic calculation steps.

Bill, walls can be evaluated in principle, but in practice the necessary calculation in combination with reading can become too complex.