I'm reading The Endgame by Tomoko Ogawa and James Davies. On page 61 is a problem that I puzzled over for a while, and was very confident I had the right answer. Apparently Tomoko disagrees with me, but her explanation wasn't clear enough for me to understand.
xela: I think chapter 2 is actually by Davies.
It says, "Assume that there is nothing else worth considering on the board and try to figure out the best sequence in which to play them... and try to assign a numerical value ... to each move."
I correctly read c as two points sente.
I thought b was one point double sente. If black takes b, white must connect by filling own territory (because we're assuming there's nothing else on the board big enough to be worth considering). But if white takes b, black must connect or else white takes the point above in sente for one point.
xela: "Nothing else" means the rest of the board outside this diagram. If black plays b then white will reply at c.
Tomoko, however, calls b 3.5 points in gote.
xela: In my copy of the book it's 3 points in gote.
xela: If then white plays at c. Whether or not black gets the marked point of territory will be decided later--that's why it's counted as a half.
However, if white moves first, , , and force white responses, though white gains one point for capture in gote.
But Tomoko says a is 1.5 points sente.
xela: 1.5 points gote in my copy of the book. Black won't play until after the rest of the position is resolved.
Would someone please help me understand? Thanks much!
Karl Knechtel: Let's look at b as an example.
The play at x removes or defends one point at y. So x is a one point gote play, and Black has half a point (we may think of it like "equity") at y. "Nothing else worth considering" does not exclude the larger plays on the first line; that may be your difficulty ;)
Black has a full point at y, and White can no longer hane at z, and the value of Black x increased. Big, but a White answer (at x?) is hardly forced (we think of x as "Black's sente" now), so not sente. After Black x, eventually White will lose the a points of territory.
That makes 3 + 0.5 points difference between the two situations, in gote either way.
Personally I think miai counting is easier to understand though :/
xela: In general I found this to be a difficult book, but well worth the effort. The important concept for this particular example (and most of the rest of chapter 2) is how to account correctly for the follow-up moves. You should carefully study the "averaging" argument given on page 51 and the analysis of diagram 17 on page 54, then look at this example again in the light of those ideas. (If you look carefully enough, there are still some tiny logical holes--explaining those issues thoroughly is the task of combinatorial game theory--but the methods in this book are good enough for almost all practical purposes.)
If , White doesn't usually answer. Next, Black gets in sente. Follow-up moves which are sente are simply added to the sequence to evaluate the score. Black makes the circled point.
Here too, is a sente follow-up. On the first line we average out the 2-point gote endgame play, assuming the territory border at x. White makes two circled points, compared to the previous diagram.
Combined, it's a 3 points double gote.
Bill: Here is the canonical sequence of play.
After , in the local region there are no points, but if Black plays in this diagram, Black gets 1 point at . That means that after Black's equity in the local region is 1/2 point. gains 1/2 point, making the local score 0, and in this diagram also gains 1/2 point, making the local score 1 (for Black).
If we back up to just before and suppose that White plays first in the local region, White can capture the stone for a local score of -1 (one point for White). Before that play we consider the local region to be worth -1/4 point. White's play gains 3/4 point, for a local score of -1, and Black's play also gains 3/4 point, for a local score of +1/2.
gains 3/4 point, but only gains 1/2 point. That's why is gote. If there were a play that would gain more than 1/2 point for White, White would play it instead of .
Similarly, in the canonical play diagram gains 1.5 points, while only gains 1 point, so is gote, too.