Go endgame values with area counting: part 4, odds and ends

I meant for part 3 to be the last entry in this series, but a few people had extra questions and I also discovered a few more subtleties while applying the theory.

Double sente

I got a couple of questions about double sente, which I had originally skipped over because I have been brainwashed by the modern endgame theorists who contend that double sente doesn’t really exist. Computing the old-fashioned deiri value is no problem; two intersections are at stake, and the local tally difference is $0$, so playing here is $2\cdot 2 - 0 = 4$ points in double sente, just like we all learned when we first saw this position. The problem, which exists with the usual method of counting as well, comes when you try to convert this into a miai value by dividing by the local tally difference. This means dividing by zero, so we get an infinite value! How can this be?

To be brief, if this is truly double sente, then whoever plays here will pick up $4$ points for absolutely free, no opportunity cost or anything, so it should be played immediately and will then disappear, like a subatomic particle that decays instantly, so it’s never really around long enough for us to bother analyzing. But what if it isn’t really free? What if the opponent could tenuki? Well, then I guess it wasn’t really globally double sente, was it?

While your head is reeling from that, I will refer you to the references (Rational Endgame in particular discusses the issue carefully) and quickly move on to other topics.

Forcing the opponent to play inside

Here’s a position that is a little easier at first to think about with territory counting: if Black plays A, they’ll get the point at B, and if White plays A, they’ll get the point at D, so playing here is $2$ points in gote.

With area counting there’s a little more to keep track of. As usual, we think in terms of intersections at stake. Here they are A, B, and C; the ownership of everything else on the board (including D) is already determined. If White goes first, they’ll get two of those three points; they’ll play A, and then the two players will split B and C. If Black goes first, White won’t get any of them, because the C-D exchange is Black’s privilege. So the difference between the two variations is $2$, and playing here is $2\cdot2 - 2 = 2$ points in gote.

So the reason for the extra point is expressed differently in the two methods, and this is also true if we score by area at the end of the game. By territory, White is penalized for having to fill in D. By area, Black gets a free dame at C instead of having only 50% equity in it.

That isn’t really any different from saying “White will have to fill in at D”, like you’re used to, but it worth realizing that Black benefits just from putting down a stone on C. You do get used to playing out these exchanges in your head, especially once you start getting obsessed with viewing everything as being about borders.

(If the concept of “Black’s privilege” is foreign to you, the references in Part 3 have good discussions of it.)

Area-territory duality

In mathematics there’s a concept that frequently comes up called duality, where you can look at a problem in two very different ways, but they’re isomorphic, so the solutions will always correspond to each other, and the concepts generally map to each other as well. These two ways of counting remind me of that. I think it’s sort of beautiful that both territory and area counting agree that because of the weakness at D, A is exactly one point more important than usual, but have very different ways of explaining why.

In fact I think it’s often fruitful to view Go concepts through an area lens, especially if you’re used to thinking using territory. For example, if you play a threat inside your opponent’s territory that must be responded to, with territory counting you note that your opponent gained a prisoner but filled in one of their own territory points, while with area counting you simply note that the borders haven’t changed. (Easy but educational question for the reader: if the opponent didn’t need to respond, the borders still haven’t changed, but now your move was terrible; why?) I also find it easier to believe that forcing moves can be discarded after they have served their purpose when I view the situation with areas.

Viewing Go as being fundamentally about borders between the two colors, rather than fundamentally about surrounding empty space, has been very eye-opening for me. Black getting to play the dame point of C for free in Diag. 2a isn’t a weird accounting trick to make the numbers come out right, it’s intrinsically valuable. I encourage you to spend some time trying to view endgame sequences as a matter of fighting over area ownership of a fixed set of points rather than as expanding and reducing territories, even if you end up coming back to the territory view.

Odd vs even

Here’s something that is hard to show unless you look at it from an area point of view: if there are no followups, gote values are even unless the difference in dame is odd. Let’s look at a pair of examples from Part 1 again.

Because we calculate the gote value by doubling the number of points at stake and subtracting $2$, it must be an even number unless there’s a half point at stake. The way that can happen is when an odd number of points belong to one side for sure in one variation, and are up for grabs in the other, as in Diag. 3b. Similarly, sente moves are usually worth an odd number of points.

Seki

Seki is a little annoying because we don’t get to depend on our usual rule of “every point belongs to one side or the other”.

By territory, Black has either $6$ points or $0$, so playing here is $6$ points in gote. By area, if we look at the five points at stake, Black will either be up $5$ to $0$ or down $0$ to $3$. That’s effectively a difference of $4$ (if you’re a baseball fan, you know that a $0$–$3$ team is four games behind a $5$–$0$ team), and $2\cdot4 - 2 = 6$.

But I think it’s easier to think of the two sides as sharing ownership of all unplayable points, so we can go back to using our favorite principle. Looking at it that way, Black owns either $5$ points or $1$ (half of the unplayable dame), giving us the desired difference of $4$.

If your ruleset does not award points for eyes in seki (Japanese does not, but AGA does, even when you use territory counting), you will have to do the same thing with those ownerless points.

Sente or gote?

In any given position, it may not be clear whether we should be doing the sente calculation or the gote calculation. I like to do this the same way as Antti Törmänen does in Rational Endgame, which is that a move is sente for Black if White wants it to be. Here’s a brief set of examples.

There are $5$ points at stake. If White goes first, Black will get none of them. If Black goes first and it’s gote, they’ll get either $2$ or $5$ for an average of $3\frac12$. That would mean that Black’s fair share in this position is the average of $0$ and $3 \frac12$, or $1 \frac34$.

What if it’s sente? Then Black’s fair share would be $2$, since they’d just make the sente exchange at some point. But that’s worse for White than if we consider it gote, so White will claim that it’s gote and not respond to Black’s first move (or possibly play here first themselves).

Let’s sweeten the pot. Now there are $6$ points at stake. As before, Black gets none of them if White goes first. If White treats it as gote and Black goes first, Black will get either $2$ or $6$ points for an average of $4$, so their fair share in the original position is $2$. If White treats it as sente, Black’s fair share is also $2$. So White has the choice of treating this position as gote or as Black’s sente.

Now there are $7$ points at stake. Black still gets $0$ if White goes first. If White treats it as gote and Black goes first, Black will get either $2$ or $7$ points for an average of $4 \frac12$, so their fair share in the original position is $2 \frac14$. If White treats it as sente, Black’s fair share is $2$ as always. So White now does better by considering this to be Black’s sente, and will respond when Black captures, and the move value is $2\cdot 2 - 1 = 3$.

The nice thing is that you can do all of these calculations with just a single type of number: Black’s equity in the points at stake. There’s no need to add and subtract territory and prisoners, or even convert to a move value until the very end. Of course, you can do exactly the same thing to decide whether followups are gote or sente.