Call it "the difference between S and 0".

But "the difference between the count of S and 0 (which is the count of a leaf)" we call the "the negative of the [net] profit" aka "the negative of the gain", right?

In CGT a game is a value, and numbers are games. Non-numerical games are fuzzy values. The count of a go position is a number, which is the mean value of its game (adjusted in modern go according to the specific rule set).

You are the one to introduce the term, profit. If you wish to use it only for numbers, be my guest. But gain and loss, in English, are not restricted to numbers. One may gain understanding or lose confidence.

The gain from playing with sente from S to 0 is 0 - S, which is the game, -S, which is greater than 0, even though its mean value is 0.

If you want to say that the profit from playing S with sente is 0, that accords with the saying, Sente gains nothing.

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

Actually, you introduced 'profit value' for one move's profit and 'net profit' for the alternating difference of a sequence's profit values quite some time ago, although you did not provide such definitions explicitly, AFAIK. I have found these terms very useful when expressing the change from a starting count to a follower's count.

'Profit' used alone is also used in different, informal go meanings ("A player takes profit here, then tenukis."). Therefore, using 'profit value' and 'net profit' as the terms for CGT or endgame theory avoids ambiguity. Besides, 'profit value' reminds us of the close relation to the term 'move value'.

There seems to be no consensus yet on whether 'gain' might occur as another term. I hope that not because profit and gain are so similar to create confusion that is better avoided.

Well, your memory is better than mine on that score. (Edit: I did a search here and did not find that I used it here.) Possibly that happened on rec.games.go or, perhaps more likely, in the lost archives of Godiscussions before 2010. Back in the 1990s I pushed for Western players to learn and use miai values for moves instead of deiri values, in an attempt to help clarify evaluation. I seem to have caused more confusion, however. John Fairbairn has pointed out that the term, deiri in accounting refers to profit and loss. It could be that I tried the term, profit value, as an English equivalent to miai value. That seems to be how you are using the term. But eventually I decided to simply use the term, gain, despite some ambiguity. Unless you get really technical, you don't have to distinguish between the difference in counts and the difference in games. Here I was getting really technical, because I wanted to show that sente in the technical sense can gain something, just not points.



Yes, I see other people using "miai value", but neither "profit value" nor "gain" seems to have generally caught on yet.

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

Maybe on Sensei‘s? You used profit value in contrast to move value to describe the different behaviour of sente and the different values of starting a sente sequence or playing reverse sente. Precision is needed and I did some related proofs because I could not find such quickly from others, except that OC you paved the conceptual way.

Yet another post on Kano's Beauty



Kano's Beauty is A + D (normalized to have a count of 0). As we know, A + D = A - D < 0. Which means that White gets the last play in Kano's Beauty, no matter who plays first. It is rather interesting that the sum of two games that are confused with 0 is less than 0. But that is not all that unusual.

Here is something that I find interesting.

A < D, so
A + A < D + D = 0

So two copies of A is less than 0. Here is an SGF file that illustrates that. Note that when White plays first he could easily make a mistake.

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

Tsumego from an actual game

Click Here To Show Diagram Code

[go]$$Wc Status?
$$ --------------------
$$ | . . X . X X . . . .
$$ | . O O X X . X O . .
$$ | X . O X X X O X X X
$$ | X O . , O O O O . ,
$$ | O X O O O X X O X X
$$ | . X X X O O X O X O
$$ | . X . X . O X O X X
$$ | O O O X . O X O X O
$$ | . X X . X O X X X X
$$ | . X . , O O O X O ,
$$ | . O O O . X . X O O
$$ | . O X X X . X X X X
$$ | . O X . . X X . O O
$$ | . O O X X . X X O X[/go]

What is the status of the Black group on the left side? Can White to play kill? (The surrounding White stones are alive, OC.)

Enjoy!

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

Black has resources in the first diagram. Think about it before peeking!

_________________
Dave Sigaty
"Short-lived are both the praiser and the praised, and rememberer and the remembered..."
- Marcus Aurelius; Meditations, VIII 21

A not so easy problem

Most of my problems and illustrations are easy. This one, however. . . .

