It is often stated that a benefit of territory scoring over area scoring is that it is more fine-grained. That is, with territory scoring, the score (difference) can be any whole number, but with area scoring, the score can usually only be an odd number. (It is only usually so, because some seki positions introduce an odd number of non-scoring points on the board.) It is likely that when most people here this they believe that territory scoring must be better in some sense. Although this isn't necessarily false, the issue is perhaps more complicated than most people realise and area scoring deserves some defense on this point.
Firstly, it should be realised that the finer granularity of territory scoring is arbitrary. Area scoring can be modified by the use of a button so that it is effectively the same as territory scoring, but by the use of multiple buttons, area scoring can be further modified so that the final score be even more fine-grained. Then, the final score might be any multiple of a half, or any multiple of a quarter, or whatever is desired.
It is generally agreed that one of the best things about go is the simplicity of the rules. With territory scoring, there must be special rules dealing with prisoners at the end of the game. Even the simplest of these add a lot of complexity. However, many people still of course prefer to use territory scoring, and so there is an argument that this must be because they prefer finer granularity at the cost of rules complexity. In fact, there are simpler ways to achieve this granularity, but more importantly because any level of granularity can be achieved, to pick the granularity of territory scoring is arbitrary.
-- The Count
The so called finer granularity is claimed but unproven. In particular, the relative frequencies of even versus odd scores (more specifically: after perfect play) under area scoring are unknown. During the now maybe decade since I have set a prize for the solution, nobody has made a serious attempt of a formal solution yet. --RobertJasiek
Bill: Cher Robert, The Count says elsewhere that he intends granularity to refer to positions without seki. As for the minimum difference between results with area scoring, my construction of a button position with area miai value of 1/2 shows that it is 1, even after the Japanese dame have been filled. Your prize is safe. ;-)
The Count: I should have made the no-seki condition clear. There was just a "usually" hanging about.
Bill: This multiple button go is equivalent to environmental go, aka token go and coupon go. The buttons range from m/n down to 1/n. For equivalence between territory and area scoring, the top area button is worth 1 point more than the top territory button. In such a case the granularity of the scores is the same.
There is a curious fact about multiple area buttons that range from 1 - 1/n down to 1/n when no ko is being contested at the time it is correct to play them. If n is even, the multiple buttons are equilavent to a single button worth 1/2. If n is odd, as n approaches infinity the multiple buttons are equivalent to a single button of 1/2.
If no kos are being fought, the players simply take the buttons in turn. With each round (pair of plays) the first player makes a net gain of 1/n. When n is even, there are n/2 - 1 rounds, after which the first player takes the last button. The first player picks up (1/n)*(n/2) = 1/2 point, the same as if she had taken a single 1/2 point button. When n is odd, there are (n-1)/2 rounds, and the first player picks up (n-1)/2n points. As n approaches infinity, that value approaches 1/2.Flower: Amazing, by introducing multiple buttons one could truly increase granularity to any value :) (like Yay I won with 0.034 points ;-) If one would still strife to prevent jigo it would be necessary though to adjust the fractional tie breaker komi in cases of very fine granularity. Other than that you asked why people would perceive territory scoring granularity as 'natural' or 'normal'. I guess this might depend on the counting method one is accustomed with. If one counts in moku one might feel awkward if only every 'second' point of the scoring range is used. The same thing might happen if one would count in Zi and use territory scoring. (But then due to the tie breaker komi people might already be accustomed to fractional results. The most important thing Granularity infuences (to my mind) is the meaningful adjustement of komi in small steps.
The Count: "Missing every second point" is what initially made me think there must be something wrong with area scoring. The fact that you can get even scores made it worse. Now though, I think it's a silly reservation.
Bill: As the game approaches the end, both kinds of scoring become more coarse grained. By territory scoring, plays that gain less than 1/3 point are rare, and there is a gap of 1/3 point at the end. By area scoring, plays that gain less than a Japanese dame are ever rarer, and there is a gap of 1 point at the end. The button sits in that gap and makes who gets the last dame irrelevant in most cases.
The Count: Embrace the one point gaps like all others! By more granular, I assume you mean the miai values of plays get closer together. I would have guessed it was the other way round...
flower: I am somewhat confused. You seem to speak about granularity of the moves by miai counting during the alternation. I had the impression that 'TheCount' *insert horror movie theme here* spoke about granularity of the possible end results of the game. (of course I realise that Miai values are getting smaller during the alternation, but I assume that this does not infuence the end result granularity as in the end only prisoners and intersections are scored which would mandate a max. granularity of 1 (territory scoring).(not assuming button go or multiple button go here). Oh wait or are you just saying: 'The button increases granularity?' (in this case we would agree).
The Count: Yes, I think Bill accidentally abused the term granular, with "as the game approaches the end ... become more granular". Perhaps this branch of the discussion could go. Perhaps Bill could alter his original post if it really is misleading.
Bill: I meant granular as grainy, as opposed to smooth. To be clear, I have changed the word to coarse grained.
Bill: But also, the point of the button is to combine aspects of both territory and area scoring. It's aim is not to smooth out results, as in coupon go, where the coupons have a certain granularity, or difference between coupon scores.
The Count: When I said "By more granular, I assume you mean the miai values of plays get closer together", the emphasis was on granular. I was suggesting you might be using granular in the wrong context. Fortunately, you mistook me for putting the emphasis on more and closer, which made us aware of a totally different issue. Yes, let's all say coarse and fine grained not more and less granular. I've definitely used more granular in the opposite sense to the way you used it. But back to the issue I meant to bring up: I suggest you don't use granularity to refer to the closeness of miai plays. Shouldn't we reserve granularity as a technical term?
