Yes, that is a feature of the stack of coupons.



Right.



Ideal environments are special cases of universal environments such that playing first in the environment (by itself) yields an eventual gain of t/2, where t is the temperature of the environment. E.g., a stack of coupons with values 2 and 1. The temperature is 2 and the eventual gain from playing first is 2/2 = 1. Note that it does not matter if the number of '2' coupons is odd or even, the eventual gain is still 1. So if you have some plays with the same temperature as an ideal environment, and consider them part of the environment, that does not change the eventual gain, even if they change the parity at the top value.



Like the stack of coupons. Gote of the form {Tx | -Tx} are desirable because they do not change the count of the position.

As I have indicated, such an environment approaches what I term ideal when D is small enough. But not having it ideal in my terms may introduce complications. For instance, D may matter.



IMO, there is a natural progression in complexity. First, there is the traditional (miai) value of a play, how much it gains on average. As a general rule the play that gains the most is best. There are exceptions, as we know. The most obvious kind, which go players have known about for a long, long time, is getting the last play at the end of the game and at certain points in the game. We can be more specific than that. Getting the last play before a significant drop in temperature can be better than simply playing the largest play. Now, CGT has developed this idea more, through the study of infinitesimals. That is one branch of study that has its own complexity. Next, there are other exceptions where the best play is neither to make the largest play nor to take the last play before a significant temperature drop. IMO, for most people the study of these positions should come after the study of the first two.

BTW, I recommend the use of ideal environments, as I use the term. Doing so makes easier, and also more generalizable (as long as we remember that the generalizations are approximations ). More on this in a later post.

_________________
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Illustration of comparing the choice of Black to play between A = {2g | 0} and B = {r || 0 | -2s} in an ideal environment

First, let’s look at traditional go evaluation. A move in A gains g. If B is a (local) reverse sente a move in it gains r; if it is gote, a move gains (r + s)/2. To choose the larger play we compare r and g in the first case or (r+s)/2 and g in the second case.

Now let us add an ideal environment with temperature, t. g > t and s > t. Why? Because otherwise {2g | 0} and {0 | -2s} disappear from view in the ideal environment, except for their mean values, g and -s. Having an ideal environment means that we can ignore anything with a temperature less than or equal to t, except for their mean values. That simplifies things. Now, it is possible that B has a lower temperature than t, but we do not know that.

OK, let’s compare lines of play.

1} Black plays the gote, A, and White replies in B, then Black responds in B to 0, then White plays first in the environment. Result: 2g - t/2

2) Black plays in B and White replies in A, then Black plays first in the environment. Result: r + t/2

So our comparison is this: r + t/2 >?< 2g - t/2

(Edit: Note that if r <= t then r + t/2 < 2g - t/2, since g > t. If we had already known that r <= t we could have saved ourselves some trouble. )

Suppose that B is a local reverse sente. Instead of comparing r >?< g we compare r >?< g + (g-t)

(g-t) is the temperature drop between A and the environment. Note by playing A first Black gets the last play before that temperature drop. We may regard it as the bonus that Black gets by getting that last play.

Now suppose that B is a local gote. Instead of comparing (r+s)/2 >?< g we compare (r+s)/2 - (s-t)/2 >?< g

(s-t)/2 is half the temperature drop between the White follower of B and the environment. We may think of it as the penalty that Black gives up by letting White get the last play when Black plays in B first.

There are other ways to think about this, OC, but we can see the value of a nodding acquaintance with infinitesimals. If A and B were infinitesimals Black should play in A first and get the last play before a play in the environment. We cannot say that in general, but there is a value to getting that last play besides the question of how much each play gains, on average.

——

When I was first studying these things, back in the 70s, without any knowledge of infinitesimals and only kind of a notion of ideal environments, I noted that the comparison in this kind of situation did not depend upon whether B was a (local) reverse sente or a gote. I called r the reverse sente value of B, since we could treat B as though it were a reverse sente, anyway. We could also treat B as though it were this gote: {r || 0 | -2t}.

Edit: This example illustrates why, in general, we should be somewhat wary of playing reverse sente. Doing so could be right, but in many ways it could be wrong. That's a lesson that we could get from studying infinitesimals, BTW.

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

Interesting study on value of playing the last move.

Studying infinitesimals lets one conclude that something might be right or wrong? Uhm, good joke.

When you studied such things in the 70s, did you do so in complete isolation or was it motivated at university and by whom?

You speak of the gain as 'traditional go evaluation' but IMO that is euphemism because tradional endgame theory uses deire counting. Rather you might say that the concept of gain in miai value terms was already there but still hidden from awareness.

The point is that getting the last move before the plays in the environment has a positive value.