Click Here To Show Diagram Code

[go]$$Wc White to play
$$ ---------------------------
$$ | X X X X X X X X X X X X . |
$$ | . O . O . O . O . O . O X |
$$ | . O . O . O . O . O . O X |
$$ | . O . O . O . O . O . O X |
$$ | . O . O . O . O O O O O X |
$$ | . O . O . O O O X O X X X |
$$ | . O . O . O X X X X . . . |
$$ | . O . O X . . . . X . X . |
$$ | . O X X X X X X X X O X . |
$$ | X O . . . . . X O O O O . |
$$ | X X X X X X X X O . . . . |
$$ | . . . . . . . . X O . O . |
$$ | . . . . . . . X . . . . . |
$$ ---------------------------[/go]

No komi. White to play and win.

Enjoy!

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

Well, nobody has bitten so far. Here is an unhelpful hint.

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

Patience William, patience! I can win, but cannot yet say why with confidence. Therefore cannot yet be sure whether Black has additional resources. Annoyingly addictive problem though.

_________________
Dave Sigaty
"Short-lived are both the praiser and the praised, and rememberer and the remembered..."
- Marcus Aurelius; Meditations, VIII 21

Glad you like it, Dave.

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

I feel confident that Dave has found the solution. But, as he says, why it is right is not easy to explain. Anyway, it has been more than a week, so here goes.

Click Here To Show Diagram Code

[go]$$Wc White to play
$$ ---------------------------
$$ | X X X X X X X X X X X X . |
$$ | . O . O . O . O . O . O X |
$$ | . O . O . O . O . O . O X |
$$ | . O . O . O . O . O . O X |
$$ | . O . O . O . O O O O O X |
$$ | . O . O . O O O X O X X X |
$$ | . O . O . O X X X X 2 . . |
$$ | . O . O X . . . . X 1 X . |
$$ | . O X X X X X X X X O X 4 |
$$ | X O . . . . . X O O O O 3 |
$$ | X X X X X X X X O . . . . |
$$ | . . . . . . . . X O . O . |
$$ | . . . . . . . X 6 5 . . . |
$$ ---------------------------[/go]

The first few moves occur at temperature 1, but Black gets the last play at that temperature. is sente, but White could also play it at 5.

Click Here To Show Diagram Code

[go]$$Wc White wins
$$ ---------------------------
$$ | X X X X X X X X X X X X . |
$$ | a O . O . O 7 O . O . O X |
$$ | . O . O . O . O . O . O X |
$$ | . O . O . O . O . O . O X |
$$ | . O . O . O . O O O O O X |
$$ | . O . O . O O O X O X X X |
$$ | . O . O . O X X X X X . . |
$$ | . O . O X . . . . X O X . |
$$ | . O X X X X X X X X O X X |
$$ | X O . . . . . X O O O O O |
$$ | X X X X X X X X O . . . . |
$$ | . . . . . . . . X O . O . |
$$ | . . . . . . . X X O . . . |
$$ ---------------------------[/go]

The winning move is . Now the rest is miai, for a score of 1 pt. for White. That is far from obvious. Besides, the play at "a" looks bigger. Normally an open corridor of length 8 is equivalent to a closed corridor of length 6. How can the best play be on a closed corridor of length 4? The answer lies in the threatened atari against the Black stones on the top edge.

Anyway, here is one way to play out the miai. Since the whole board is miai, there are several ways to play, OC.

Click Here To Show Diagram Code

[go]$$Wc White wins, cont.
$$ ---------------------------
$$ | X X X X X X X X X X X X . |
$$ | . O . O . O O O 8 O 9 O X |
$$ | . O . O . O . O 0 O . O X |
$$ | . O . O . O . O . O . O X |
$$ | . O . O . O . O O O O O X |
$$ | . O . O . O O O X O X X X |
$$ | . O . O . O X X X X X . . |
$$ | . O . O X . . . . X O X . |
$$ | . O X X X X X X X X O X X |
$$ | X O . . . . . X O O O O O |
$$ | X X X X X X X X O . . . . |
$$ | . . . . . . . . X O . O . |
$$ | . . . . . . . X X O . . . |
$$ ---------------------------[/go]

and are obviously miai, even if they may not appear to be best. eliminates the atari before the dame stage, so the remaining plays continue as usual.

Click Here To Show Diagram Code

[go]$$Wcm11 White wins (ii)
$$ ---------------------------
$$ | X X X X X X X X X X X X . |
$$ | 1 O 5 O . O O O X O O O X |
$$ | . O . O . O . O X O . O X |
$$ | . O . O . O . O . O . O X |
$$ | . O . O . O . O O O O O X |
$$ | . O . O . O O O X O X X X |
$$ | . O 7 O . O X X X X X . . |
$$ | 3 O 6 O X . . . . X O X . |
$$ | 2 O X X X X X X X X O X X |
$$ | X O 4 . . . . X O O O O O |
$$ | X X X X X X X X O . . . . |
$$ | . . . . . . . . X O . O . |
$$ | . . . . . . . X X O . . . |
$$ ---------------------------[/go]

- closes off the longest corridor. Then - are miai.