Bill: OK. Granularity = difference between miai values?
The Count: No! Can we reserve granularity to mean the smallest possible difference between scores in a particular scoring system (excepting seki). So, for area scoring, granularity = 2; for territory scoring, granularity = 1.
Flower: Given the wide and general meaning of Granularity I would not try to introduce a new specialised meaning of it. I think it would be better to always use granularity together with e.g. 'granularity of miai values' 'granularity of possible game results'. Also, as Bill said it seems to be wise to avoid the use of 'increased and decreased granularity' in favor of 'more/less fine grained and more/less coarse grained'.
Flower: Replying to myself.. oh well.. :) As such I would not claim it is necessary to change the page title though. One could add a small note that it is about Granularity of possible game results in the top. (that or add a small disambiguation part where it links to miai granularity should pages of it exist)
The Count: Yes, no more more/less grainy. Oh, is it bad to try to introduce granularity as a technical term? Hmm, perhaps. It's hard to write about it without an easy expression. I guess I'll just say granularity of the score and make sure the phrase is introduced on the relevant pages.
flower: I do not think that it is 'per se' bad to introduce granularity as a technical term. I just think it is inopportune as granularity already has a meaning in 'common language' as well as several subcultural niches ( here1, here2). A newcomer to Go is already puzzled by all the termini, if we can avoid we should not add more. That much said: If you were to note at the top of a page to which granularity your refer to exclusively on this page then you could use the convenient short 'granularity' for the remainder of the page.
flower: As far as I can see both your examples show differences in strategy or possible results of area and territory scoring but fail to address Ikeda's idea of free mending in case of even dame. I will try to post an example some day tomorrow :)
The Count: The first example has zero dame, which is of course an even number of dame. Black mends his group for free under area scoring.
flower: Aye I agree, it might be that either my tired mind played tricks on me or that the thought slipped my mind while sleeping. I rationalize that I was confused by the zero dame (which are of course a case even dame) as I always pictured it as being punished for making unecessary repairs when one truly should occupy a dame. (the curse of imprecise mnemonics :)
Flower: Ahh mayhap it just came back to me now. Your examples demonstrate indeed the 'free mending' problem. But they do not show how the game gets more interesting to remedy it. Or at least your version is unintuitive as one would have to be pretty braindead in order not to reinforce one's territory in both cases. There is no added skill and decision element involved (and this added element is what ikeda referred to as 'more interesting')
The Count: I don't quite understand what you mean by "they do not show how the game gets more interesting to remedy it." That is, if there is currently a problem with the example, I don't know what to do to change it.
The Count: A reinforcement is either necessary or it isn't. Maybe the rectangular six problem is obvious to you, but it doesn't really matter, as long as you pretend there is a question of whether the reinforcement is necessary or not. I'll make it clear on the page.
flower: Btw. it would be interesting to mention somewhere in your examples that Area Scoring will yield the same result as Territory scoring if one uses Ikeda's (Area III) Rule (of Scoring) (the one that followed the recognition of the free mending phenomenon).
Flower: Say would it not be good to move the in document discussion here, discuss the whole thing, find some consent and only then edit the main page? (most pages I see on SL where In-document discussion occurs are a mess sooner or later :)
willemien: I would like to add some definitions to this discussion What is granularity exactly? (where are we talking about?)
as starting point 2 incomplete definitions:
ruleset A has a higher granularity than ruleset B if
End score granularity
Relevant-1 and relevant-2 need to be worked out.
Relevant 1 so that damepoints (where for territory scoring passing doesn't influence the outcome while for area scoring it are losing moves are excluded and similar in relevant 2 positions.
RobertJasiek: Granularity, by Robert Maas, is one of the worst words for a term ever. Why? It is the same for all practically useful scoring systems! It is 1 for each of them: The smallest score difference is 1.
Instead a different term should be used as follows:
A ruleset's (1) even score fraction is the number of legal sequences (2) ending in an even score divided by the number of legal sequences.
(1) The rulesets should have suitable characterists: Well-defined, 19x19 board (odd), 0 komi (even), allowing finite sequences only, always creating a game end by means of a succession of a certain number of passes. Otherwise the theory would have to become broader.
(2) Starting from the game start's empty position and ending with a game ending succession of passes.
Now the problem is: Given an (3) area scoring ruleset A and a (3) territory scoring ruleset T, is it correct that A's even score fraction is (4) smaller than T's even score fraction?
(3) Arbitrary rulesets would be the generalized problem. But we can start with specific rulesets; this is already difficult enough.
(4) Optionally insert "much".
The yet more advanced problem is to allow only sequences that constitute perfect play.
willemien: Thanks for your comment.
The description that you give is hardly calculatable. the number of possible games on a 19x19 board is around 1.2604e+175 (the lowest number on Number of Possible Go Games and if even 99.99999% of them are odd the number of even outcomes is still more than fit in my little brain.
Another problem is that If we ever are able to allow only sequences that constitute perfect play under a certain ruleset. they will all lead to the same result. (otherwise some of them are just not perfect) and so the even score fraction will be 0 or 1 depending of of perfect play leads to an odd or even board score and with that the even score fraction becomes a meaningless statistic.
The idea behind the strategic granulariy is that it looks at the amount of perfect games. (the lower the number of perfect games the higher the granularity, perfect games by definition end in a draw)
RobertJasiek: Nobody has said it is easy:) In fact, I have offered a prize precisely because it is so difficult that I cannot solve the problem alone. However, it is not necessarily so that exact numbers of sequences will have to be calculated. Maybe an indirect proof is possible. In mathematics, such is often possible. Don't feel discouraged by the definition! :)