Jean-loup Gailly wrote:
m is my definition of the value of the first move.
Willemien wrote:
in 7x7 go under perfect play and without komi black wins by 9 points. if Black passes on the first move then under the same perfect play white wins with 9 points (again without komi) Does that mean that the value of the first move is 18 points?
The problem is that there is not "the" single value. Jean's definition is nice and under that definition the related value Willemien refers to would be the same as his. The crucial question now is whether that value is the deire endgame value (or, since the local-tally is 2, twice the miai value) of the first move or is some other value (which accidentally might or might not be the same when practically determining it for a board).
According to http://senseis.xmp.net/?MiaiCounting, since by symmetry (either Black first stone or White first stone) we may assume that the value does not depend on ko threats and since the first play does not threaten to raise the temperature, we may as well assume that the first play on the board is gote and, using Jean's construction, determine both the count and the local-tally to determine the miai value (and, multiplying by 2, the deire value).
I would like to make a minor improvement: The addition of 0.5 to a komi is artificial. Better omit it and revise Jean's calculation accordingly. This gives k = m-k or m = 2*k.
So apart from some ambiguity of what is a ko threat and what is a gote (which can be ignored by using a simplified substitute for the usually used term of "miai value"), I am tempted to believe that Jean-loup's construction embedded in the above context is indeed a proof now. The construction via maximal / minimal value under assumed perfect play and the parameter x relate the 1990ies' raw sketch along the idea "Black passes, then instead White makes the first play" to perfect play.
Note that the "miai value" of the second play on the board in relation to komi is still unknown.
-- robert jasiek
Assume a 19x19 no-handicap game played by perfect players with New Zealand rules with komi k. k is chosen as the largest value N+0.5 (with N integer) which guarantees a win by black. k exists because with New Zealand rules the game is finite. Black cannot win by an infinite amount of points, so there is an upper bound for k, therefore k exists. Now give the choice to black to either play the first move, or pass and receive x extra points, x constrained to be an integer. Let m be the smallest value of x that black will accept to pass instead of playing, and still be sure of winning. m exists because the game is finite, and black being perfect can determine m exactly. m is my definition of the value of the first move. I am not attempting to define the value of subsequent moves, and not assuming that these values decrease constantly. I am only interested in the relation between m and k. If black passes, black has m points, white has k points, white to play. White playing perfectly for maximum score will lose by exactly 0.5 point. If white is given one extra point, white would be exactly in the position that black was at initially: next to play and guaranteed to win by 0.5. The player next to move on the empty board and with a guaranteed 0.5 win has a cash deficit of k points in the first case (black to play) and m-(k+1) points in the other (black passed and white to play). So k = m-(k+1) or m = 2*k+1. Again the above analysis only considers the game theoretical value of the empty board, not any subsequent position. I welcome any correction if there is a flaw in my reasoning (which is quite possible). Jean-loup
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