One of my Go activities while L19 was down was to work through how to use area counting rather than territory counting to calculate endgame values. I had always guessed that this might be easier than using territory counting because there are fewer things to keep track of, but had never actually worked it out. My findings are presented in the following blog posts. I have pretty much switched over to this method of calculating endgame values now; the benefits and drawbacks are presented in the posts, and for me the benefits are generally well worth it.
Endgame values with area counting
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RobertJasiek
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Re: Endgame values with area counting
Here are comments on part 1.
You study area deiri counting. Usually, one would use area miai counting.
For area miai counting, I (and others before me) have stated the principle "Usually, an area move value is 1 larger than a territory move value."
Unsurprisingly, for area deiri counting, you need "is 2 larger", wherefore you subtract by 2 to calibrate your area move values for area deiri counting to those of territory deiri counting. If you did not want to calibrate your area move values to territory deiri counting, you could omit subtracting 2 and simply use area move values as they are: calibrated to area deiri counting.
You argue that, for the endgame, area counting can be more difficult than territory counting. However, if you calibrated to area deiri counting instead of territory deiri counting, you might instead argue that territory counting would appear more difficult than area counting.
You proclaim a formula of the type V_A = term - 2. While I understand why you are doing this, in maths we need justification. From what is the -2 derived and why is it exactly -2? You use the -2 as if it would be an axiomatic number out of nowhere. Rather, you might have started with a still unknown parameter, say D, transformed algebraically and only then discovered that D = -2. Then it would not be "magic" but maths! You put your discovery of the -2 at the start of your algebraic study. Instead, it ought to have emerged at the end! Alternatively, declare the presupposition D = -2, state the gote conjecture V_A + D = V_T, do the algebra, find truth and thereby confirm that D has been chosen well.
"Black will own either 2 1/2 or 0 points; 2 * 2 1/2 - 2 = 3." was a bit rough to read because the reader has to understand that you omit the calculation step 2 1/2 - 0 = 2 1/2 before inserting that in the 2 * 2 1/2 term.
I disapprove your use of the word "gain". Gain is the term for a move's change to the counts, comparing the count before the move to the count after the move. You use "gain" carelessly with a different meaning, which is not even directly related to the terms you write. You want to express with too few words what your terms express.
EDIT: teiri -> miai
You study area deiri counting. Usually, one would use area miai counting.
For area miai counting, I (and others before me) have stated the principle "Usually, an area move value is 1 larger than a territory move value."
Unsurprisingly, for area deiri counting, you need "is 2 larger", wherefore you subtract by 2 to calibrate your area move values for area deiri counting to those of territory deiri counting. If you did not want to calibrate your area move values to territory deiri counting, you could omit subtracting 2 and simply use area move values as they are: calibrated to area deiri counting.
You argue that, for the endgame, area counting can be more difficult than territory counting. However, if you calibrated to area deiri counting instead of territory deiri counting, you might instead argue that territory counting would appear more difficult than area counting.
You proclaim a formula of the type V_A = term - 2. While I understand why you are doing this, in maths we need justification. From what is the -2 derived and why is it exactly -2? You use the -2 as if it would be an axiomatic number out of nowhere. Rather, you might have started with a still unknown parameter, say D, transformed algebraically and only then discovered that D = -2. Then it would not be "magic" but maths! You put your discovery of the -2 at the start of your algebraic study. Instead, it ought to have emerged at the end! Alternatively, declare the presupposition D = -2, state the gote conjecture V_A + D = V_T, do the algebra, find truth and thereby confirm that D has been chosen well.
"Black will own either 2 1/2 or 0 points; 2 * 2 1/2 - 2 = 3." was a bit rough to read because the reader has to understand that you omit the calculation step 2 1/2 - 0 = 2 1/2 before inserting that in the 2 * 2 1/2 term.
I disapprove your use of the word "gain". Gain is the term for a move's change to the counts, comparing the count before the move to the count after the move. You use "gain" carelessly with a different meaning, which is not even directly related to the terms you write. You want to express with too few words what your terms express.
EDIT: teiri -> miai
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kvasir
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Re: Endgame values with area counting
I have a different perspective to what was expressed above. I think the articles would have been much improved if the ratio of arithmetic and algebra to examples and good arguments was much lower. There could be much more of the other things.
I thought that the articles largely glossed over how this theory might only be useful in certain situations (when it does simplify the calculation). There were some examples of how it can simplify some calculations and complicate others but I tried to think of more examples were it complicates things. I'm undecided if the following example show how the theory does complicate things or if it just show how it can work.
I'd normally say that this endgame is 2 points sente for black.
If we look at this closer we can see that control changes for the labeled intersections. However, the numeric values needed to make the theory work are not only different for the intersections marked a and b, they have a different sign. Control of the intersections marked a is worth 1 unit each and control of b is worth -0.5.
