A while back there was a series of posts on "Boundary Plays", based on a book by O Meien. I think if this series had continued, we might have an authoratative approach to analyzing positions like this. I for one would love to see that series revived.
Here is my take on this. The value of a position is an interesting concept, which has two related uses: 1) estimating the score, 2) calculating the value of a local move. The second case is probably more interesting.
The position below is a good example, simple enough to analyze, but complex enough to have many variations.
- Click Here To Show Diagram Code
[go]$$W
$$ ---------------------------------------
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . O . . O . . . . . O X . . . . . . |
$$ | . . . , . . . O . , . O X X X X X X X |
$$ | O O O O O O O O . . . O O O O O O O O |
$$ | . . . . . . . . . . . . . . . . . . . |[/go]
For purposes of calculation, let's compare all future results to the following symmetric base result. By the way, this ending position also provides a fair estimate of the value of the starting position, assuming different plays by both sides are about equally large, even though it will never occur in an actual game.
- Click Here To Show Diagram Code
[go]$$W
$$ ---------------------------------------
$$ | . . . . . . . . . . . O X . . . . . . |
$$ | . . . . . . . . . . . O X . . . . . . |
$$ | . . O . . O . . . . . O X . . . . . . |
$$ | . . . , . . . O . , . O X X X X X X X |
$$ | O O O O O O O O . . . O O O O O O O O |
$$ | . . . . . . . . . . . . . . . . . . . |[/go]
First, suppose there are no other large plays on the board. If W starts with the hane and hanging connection, B will have to block, letting W end in sente:
- Click Here To Show Diagram Code
[go]$$W
$$ ---------------------------------------
$$ | . . . . . . . . . . . 3 . 4 . . . . . |
$$ | . . . . . . . . . . . . 1 2 . . . . . |
$$ | . . O . . O . . . . . O X . . . . . . |
$$ | . . . , . . . O . , . O X X X X X X X |
$$ | O O O O O O O O . . . O O O O O O O O |
$$ | . . . . . . . . . . . . . . . . . . . |[/go]
Later we can assume (see digression below) that B will get to atari and W will have to connect. Compared to the base diagram, W has the same points, while B has 2 fewer points.
(Digression -- this is one of many simplifying assumptions we will make in this analysis. If we were mathematical purists, we might consider that there is a finite probability for W to connect before B gets to atari, and we could then credit W with some small fraction of that one point reverse sente play. But as practical Go players, let's agree to dispense with complicated fractional points.)
W could also start with the straight descent. Again if there are no other large plays on the board, B will have to block, letting W end in sente:
- Click Here To Show Diagram Code
[go]$$W
$$ ---------------------------------------
$$ | . . . . . . . . . . . 5 3 4 . . . . . |
$$ | . . . . . . . . . . . 1 2 6 . . . . . |
$$ | . . O . . O . . . . . O X . . . . . . |
$$ | . . . , . . . O . , . O X X X X X X X |
$$ | O O O O O O O O . . . O O O O O O O O |
$$ | . . . . . . . . . . . . . . . . . . . |[/go]
The net result here is the same as for the hanging connection.
So we may be tempted to say that either side has a 4 point sente play in this position. But things are not so simple ....
Suppose there are other large plays on the board, so that taking sente has significant value. This is the usual assumption for endgame position analysis. Again we have to consider two possible moves for each side, hane and straight descent, but with more involved continuations.
B4 tenuki
- Click Here To Show Diagram Code
[go]$$W B4 tenuki
$$ ---------------------------------------
$$ | . . . . . . . . . . . 3 . . . . . . . |
$$ | . . . . . . . . . . . . 1 2 . . . . . |
$$ | . . O . . O . . . . . O X . . . . . . |
$$ | . . . , . . . O . , . O X X X X X X X |
$$ | O O O O O O O O . . . O O O O O O O O |
$$ | . . . . . . . . . . . . . . . . . . . |[/go]
The hane and connection may be gote, but it leaves a large follow-up for later. Let's assume that this follow-up is not so large that B must prevent it now (B wants sente to play somewhere else), but large enough that W will get to play it later in sente. This is a judgement call, but again it is a common assumption in endgame analysis. Then the hane it is worth more than our previous calculation, because we assume that taking gote now earns a follow-up sente play:
B tenuki
- Click Here To Show Diagram Code
[go]$$W B tenuki
$$ ---------------------------------------
$$ | . . . . . . . . . . . O 5 1 3 4 . . . |
$$ | . . . . . . . . . . . . O X 2 6 . . . |
$$ | . . O . . O . . . . . O X . . . . . . |
$$ | . . . , . . . O . , . O X X X X X X X |
$$ | O O O O O O O O . . . O O O O O O O O |
$$ | . . . . . . . . . . . . . . . . . . . |[/go]
Note that the hanging connection is better than a solid connection, because it leaves a larger follow-up for later, under the usual assumption that the other side cannot afford a large ko.
Compared to the base position, W has 1 more point and B has 6 fewer points. The sequence is similar if B gets to play first, so we can say that in this situation the hane is worth 14 points in gote for either player.
Now let's consider the straight descent. What happens if the other side ignores the descent? Then there is an enormous follow-up, either a one-space jump or a monkey-jump. These continuations are probably sente, because they threaten further large incursions (or even death). For example:
- Click Here To Show Diagram Code
[go]$$W
$$ ---------------------------------------
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . O . 1 . 2 . . . |
$$ | . . O . . O . . . . . O X . . . . . . |
$$ | . . . , . . . O . , . O X X X X X X X |
$$ | O O O O O O O O . . . O O O O O O O O |
$$ | . . . . . . . . . . . . . . . . . . . |[/go]
- Click Here To Show Diagram Code
[go]$$B
$$ ---------------------------------------
$$ | . . . . . . . . . 1 . . . . . . . . . |
$$ | . . . . . . . . 2 . . . X . . . . . . |
$$ | . . O . . O . . . . . O X . . . . . . |
$$ | . . . , . . . O . , . O X X X X X X X |
$$ | O O O O O O O O . . . O O O O O O O O |
$$ | . . . . . . . . . . . . . . . . . . . |[/go]
We can debate the proper continuation in either case, and we would have to work out detailed sequences to calculate the numerical result. (This might make an interesting discussion by itself, maybe for another thread.) But in practice this is probably not necessary -- we can assume that by the time a play here becomes profitable, the territorial loss incurred by not answering will be too large for either side to allow. Put another way, it is likely that someone will be able to time their play here to make the descent in sente.
So to summarize:
Assuming no other large plays available (unlikely):
Hane and hanging connection = 4 points sente
Straight descent = 4 points sente
Assuming other large plays available (normal situation):
Hane and hanging connection = 14 points gote
Straight descent = even larger gote, probably so large that it is actually sente
Exercise for the reader: construct a full board position, with B to play, where the descent is correct but gote.
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Kirby
- 
come later.)
before, 
, which does not affect the estimate of Black territory.
White has a sente follow-up. As per the usual assumption, Black avoids ko. Black has 12 points.
will Black follow up with 
? Of course not. She will jump to 4 or slide to "a". 
