Life In 19x19

I am trying to get a grasp on evaluating board positions, for example, for endgame play.

I know that there are counting techniques, and I'd like to see someone demonstrate this with the example shown below.

It's black's turn to play in this position, somewhat early in the game:

Click Here To Show Diagram Code

[go]$$Bc
$$ | . . . . . . . . . . .
$$ | . . . X X X X O O . .
$$ | . . . . X O O X O . .
$$ | . . . . . . O X O . .
$$ | . . . . . . . . . . .
$$ -----------------------[/go]

I believe that the "correct" sequence is as follows:

Click Here To Show Diagram Code

[go]$$Bcm1
$$ | . . . . . . . . . . .
$$ | . . . X X X X O O . .
$$ | . . . . X O O C O . .
$$ | . . . . . 1 O C O . .
$$ | . . . . . . . 2 . . .
$$ -----------------------[/go]

And then it is black sente to play elsewhere on the board.

I used to wonder about the real point behind . If black wants sente, why not take it immediately? That is, don't play and immediately tenuki. Of course, you can then consider the following:

Click Here To Show Diagram Code

[go]$$Wcm1
$$ | . . . . . . . . . . .
$$ | . . . X X X X O O . .
$$ | . . . . X O O X O . .
$$ | . . . 2 1 . O X O . .
$$ | . . . C . . . . . . .
$$ -----------------------[/go]

Now, white can play on the marked area, but he wouldn't do this, since it would get captured. Therefore, I believe that, after the sequence above, white would tenuki, giving white sente (after b's sente from not playing the original ).

So now I can compare what happens if black DID play the original , but white captures:

Click Here To Show Diagram Code

[go]$$Wcm1
$$ | . . . . . . . . . . .
$$ | . . . X X X X O O . .
$$ | . . . . X O O . O . .
$$ | . . . 2 1 C O . O . .
$$ | . . . . . 3 . O . . .
$$ -----------------------[/go]

And again, probably black's sente... So considering these various diagrams, I can make the following intuitive argument:

By playing the original , black gets two benefits:
1.) Sente immediately.
2.) Sente again if white decides to capture.

If black omits the original , black gets:
1.) Sente immediately.
2.) Gote if white decides to continue on in the local position.

By this argument, I can convince myself, "Yes, I should play "... But this type of reasoning is somewhat of a trial and error approach. It seems sloppy and imprecise.

Please illustrate how I can mathematically analyze playing vs. not playing and taking sente immediately.

Kirby, cannot help you with the "math" part -- that is, the exact & precise yose calculations;
but everything you analyzed is correct. Here are a few more tidbits:

This is the best yose for B (if connects at (a), B may lose 0.5 points later; but descend gets everything for B) :

Click Here To Show Diagram Code

[go]$$B
$$ | . . . . . . . . . . .
$$ | . . . X X X X O O . .
$$ | . . . . X O O . O . .
$$ | . . . . a 1 O . O . .
$$ | . . . . . 3 . 2 . . .
$$ -----------------------[/go]

More issues with the sente: the aji of W clamp(b) and W peep(c) -- these are additional reasons for your atari:

Click Here To Show Diagram Code

[go]$$W
$$ | . . . . . . . . . . .
$$ | . . . X X X X O O . .
$$ | . . c . X O O X O . .
$$ | . . b 2 1 . O X O . .
$$ | . . . . . . . . . . .
$$ -----------------------[/go]

Another issue: if W is desperate in yose, W can make a ko with (if W has enough ko threats) --
then B may not be able to block with , but has to pull back at (b), losing points:

Click Here To Show Diagram Code

[go]$$W
$$ | . . . . . . . . . . .
$$ | . . . X X X X O O . .
$$ | . . . . X O O X O . .
$$ | . . b 2 1 . O X O . .
$$ | . . 4 3 . 5 . . . . .
$$ -----------------------[/go]

But wait! There's more! (W can play later, KNOWING about the possibility of peep, depending on the whole board) :

Click Here To Show Diagram Code

[go]$$W
$$ | . . . . . . . . . . .
$$ | . . 9 X X X X O O . .
$$ | . . . 7 X O O X O . .
$$ | . . 5 2 1 6 O X O . .
$$ | . . . 3 . 4 8 . . . .
$$ -----------------------[/go]

What about this tesuji? Seems better yose as you get d1 in sente if white takes the stone later (but is worse yose than atari if ignored as white gets j1 sente).

Click Here To Show Diagram Code

[go]$$Bcm1
$$ | . . . . . . . . . . .
$$ | . . . X X X X O O . .
$$ | . . . . X O O C O . .
$$ | . . . . . . O C O . .
$$ | . . . . . 1 . 2 . . .
$$ -----------------------[/go]

P.S. It can be dangerous to over-analyse a position as you may focus so much on precise counting you might miss the big picture and not look for better moves; I'm sure I suffer from this.

