John Fairbairn wrote:Abe's formula is only a rule of thumb, which amounts to drawing a sector line and counting the intersections inside the created imagined triangle. Instead, determining territory should rely on imagined reduction sequences.
There are several things wrong with this, all showing RJ has not read Abe's book - usually a good reason for not criticising or caricaturing it.
First, Abe counts the value of the thickness, not the territory, and so the extension stone does not come into his equation.
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.
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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.
John Fairbairn wrote:Ultimately, the value of the thickness has to come down to territory points, but not necessarily in their own vicinity. Therefore his method is not so much a means of evaluating a local exchange (though it can do that) but of providing help in evaluating the game as a whole.
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.
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John Fairbairn wrote:Second, his method cannot be characterised as drawing a sector line dividing a rectangle because the point is that he allows wraparound and irregular walls facing two or even three directions (and also allows for intersection of walls).
Very good. But these walls should have different formulae associated with them, right?
John Fairbairn wrote:Third, although with an entirely difference emphasis he does allow imagined sequences to influence the count. You'll have to read the book to see how.
Of course.
John Fairbairn wrote:As a general point, I think we need to remember the hoary chestnut of the difference between precision and accuracy. . . . Abe's method probably has very low precision but he must regard it as being accurate enough to publicise it, and he is recommending it as a whole-board measure.
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.