Click Here To Show Diagram Code

[go]$$Bcm18 White wins (iii)
$$ ---------------------------
$$ | X X X X X X X X X X X X . |
$$ | O O O O 2 O O O X O O O X |
$$ | . O . O . O . O X O . O X |
$$ | . O . O . O . O . O . O X |
$$ | . O . O . O . O O O O O X |
$$ | . O . O 4 O O O X O X X X |
$$ | . O O O 3 O X X X X X . . |
$$ | O O X O X 1 . . . X O X . |
$$ | X O X X X X X X X X O X X |
$$ | X O X . . . . X O O O O O |
$$ | X X X X X X X X O . . . . |
$$ | . . . . . . . . X O . O . |
$$ | . . . . . . . X X O . . . |
$$ ---------------------------[/go]

Then - are miai. Black scores 25 pts., but White scores 26 pts. to win by 1 pt.

Let's try a variation for Black at move 10.

Click Here To Show Diagram Code

[go]$$Bcm10 Variation
$$ ---------------------------
$$ | X X X X X X X X X X X X 9 |
$$ | 6 O 4 O 8 O O O X O O O X |
$$ | . O . O . O . O 2 O . O X |
$$ | . O . O . O . O . O . O X |
$$ | . O . O . O . O O O O O X |
$$ | . O . O . O O O X O X X X |
$$ | 7 O 0 O . O X X X X X . . |
$$ | 3 O 5 O X . . . . X O X . |
$$ | 1 O X X X X X X X X O X X |
$$ | X O . . . . . X O O O O O |
$$ | X X X X X X X X O . . . . |
$$ | . . . . . . . . X O . O . |
$$ | . . . . . . . X X O . . . |
$$ ---------------------------[/go]

enters the longest corridor, and closes off the shortest one. Then the two White corridors on the left are plainly miai. After - they still are. After gets in the sente and then comes back to .

Click Here To Show Diagram Code

[go]$$Bcm20 Variation, cont.
$$ ---------------------------
$$ | X X X X X X X X X X X X X |
$$ | O O O O O O O O X O O O X |
$$ | . O . O . O . O O O . O X |
$$ | . O . O . O . O . O . O X |
$$ | . O . O . O . O O O O O X |
$$ | 4 O . O . O O O X O X X X |
$$ | X O O O 2 O X X X X X . . |
$$ | X O X O X 3 . . . X O X . |
$$ | X O X X X X X X X X O X X |
$$ | X O 1 . . . . X O O O O O |
$$ | X X X X X X X X O . . . . |
$$ | . . . . . . . . X O . O . |
$$ | . . . . . . . X X O . . . |
$$ ---------------------------[/go]

and are miai, as are and . Black gets 25 pts. and White gets 26 pts., as above.

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

White failure.

Click Here To Show Diagram Code

[go]$$Wc White failure, jigo
$$ ---------------------------
$$ | X X X X X X X X X X X X . |
$$ | 7 O . O . O 8 O . O . O X |
$$ | . O . O . O 0 O . O . O X |
$$ | . O . O . O . O . O . O X |
$$ | . O . O . O . O O O O O X |
$$ | . O . O . O O O X O X X X |
$$ | . O . O . O X X X X X . . |
$$ | . O . O X . . . . X O X . |
$$ | 9 O X X X X X X X X O X X |
$$ | X O . . . . . X O O O O O |
$$ | X X X X X X X X O . . . . |
$$ | . . . . . . . . X O . O . |
$$ | . . . . . . . X X O . . . |
$$ ---------------------------[/go]

looks correct, but it is a very small error. is correct, OC. Then is correct if was correct. (Edit: is actually correct now, but we don't have to show that. ) eliminates the sente against the Black stones on the top edge. The rest of the play is as usual.

