: Re: answering Bill
(2006-12-18 15:32) [#2698]
For me, the relationship is not easy to show in general, even though by area scoring you gain a point for the stone played which you do not gain by territory scoring. My difficulty is that I do not know what exactly the plain, precise, formal definitions are of "area miai value" and "territory miai value".
Formally, the miai value corresponds to the temperature of a position, in CGT terms. It is determined by the thermograph of the position. Each thermograph has a vertical mast at its top, which indicates the count of the position. The temperature of the position is the temperature at the base of that mast. (For more details, see On Numbers and Games, Winning Ways, or my paper on Extended Thermography.)
I might create my own ad hoc definitions, but then I am not sure if we speak of the same. So what are the generally used precise and formal definitions of these two terms? How then is it easy to prove?
Well, it's not easy to prove, because it is not true. There are exceptions involving seki. (That is true even for my territory rules, because they do not have fractional scores.) It is easy to show in general, because miai value represents how much a play gains, for gote or reverse sente, or how much the reverse sente would gain, for sente. Counting the played stone gains one more point by area scoring.
Of course, I am aware of some special scoring differences between area scoring and territory scoring in special positions. I am glad to ignore those so far. However, since they exist, my equations do not always hold. This is my question: Can we define characteristics of a (an as complete as possible) class of "normal" (local) positions where my equations hold?
Why, in a button position, does a territory miai value of -1/2 imply a fractional territory score (count) on the board?
At temperature -1, all we have left are integer scores. If a play to one of those score loses 1/2 point, then the position it was played in has a count of an integer +/- 1/2 at higher temperatures. Temperature 0 is the temperature of territory scores, and that position has the same fractional count at that temperature.
In the seki or ko position, why do you say that under territory scoring its miai value should be -1?
All I had in mind was, "Given an area miai value of 0, the territory miai value should be -1." Under my territory rules that is so. White, if forced to make a local play or to give up a pass stone ("gaining" -1), will prefer to give up the stone.
Of course, that is elegant in a sense, but currently I am interested not in elegant rulesets but in explaining things for real world scoring rulesets, i.e. in particular Traditional Territory Scoring. I want to enable players to determine biggest moves and scores in practical games, not to provide the most elegant theory for theoretically the most elegant (here: territory) rules possible.
For non-kos, that is easy. As I said, the miai value is how much a gote or reverse sente gains. To find that out, you have to figure the values (counts or scores) of the positions involved. For gote plays, the value of the original position is the average of the values of the position resulting after Black plays first and the one after White plays first. For sente plays, the result after the sente has the same value as the original position.
Kos are trickier, and I do not think are easy to understand fully without thermography. However, for regular kos you can find the miai values by dividing the difference in values between resulting positions by the number of net plays between them. E. g., if Black can win a ko in one play, but it takes two plays for White to take and win it, the miai value is the difference in results divided by three.