moha wrote:
Bill Spight wrote:
Black to play rounds up the score
This was the point that seemed problematic the other thread. If rounding direction depends on sente, then 6.1+sente is better than 6.7+gote. If there is no rounding in chilled go, then correct play there (which presumably maximizes the chilled/fractional score) is not necessarily correct play in territory even without kos. OC not likely for two similar scores to have different dame parities and roundings, but still seems possible if an early decision is between choices that lead / doesn't lead to seki. Where is the flaw in this reasoning?
How do you define a fractional error under territory or area scoring? I can define a fractional error as one that results in a fractional difference in the final score, assuming correct play otherwise. IOW, I can define it only for chilled go, because chilled go has fractional scores.
What is true is that, without ko complications, the fractional scores of chilled go are equal to the mean values of territory and area scoring. Those mean values are estimates of the final territory or area scores, which are integers before applying komi. I do think, as estimates, that the size of the fraction indicates which way the final integer score will likely turn out. An estimate of 0.2 is, given correct play up to the end of the game, more likely to become an integer score of 0 than an integer score of 1, and an estimate of 0.6 is more likely to become an integer score of 1 than an integer score of 0. (I haven't tried to check that out, but it is an empirical question.) So, yes, sometimes a fraction in chilled go will "round" to the more distant integer in territory and area scoring.
However, it is not true that, given correct play after the fractional errors have occurred, and no ko complications, a smaller chilled go fraction will round up while a larger chilled go fraction will round down. The reason is that each possible fraction between the same integers will round in the same direction. And the reason for that is the phenomenon of reverses. Suppose, with correct play, that Black has the move and makes a play that gains so much and then White makes a play that gains more than that. That is something that would be necessary for the fractional chilled score to round down instead of up. But in that case Black has a play or sequence of play that will gain at least as much as she has lost so far. Were that not so, Black's initial play would have been a mistake. Now, kos can cause Black to lose points, even with correct play, but that's another question. There is an example in the Mueller, Berlekamp, Spight paper of 1996. I think it is available on Martin's web site.
Edit: In terms of the thermograph, with no kos, the Left wall, which indicates the minimax results at each temperature when Black plays first, consists of 0 or more vertical line segments and 1 or more line segments at 45° with the Black score increasing as the temperature drops. If the chilled score rounds down, the final line will be vertical at the final integer score, with an intersection below temperature 1. If the chilled score rounds up, the final line will be at 45°. again with an intersection below temperature 1. For the chilled score that rounds down to be greater for Black than the chilled score that rounds up, these two lines would have to intersect below temperature 1. No es possible.
Using rounding to 0 or 1 the possible thermographic lines that round down look like this up to temperature 1.
Code:
Rounding down to 0
______________________ T = 1
/
|
|
___________|___________ T = 0
0
Code:
Rounding up to 1
______________________ T = 1
|
/
/
_______/_______________ T = 0
1
At T = 1 the Left scaffold will be at a fraction less than 0 when the fraction rounds down to 0 and at a fraction greater than 0 when the fraction rounds up to 1. That mean that when both are possible from the same starting position, the sequence that rounds to 1 will dominate the sequence that rounds to 0 at these temperatures, and the chilled go score will be for the score that rounds to 1.
Edit2: For people who are unfamiliar with thermography, which will probably be most of you, the thermographic lines represent plays or lines of play in the game. The two parts of scaffolds here represent lines of play starting from two different options that Black can choose to play, to which White has already chosen his replies to minimize the results for Black at each temperature. Black then chooses her options to maximize the results at each temperature. See
https://senseis.xmp.net/?TemperatureCGT .
At territory scoring Black tries to maximize the result at temperature 0, i.e., when any plays in the background are dame. In chilled go Black tries to maximize the result at temperature 1. The graphs show that when the choice Black has is between an option that "rounds" to 0 at territory scoring (temperature 0) and one that rounds to 1 at territory scoring, which Black will choose at territory scoring, Black also maximizes her score at chilled go (temperature 1) by making the same choice. (Assuming no ko complications, OC.
)
Edit3: There is an apparent paradox here, in that the estimate of the final territory score, which we get by projecting the lines at temperature 1 down to temperature 0, may be greater for the scaffold that rounds to 0 than for the estimate for the scaffold that rounds to 1. How can that option be less for chilled go than the one that gives a lower estimate? The answer lies in the inclination of the line at temperature 1. The line for the result that gets rounded to 1 is vertical, which means that White has made the same number of moves and Black, but the line for the result that gets rounded to 0 is inclined, which means that Black has made one more move than White. Chilling penalizes Black one point for that extra play. (
https://senseis.xmp.net/?Chilling ). That's why the chilled go result is less.
For instance, suppose that one line of play estimates the territory result to be ¾ but gets rounded down to 0 and other line of play estimates the territory result to be ½ but gets rounded up to 1. The one that estimates a result of ¾ by territory scoring has a chilled go score of -¼, while the one that gets rounded up to 1 has a chilled go score of ½.
This failure to distinguish between the estimated territory score and the chilled go score is my fault. I just never thought about it before.
It is the estimate that gets rounded up or down, not the chilled go score. I misunderstood David Wolfe's comment to me years ago.
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Bill Spight
So what's the best strategy and proper komi? 
points. The Black corridor corresponds to the stack with 1 Blue chip, the White corridor corresponds to the stack with 3 Red chips. OC, go players know how to play this game. 
This is why the rounding of mean values is possible, unless kos widen the variation.