Just my own study.



By the time I learned go the concept of miai values was well established, textbook material.

_________________
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At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

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Established textbook material: ah, I recall you have had access to Japanese books. There was absolutely nothing like that in English books before 1994 (when Mathematical Go Endgames appeared). Instead, Ogawa / Davies put us on the wrong track. Even fairly recently, most professionals still seemed to have known / preferred deire counting: it is what they teach and some were surprised when I mentioned miai counting.

I know it's foolish of me to tread in the world of numbers, but I think you are on the wrong track, Robert, as least as far as understanding Japanese counting goes. Bill seems spot on to me, with the proviso that I can't claim to follow him 100% as I suspect he's using some terms for which I only know the Japanese. The only quibble I'd have with Bill is that I don't think "textbook" is the right word to use about miai counting. Miai counting was known at least as far back as Meiji times, and almost certainly well before that. But what he is referring to, I suspect, is an advanced exposition of it initiated around 1955 by Sakauchi Junei, who was a strong amateur (good enough when over 80 to play Go Seigen on 3 stones). These were articles in magazines rather than books, and as far as I know Sakauchi's contributions did not appear in book form until much, much later (and have caused major confusion). In any case, we should really make a distinction between "miai counting" and "Sakauchi miai counting" (and maybe also Berlekamp miai counting").

Here are the points where I think you need to either make corrections or re-think.

1. There is no such word as deire. It is deiri.

2. Deiri counting and miai counting count different things but also are used just as launching pads for other ways of counting. They are not ends in themselves. They are actually accountancy terms and it would surprise no-one in Japan to hear them spoken of together. And just as in business, each method simply provides a snapshot of what is going on. Actual decisions are based on other factors as well. Each method may have a theoretical basis, but for CEOs and pros they are just tools to help make practical choices.

3. As to which tools are used most, deiri is indeed used most because it measures the size of a move. The earliest exposition is Genan Inseki. However, the way that measurement is used varies among pros. Deiri means surplus-and-deficit. A typical boundary-play calculation would be based on making hypothetical gote plays for each side in turn and comparing the resulting respective profit and loss in territories. So if we end up with Black making a territory of 0 points or 4 points (depending on whether he plays first or second) and White gets 0 points or 2 points, the result of a deiri calculation would typically be expressed as e.g. 出入目数は両後手６もく - "the deiri count is 6 points in double gote." Note that the deiri count is always distinguished by adding the extra info such as double gote, reverse sente, etc. Moves in various parts of the board can be compared on the basis of that measure, but when it comes to using it to count overall, this measure is not used directly, although it is my impression that westerners (not Bill, of course) generally try to do it that way. Instead one of two sub-results of the calculation is used by pros. The commonest way seems to be to take the average count (平均目数) and this would be expressed as a territorial count, in this case as e.g. 平均目数は白地１目 - the average territory is 1 point of territory for White. But there is another "net value" method of using the deiri count and that would be expressed in this case as 正価目数は３目 - the net value count is 3 points.

Incidentally, even when a count is made, by whichever method, it is often also qualified according to the stage of the game. Bare counts as above would apply only at the late boundary-play stage. Before that, plays will typically be qualified either in general terms (... but Black is thicker) or the move score itself may be rounded up or down a little (adding a point or two to take account of aji).

4. The purpose of miai counting nowadays (post Sakauchi) is to provide a convenient way to count territory rather than the size of a move. But originally it just meant something simpler (and this simpler meaning is still in use, which complicates things a little). The simple use is to take the accountancy meaning of offsetting, so that you count move sizes (based on deiri counts) and any that are the same are simply cancelled one against the other - or, essentially, ignored. Sakauchi brought tiny fractions into the mix, which meant that offsetting was less likely to occur, and so in a way undermined the very meaning of miai. The modern miai count also depends on (or "gives" if you prefer) the "net count" of deiri, so the two methods have become intertwined.

5. On top of all that there is absolute counting as used by O Meien and, he implies, most pros in Korea and China. As I understand it, he is not claiming that Japanese players don't know how to play the endgame, though he may be implying that they use "average value" more often than "net value". What he is mainly saying, I think, is that the Japanese method has become a mess of terminology (see above - but there's more, such as is sente countable?) and has rendered what is really a simple concept into something approaching quantum physics. He specifically says that his book is an attempt to sort out "what has been made confusing about endgame plays hitherto." To repeat, he is rejecting the terminology rather than the traditional method because the terminology has turned the method into methods (plural) and turned the field into a pig's breakfast.