That is
2 (a + a + b) - 1 = 2
or
2 * 1.5 - 1 = 2
I imagine that all kinds of other signed and unsigned fractional values are needed in more complex examples but I find it moderately paradoxical that a theory that attempts to simplify calculations for territorial scoring has signed dame. Maybe that was overly dramatic but I think I could easily miss that b subtracts something here.
Is this an example of how things get overly complicated or of how this just works?
I thought that the articles largely glossed over how this theory might only be useful in certain situations (when it does simplify the calculation). There were some examples of how it can simplify some calculations and complicate others but I tried to think of more examples were it complicates things. I'm undecided if the following example show how the theory does complicate things or if it just show how it can work.
I'd normally say that this endgame is 2 points sente for black.
If we look at this closer we can see that control changes for the labeled intersections. However, the numeric values needed to make the theory work are not only different for the intersections marked a and b, they have a different sign. Control of the intersections marked a is worth 1 unit each and control of b is worth -0.5.
That is
2 (a + a + b) - 1 = 2
or
2 * 1.5 - 1 = 2
I imagine that all kinds of other signed and unsigned fractional values are needed in more complex examples but I find it moderately paradoxical that a theory that attempts to simplify calculations for territorial scoring has signed dame. Maybe that was overly dramatic but I think I could easily miss that b subtracts something here.
Is this an example of how things get overly complicated or of how this just works?
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dfan
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Re: Endgame values with area counting
Thanks for the comments and your prior work on the subject, Robert. I did read them carefully but I'm not going to respond to them point by point because they are all addressed to my own satisfaction (obviously not yours!) somewhere in the series, and I've learned from experience what happens when people get into technical back-and-forths with you here. 
On to kvasir! I love worked examples too and I wish that most books had more of them. I was actually worried that I had too many examples considering that it was just a bunch of blog posts, so it's nice to see that I also had too few. I tried to be clear that the area method is not always simpler (e.g., the "Is this always more convenient?" and "Dame" sections in Part 1) but of course there could be even more examples of when you might not want to use this method.
Your example is a good one, and I'm weak enough that it took me a minute to even see what the point is (for those weaker or lazier than me, after
, Black is threatening to play to the left of 
). The way I count this is that there are 3 points being contested, and Black's share will be either 2 or ½, a difference of 1½.
This is in fact the general process I use presently ("Of these X points, Black will own A if they go first and B if White goes first, for a difference of A - B"), and I think you can see me subconsciously drifting towards it over the course of writing the series (for example, the gote example in Part 3). In fact I'm already using non-zero B in the Dame section in Part 1 without really calling it out. I agree that it would be a good idea to be more explicit about that, since the first examples implicitly have a B of 0 and make the math seem even simpler than it really is.
I still do find that in the general case this method requires me to juggle fewer numbers — two values of "number of contested points controlled by Black", rather than two values each of Black territory, White territory, Black prisoners, and White prisoners, not to mention that there is a buffer of Black and White stones between the two territories. In any particular situation, many of the latter eight values may be trivially zero (or very easy to calculate), or you may be able easily visually assess derived values such as "difference in Black territory between the two variations" without computing the underlying values, which can make the territory method simpler in those cases. (Another case is if there are one-sided dame, which can totally be ignored with territory but not with area.) Your example is a good one, where one can very quickly evaluate "difference in Black territory" and "difference in White territory", and the dame is relevant to the area method but not the territory method.
On to kvasir! I love worked examples too and I wish that most books had more of them. I was actually worried that I had too many examples considering that it was just a bunch of blog posts, so it's nice to see that I also had too few. I tried to be clear that the area method is not always simpler (e.g., the "Is this always more convenient?" and "Dame" sections in Part 1) but of course there could be even more examples of when you might not want to use this method.
Your example is a good one, and I'm weak enough that it took me a minute to even see what the point is (for those weaker or lazier than me, after
, Black is threatening to play to the left of ). The way I count this is that there are 3 points being contested, and Black's share will be either 2 or ½, a difference of 1½.This is in fact the general process I use presently ("Of these X points, Black will own A if they go first and B if White goes first, for a difference of A - B"), and I think you can see me subconsciously drifting towards it over the course of writing the series (for example, the gote example in Part 3). In fact I'm already using non-zero B in the Dame section in Part 1 without really calling it out. I agree that it would be a good idea to be more explicit about that, since the first examples implicitly have a B of 0 and make the math seem even simpler than it really is.
I still do find that in the general case this method requires me to juggle fewer numbers — two values of "number of contested points controlled by Black", rather than two values each of Black territory, White territory, Black prisoners, and White prisoners, not to mention that there is a buffer of Black and White stones between the two territories. In any particular situation, many of the latter eight values may be trivially zero (or very easy to calculate), or you may be able easily visually assess derived values such as "difference in Black territory between the two variations" without computing the underlying values, which can make the territory method simpler in those cases. (Another case is if there are one-sided dame, which can totally be ignored with territory but not with area.) Your example is a good one, where one can very quickly evaluate "difference in Black territory" and "difference in White territory", and the dame is relevant to the area method but not the territory method.