INTRODUCTIONWhat are the move value, gains and count?
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[go]$$B Initial position
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$$ | . X . X . X .
$$ | X O X X X X X
$$ | O O O O . . .
$$ | . O . O . . .[/go]
Transformation to a follow-up:
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[go]$$W Transformation
$$ ---------------
$$ | 1 X 2 X . X .
$$ | X O X X X X X
$$ | O O O O . . .
$$ | . O . O . . .[/go]
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[go]$$B Prisoners = -1, count = -2/3
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$$ | O X X X . X .
$$ | . O X X X X X
$$ | O O O O . . .
$$ | . O . O . . .[/go]
The prisoner difference is -1 and the ko has the count 1/3 so the follow-up position has the count -1 + 1/3 = -2/3.
Now, Bill has used this count of the follow-up to derive the alleged count -2/3 of the initial position.
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[go]$$B Miai?
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$$ | W X B X . X .
$$ | X O X X X X X
$$ | O O O O . . .
$$ | . O . O . . .[/go]
His argument has been that the circled stones could be exchanged in either order in a sente-like sequence so would be miai and therefore the initial position would inherit the count -2/3 from its sente-like follower.
However, "miai" is only an informal term. In the transformation above, Black need not play at 2 but he has the alternative option of playing elsewhere and fighting the stage ko. There is no a priori justification why the initial count would have to be determined by the basic ko fight due to the connection Black 2.
The initial position allows a basic ko fight or a stage ko fight. The ko(s) can be fought differently. Such kos can be 'active' or 'hyperactive', their move value, gains or count can depend on the environment and possibly its ko threats. Due to different fights (here: basic ko fight or stage ko fight) in different environments, the initial values can differ. There need not be only one move value, only one black gain, only one white gain and only one count.
BASIC KO FIGHTIf there is the basic ko fight, the initial count is -2/3 indeed, the move value is 1/3 and each gain is 1/3.
STAGE KO FIGHTHowever, if there is the stage ko fight, we must not use the initial count -2/3 of the basic ko fight and must not derive the gain (-2/3 - (-3)) / 3 = 7/9 from this abused initial count and the count -3 of the settled white follower after three successive white plays.
Instead, for the stage ko fight, we must calculate its own values.
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[go]$$B Stage ko fight, Black starts, count = -1/3
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$$ | . X 1 X . X .
$$ | X O X X X X X
$$ | O O O O . . .
$$ | . O . O . . .[/go]
If Black starts in the stage ko fight, the local sequence comprises one play. It does not comprise two plays (with White answering by capturing the ko) because the gain 1/3 of White's ko capture would be smaller than the move value calculated then. Black's start results, after the only one play, in the count -1/3 for the ko with its black ko stone.
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[go]$$W Stage ko fight, White starts, Black 2 and 4 elsewhere
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$$ | 1 5 3 X . X .
$$ | X O X X X X X
$$ | O O O O . . .
$$ | . O . O . . .[/go]
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[go]$$W Prisoners = -2, count = -3
$$ ---------------
$$ | O O O X . X .
$$ | C O X X X X X
$$ | O O O O . . .
$$ | . O . O . . .[/go]
The stage ko fight and White's start result in the prisoner difference -2 and the count -3.
Now, for the stage ko fight, we know that Black's start leads to 1 black excess play and the count -1/3 while White's start leads to 3 white excess plays and the count -3. The tally of both numbers of excess plays is 1 + 3 = 4. We also use the difference of resulting counts to calculate the move value and count of the initial position in the case of a stage ko fight:
Move value = (-1/3 - (-3)) / 4 = 2/3.
We can derive the count from Black's or White's follower:
Count = -1/3 - 1 * 2/3 = -3 + 3 * 2/3 = -1.
Gain of Black's play: -1/3 - (-1) = 2/3.
The gains of White's plays need not all be the same. First, we would have to determine the counts after White 1 and White 3 in the stage ko fight.