mitsun wrote:
W to play (reverse-sente) can create a final position worth 0 points. It is then B turn to play, somewhere else on the board.
B to play (sente) can create a final position worth 2 points after W replies. It is then B turn to play again, somewhere else on the board.
(The B move, which we are calling sente, leaves a follow-up move worth 100 additional points.[...]
1) What is the value (count) of the initial position (I believe the answer is 2)
2) What is the value of the W reverse-sente move? (I suspect the answer is 2)
3) What is the value of the B sente move? (I know Bill's answer is 0)
Bill or Robert, are these indeed the values you would calculate?
I take it that you mean that, after the B move creating the intermediate position being a local gote, the follow-up move is a 100 points gain for the next moving player.
***
1) The count of the initial (local endgame) position is 2. This is the inherited count 2 of the black sente follower created after his sente sequence.
2) The value of the white reverse sente move is 2. This value is the sente move value (and equals the gain of the white move).
Calculated as the sente move value, it is the difference value of the count 2 of the black sente follower (created after Black's sente sequence) and the count 0 of the white reverse sente follower (created after White's reverse sente move). Therefore, we calculate 2 - 0 = 2.
Calculated as the gain of the white move, it is the difference of the count 2 of the initial position and the count 0 of the white reverse sente follower (created after White's reverse sente move). Therefore, we calculate 2 - 0 = 2.
The two calculations are equal because the count 2 of the initial position equals the count 2 of the black sente follower.
3) The value of the black sente move is its gain and does NOT equal the sente move value. The gain of the black sente move is the difference of the count of the intermediate position created by it and the count of the initial position. We do not know the count of the intermediate position yet but we already know the count 2 of the initial position.
The intermediate position is a local gote, whose count is calculated as the average of its followers. Its black follower is created after Black moves twice successively from the initial position; its white follower is the initial position's black sente follower (created by the sente sequence started by Black and concluded by White's immediate reply). Since you have not specified an absolute count of its black follower, we need a variable, say A, for the count of the intermediate position. You say that a move in it gains 100 points for the next moving player. Therefore, the count of its black follower is (A + 100) and the count of its white follower (which is the initial position's sente follower) is (A - 100). We subtract when White moves and gains because Black incurs a loss. White's gain is Black's loss.
Since the intermediate position is a local gote, its gote count A is calculated as the average of the counts of its followers. We have the count (A + 100) of its black follower and the count (A - 100) of its white follower. Therefore, we can now calculate the count A of the intermediate position as the average:
A = ( (A + 100) + (A - 100) ) / 2 = (A + 100 + A - 100) / 2 = ( 2 * A ) / 2 = A.
Surprise. We have learnt nothing new for the value of A.
We can, however, derive the numerical value of A from the count 2 of the sente follower. To do so, we use the gain of the white move played in the intermediate position and creating the sente follower. We know that the gain of the white move is 100 (a positive number because White gains points when playing his move). White's gain is Black's loss. We view the count A of the intermediate position from Black's perspective. When White gains 100, Black loses 100. Therefore, White's move from the intermediate position to the sente follower decreases the count.
However, we want the converse: the impact of undoing White's move. The count A of the intermediate position is larger than the count 2 of the sente follower. It is larger by the amount 100 of the gain. Therefore, to the count 2 of the sente follower, we add the gain 100 to calculate the count A of the intermediate position:
A = 2 + 100 = 102.
Next, we can calculate the gain of the black move in the initial position. The gain is the difference of the count A = 102 of the intermediate position and the count 2 of the initial position. The gain of the black move in the initial position is 102 - 2 = 100. This is the value of the black move considered as a single move.
4) Let us study the sente sequence. Black's first move gains +100, then White's reply lets White gain 100 or Black lose 100 so the impact of White's reply is -100. Now, we can calculate the 'net profit' of the sente sequence by summing up the impacts of its moves. The net profit of the sente sequence is +100 - 100 = 0.
Where informal descriptions say "sente gains nothing", what they should mean is: the sente sequence of a local sente endgame has the net profit 0.
It is NOT the single first black move played in the initial position that you might have guessed to have the "value" 0 (what value?). Instead, it is the net profit of the sente sequence that has the value 0. The sente sequence is started by the single first black move but this does NOT mean that the move itself would have the value of the whole sente sequence.
If Black's first move is ignored, the local impact is a change of +100. If Black's first move is replied, the local impact of the sente sequence is 0.
Code:
2
/ \
+100 / \ -2
/ \
/ \
A = 102 0
/\ ^
/ \ |
+100 / \ -100 |
/ \ |
/ \ |
202 2 <--- move value calculated from these two counts as 2 - 0 = 2
The leftwards / rightwards lines are Black's / White's moves. The numbers next to these lines are the gains / losses from Black's value perspective. The numbers at the nodes linked by these lines are the counts of the positions. The arrows in the lower right of the diagram are commentary only. The sente sequence is from 2 via A to 2.