Click Here To Show Diagram Code

[go]$$Wcm11 White failure (ii)
$$ ---------------------------
$$ | X X X X X X X X X X X X . |
$$ | O O 1 O 5 O X O . O . O X |
$$ | . O . O . O X O . O . O X |
$$ | . O . O . O . O . O . O X |
$$ | . O . O . O . O O O O O X |
$$ | . O . O 7 O O O X O X X X |
$$ | . O 3 O 6 O X X X X X . . |
$$ | . O 2 O X 8 . . . X O X . |
$$ | O O X X X X X X X X O X X |
$$ | X O 4 . . . . X O O O O O |
$$ | X X X X X X X X O . . . . |
$$ | . . . . . . . . X O . O . |
$$ | . . . . . . . X X O . . . |
$$ ---------------------------[/go]

- are miai, as are - .

Click Here To Show Diagram Code

[go]$$Wcm19 White failure (iii)
$$ ---------------------------
$$ | X X X X X X X X X X X X . |
$$ | O O O O O O X O 1 O 2 O X |
$$ | . O . O . O X O . O 4 O X |
$$ | . O . O . O 3 O . O . O X |
$$ | . O . O . O . O O O O O X |
$$ | . O . O O O O O X O X X X |
$$ | . O O O X O X X X X X . . |
$$ | . O X O X X . . . X O X . |
$$ | O O X X X X X X X X O X X |
$$ | X O X . . . . X O O O O O |
$$ | X X X X X X X X O . . . . |
$$ | . . . . . . . . X O . O . |
$$ | . . . . . . . X X O . . . |
$$ ---------------------------[/go]

Finally and are miai, as are and . Black gets 25 pts., as usual, and White also gets 25 pts., for jigo.

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

Click Here To Show Diagram Code

[go]$$Wc White wins
$$ ---------------------------
$$ | X X X X X X X X X X X X . |
$$ | a O . O . O 7 O . O . O X |
$$ | . O . O . O . O . O . O X |
$$ | . O . O . O . O . O . O X |
$$ | . O . O . O . O O O O O X |
$$ | . O . O . O O O X O X X X |
$$ | . O . O . O X X X X X . . |
$$ | . O . O X . . . . X O X . |
$$ | . O X X X X X X X X O X X |
$$ | X O . . . . . X O O O O O |
$$ | X X X X X X X X O . . . . |
$$ | . . . . . . . . X O . O . |
$$ | . . . . . . . X X O . . . |
$$ ---------------------------[/go]

"Now the rest is miai"

Once more I wonder: What IS 'miai'? (And, applied to this position, why do we have a miai here?)

Click Here To Show Diagram Code

[go]$$Wc White wins, cont.
$$ ---------------------------
$$ | X X X X X X X X X X X X . |
$$ | . O . O . O O O 8 O 9 O X |
$$ | . O . O . O . O . O . O X |
$$ | . O . O . O . O . O . O X |
$$ | . O . O . O . O O O O O X |
$$ | . O . O . O O O X O X X X |
$$ | . O . O . O X X X X X . . |
$$ | . O . O X . . . . X O X . |
$$ | . O X X X X X X X X O X X |
$$ | X O . . . . . X O O O O O |
$$ | X X X X X X X X O . . . . |
$$ | . . . . . . . . X O . O . |
$$ | . . . . . . . X X O . . . |
$$ ---------------------------[/go]

":b8: and :w9: are obviously miai"

Why are they miai? What IS miai? It must be something more specific than 'same move value'.

These corridors are obviously miai because, sans ko, we may consider that if one player plays in one the other player will play in the other. There are only two of these, so that is obvious. Now suppose that there are more than two of them. In actual play we do not know which one our opponent may play in, but we can decide which pair or pairs to regard as miai.



Like most words, miai has more than one meaning. The basic one is that there is a pair of local positions such that we may assume, in correct alternating play sans ko, that one player will play in one and the other player will play in the other. We may also have a pair of plays in a single local position where the plays are not independent, such that we may assume that one player will make one play and the other player will make the other. Sometimes we know which player will make which play. It does not matter which player plays in a miai first, and, as I have indicated above, we may regard a pair of plays or positions as miai without consulting the other player.

Suppose that we have two independent, mirror positions on the board. Sans ko, they are miai. They are miai, no matter how complex they are, because if one player plays in one, the other player may make the corresponding opposite and equal play in the other, to produce another pair of mirror positions.

In this case, although it is complicated, play proceeds through a number of miai sequences. It does not matter who plays first, the result will be the same. Let White play first, and the result will still be a one point win for White.