If the rest of the board is empty, a complete mathematical analysis is too complex. To do some, one must set a locale and assumptions such as "The black sphere of influence cannot be attacked from the left side or inside." and "Any move with a BIG local threat is answered immediately.". The size of BIG needs further specification; e.g., it can depend on an assumption of a particular global temperature.

There is more than one possible kind of analysis. To get a precise analysis, CGT-style endgame calculation is useful. If an approximation suffices, then a territorial positional judgement is possible, but you might insist on not calling it a (precise) mathematical analysis. OTOH, you mention that the position occurs during the middle game. So what do you want? Approximation useful during actual games or exactness useful only for the sake of theory study?

Thanks to all who have replied. Ed, that's some useful perspective. Uberdude, I hadn't considered that play, and mist admit that I see the one I provided more often.

Perhaps my question of "mathematical analysis" was too broad. How about a more specific question: Could you illustrate how to calculate the value of playing in the original position (eg. numerically assign a point value approximation).

Click Here To Show Diagram Code

[go]$$Bc simplified position, outer stones alive
$$ | . . . . . . . . . . .
$$ | X X X X X X X O O . .
$$ | . . . . X O O X O . .
$$ | . . . . . . O X O . .
$$ | . . . . . . . . O . .
$$ -----------------------[/go]

A single value calculation makes sense only if we simplify the position and assume a locale such as

Click Here To Show Diagram Code

[go]$$Bc locale
$$ | . . . . . . . . . . .
$$ | X X X X X X X O O . .
$$ | C C C C X W W B O . .
$$ | C C C C C C W B O . .
$$ | C C C C C C C C O . .
$$ -----------------------[/go]

Is this kind of simplification ok for you or would you like to suggest a different simplification?

Furthermore, we still need additional assumptions about global temperature and whether BIG threats must be answered immediately or whether we shall calculate the then still too complex complete game tree.

For such a locally tightly enclosed shape, endgame calculations become possible, so, if you like, we can then forgo alternative approximations of middle game positional judgement.

Please clarify!

Clarification: How would you decide whether to play in a game, or to play somewhere else?

Click Here To Show Diagram Code

[go]$$W
$$ | . . . . . . . . . . .
$$ | . 1 . X X X X O O . .
$$ | . . . . X O O X O . .
$$ | . . . . . . O X O . .
$$ | . . . . . . . . . . .
$$ -----------------------[/go]

Part of the considerations as to possibly playing elsewhere is a comparion with moves such as 1, which surely is more interesting than White's start on the lower side...

It's black's turn.

Usually, evaluation of values of local moves considers Black's first local move AND (alternatively!) White's first local move to THEN compare both and calculate a COMBINED value (e.g., the so called miai value). With that, one can then better decide whether Black, having the turn, should use it to play locally or elsewhere.

Kirby wrote:Clarification: How would you decide whether to play in a game, or to play somewhere else?

This was already answered by your own Post #1, and Post #2.

EdLee wrote:

Kirby wrote:Clarification: How would you decide whether to play in a game, or to play somewhere else?

This was already answered by your own Post #1, and Post #2.

Hmm. Maybe it was my misconception that there is a more precise method of evaluation. I can imagine possible scenarios and try to pick from them, but it just seems so arbitrary. Seems like there should be something more precise, like what Robert's mentioning with miai value. I dunno.

Maybe there's not a better way.

There are precise methods, but, as I have said, they require assumptions so that their application has a suitable context in which they can express good meaning.

- A privilege requires an assumption of sente.
- An unrest model or a joseki evaluation model require an assumption of known positional environment so that group stability can be assessed correctly.
- A territorial positional judgement relying on reduction sequences requires an assumption of reduction being better than invasion.
- A CGT-style endgame or unsettled group average evaluation requires an assumption of the local sitation being an endgame or allowing a similar analysis.
- Etc.

We need some such assumptions because we do not have a general evaluation theory for an arbitrary given local shape in the context of an arbitrary unknown global context. We "only" have a few mighty theories or approximative models for certain identified assumptions. This is how mathematics works; propositions work for fulfiled assumptions. You can count visible stones but you cannot count stones hidden in fog.

Kirby wrote: Maybe there's not a better way.

Yes. Not worrying about meaningless count and ok.
Lets worry about what make you strong not a research paper.

Counts can be very meaningful to decide about good or even correct move order. E.g., if one gote endgame move has the count 10 and another gote endgame move has the count 9, then one chooses the bigger move.

For improving, it is important to distinguish 10 from 9, but one need not distinguish 10 from 9.99. Therefore, one does not need research papers for determining counts in actual games. Approximations suffice. During the middle game, +-0.5 points is good enough. During the endgame, 1/3 or 1/4 of a point can sometimes be useful. Greater precision tends to be abuse of one's thinking time.

However, one should not neglect counts. 15 mistakes, each losing 0.5 points, during 150 moves loses 15 * 0.5 = 7.5 points. This is a lot. Needless to say, avoiding bigger mistakes is even more important.