> Yours is not a proof because what follows is not just a single move > of value x but a game tree of moves of various sizes,
So let me try to be more precise. 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 2010/2/11 Robert Jasiek <jas...@snafu.de> > Jean-loup Gailly wrote: > >> I would write the proof as follows. >> >> Assume x is the value of one move >> > > Yours is not a proof because what follows is not just a single move of > value x but a game tree of moves of various sizes, which need not even > decrease constantly. Many years ago, Barry Phease was a bit farther by > assuming a constant decrement and forming a sum. His was not a proof either > because sizes of moves in the paths of the game tree need not decrease > constantly. > > Rather than being proofs, such arguments are plausible approximations. > > -- > robert jasiek > > _______________________________________________ > computer-go mailing list > computer-go@computer-go.org > http://www.computer-go.org/mailman/listinfo/computer-go/ > >
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