Click Here To Show Diagram Code

[go]$$Wc White wins by one point
$$ ---------------------------
$$ | X X X X X X X X X X X X . |
$$ | 3 O . O . O W O 1 O 2 O X |
$$ | . O . O . O . O . O 6 O X |
$$ | . O . O . O . O . O . O X |
$$ | . O . O . O . O O O O O X |
$$ | . O . O . O O O X O X X X |
$$ | . O . O . O X X X X X . . |
$$ | 5 O . O X . . . . X O X . |
$$ | 4 O X X X X X X X X O X X |
$$ | X O . . . . . X O O O O O |
$$ | X X X X X X X X O . . . . |
$$ | . . . . . . . . X O . O . |
$$ | . . . . . . . X X O . . . |
$$ ---------------------------[/go]

and are miai. - is a miai sequence.

Click Here To Show Diagram Code

[go]$$Wcm7 White wins by one point
$$ ---------------------------
$$ | X X X X X X X X X X X X . |
$$ | O O 1 O 5 O W O O O X O X |
$$ | . O . O . O . O . O X O X |
$$ | . O . O . O . O . O . O X |
$$ | . O . O . O . O O O O O X |
$$ | . O . O 7 O O O X O X X X |
$$ | . O 3 O 6 O X X X X X . . |
$$ | O O 2 O X 8 . . . X O X . |
$$ | X O X X X X X X X X O X X |
$$ | X O 4 . . . . X O O O O O |
$$ | X X X X X X X X O . . . . |
$$ | . . . . . . . . X O . O . |
$$ | . . . . . . . X X O . . . |
$$ ---------------------------[/go]

- and - are miai sequences. Black gets 25 pts. but White gets 26 pts. to win by 1 pt.

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

Question for Bill - assuming no ko is involved anywhere, if you have two copies of a subgame G such that the temperature never increases as you descend any branch of the game tree of G (for example, pushing down a corridor that terminates in nothing or that terminates in only saving a single stone at the end satisfies this condition), is it always the case that you can treat the first move in the two copies of G as "miai" when determining what best play is?

And if so, is there a more general condition than this?

With no kos, if the temperature of G is less than 1, you can treat two copies of G as miai. Otherwise, maybe.

In the following position, consider play only in the corridors.

Click Here To Show Diagram Code

[go]$$ Miai, but . . .
$$ -------------------
$$ | O X . . X . . X O |
$$ | O O O O X O O O O |
$$ | . . . . X . . . . |
$$ | . . . . . . . . . |
$$ | . . . O . . . . . |
$$ | . . . O X X X . . |
$$ | . . . O . O X . . |
$$ | . . . O X X X X X |
$$ | . . . O . . . O X |
$$ -------------------[/go]

Here the top two corridors are not miai.

Click Here To Show Diagram Code

[go]$$ Black wins
$$ -------------------
$$ | O X . 1 X 2 . X O |
$$ | O O O O X O O O O |
$$ | . . . . X . . . . |
$$ | . . . . . . . . . |
$$ | . . . O . . . . . |
$$ | . . . O X X X . . |
$$ | . . . O . O X . . |
$$ | . . . O X X X X X |
$$ | . . . O 3 . . O X |
$$ -------------------[/go]

is a mistake. In the corridors Black wins by 1 pt.

Click Here To Show Diagram Code

[go]$$ Miai play
$$ -------------------
$$ | O X . 1 X . . X O |
$$ | O O O O X O O O O |
$$ | . . . . X . . . . |
$$ | . . . . . . . . . |
$$ | . . . O . . . . . |
$$ | . . . O X X X . . |
$$ | . . . O . O X . . |
$$ | . . . O X X X X X |
$$ | . . . O 2 . . O X |
$$ -------------------[/go]

and are miai. The rest is a mirror position, for jigo.

Edit: Note that and are not interchangeable. We expect White to play will play at 2, and then Black will reply at 1. If instead White to play plays at 1, Black will not reply at 2.

Click Here To Show Diagram Code

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

White wins one point in the corridors.

Instead Black replies this way.

Click Here To Show Diagram Code

[go]$$W Miai play
$$ -------------------
$$ | O X . 1 X 2 . X O |
$$ | O O O O X O O O O |
$$ | . . . . X . . . . |
$$ | . . . . . . . . . |
$$ | . . . O . . . . . |
$$ | . . . O X X X . . |
$$ | . . . O . O X . . |
$$ | . . . O X X X X X |
$$ | . . . O 3 4 . O X |
$$ -------------------[/go]

Jigo.

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

Hmmm, I'll need to think about that a bit. Thanks.