Granularity
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.[1] 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.
- See the discussion.
[1]
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.