6. In his book O Meien tries to avoid either deiri or miai. But he does follow the traditional method (singular), though adds to that his own very useful and simple empirical formula (which I have described elsewhere here) to deal with the fuzzy counts. In fact, he avoids miai altogether except in its fuseki/middle game options meaning, and while deiri is mentioned it is mainly to explain why he prefers to avoid it, in favour of the concept "value of one move." To quote him again directly: "The reason why many people fall under a misapprehension is that instead of the value of one move they are using, as is, the figure produced by a deiri calculation." Bill may wish to correct me, but I think "value of one move" is basically "net value."

My recommendation, which I suspect Bill might back whether or not I've misunderstood, is to forget the Japanese deiri/miai mish-mash and follow O Meien.

John, thank you for your, as always, very helpful background on history and Eastern usage (and the correction of spelling).

I have come to the same conclusion of avoiding the deiri / miai confusion and instead focussing on value per move, but also value (count) of positions or local positions. I only know about O Meien's theory from descriptions by others and think he avoids assigning values to local sente. Presumably, I am more a follower of Bill because I have found values of moves in a local sente sequence to be very useful: they express local move values even if the opponent plays elsewhere, and they are consistent with the move value of a local follow-up.

Also Western endgame theories have become a mess:)

Endgame theory can be restricted to a simple core, which involves approximations, or unfold as quantum physics if all those infinitesimals, mathematical proofs, thermographs etc. are brought in to seek the last fraction of a point. Both the simple and the quantum side have their justifications. Kyus may stick to the simple concepts but ambitious dans can profit much from some quantum physics to greatly accelerate endgame evaluation, have tools for luring the opponent into mistakes or get the last point.

I am not sure I understand what you refer to as net value count, but if it is just part of the mess to be ignored, it might be immaterial.

Wrong track? That's a bit strong. Deiri values are OK for comparing plays or lines of play -- as long as you make the adjustments and don't use double sente values. For instance, the comparison above could be written with deiri values as

R + T/2 >?< G

However, I noted back in the 70s that people who I thought should have known better were trying to use deiri values to represent gains and losses. You need miai values for that.



Well, deiri values are probably what they were taught, and they may have been surprised that you knew about miai values.

For a long time the Nihon Kiin's Small Yose Dictionary had a section of plays grouped by size. They used deiri values and mixed sente and gote together without multiplying the sente values by two. That was definitely misleading. I guess somebody pointed out that that was wrong, since the latest edition I have seen leaves that section out entirely. Also, AFAIK, texts still talk about double sente as though it were local. You can't do that with miai values, OC.

_________________
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— Winona Adkins

Visualize whirled peas.

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Sente can be analysed locally but must be interpreted in a global context. Same for double sente.

The problem with having been taught deiri counting is to have been put into a restrictive framework with limited scope of application. Therefore my strong negative comment.

Did I say not to use double sente values in comparisons? You can if you know what you are doing.

Consider the play {2s | 0 || -r | -r - 2p} in an ideal environment of temperature, t, with r > 0, p, s > t.

Should Black play in the environment or make this play?

Let's do the comparison.

1) Black makes this play, White replies, and then Black plays in the environment. Result: t/2

2) Black plays in the environment, White makes this play, Black replies, and then White plays in the environment. Result: t - r - t/2 = t/2 - r

Comparison: t/2 >?< t/2 - r

or r >?< 0

OC, Black should make the play instead of playing in the environment. r is the double sente value of the play.

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

Oh, no, John. I remember when we met at the International Conference on Baduk in '06. When I gave my talk on ko evaluation you made some very astute observations.



Actually, I learned about miai counting from Takagawa's Igo Reader and Sakata's book on tsumego and yose in the Killer of Go series. Takagawa seemed quite clear to me, and maybe he was influenced by Sakauchi. Sakata's few pages on miai counting and tedomari seemed something of a muddle to me.



I can see how that might happen. I went on a campaign in the '90s to introduce miai counting to the West, at least in rec.games.go and later Sensei's Library. I meant to shed some light and I am afraid I caused confusion. Not that there wasn't already confusion, in both the East and the West. Now I just talk about how much a play gains or loses on average.



I rather suspect not. They all should come down to average gains and losses.



I am sure that Robert and I and quite a few others would disagree. IMHO, deiri counting is used because it is usually easier to calculate, and it may be used to compare two plays, as long as you make the adjustments, like doubling the sente value and taking 2/3 of the simple ko value.



I bet it does. If they use it for something other than comparing two plays, they may well be foundering.



Yes, that information may be necessary to make comparisons with other plays.



What, following Berlekamp, I have taken to calling the count (or territorial count) of a region.



This sounds like a conversion to the miai value, dividing by the net move difference between the counts used for the deiri value.



Ah!



O Meien's absolute counting is equivalent to what I learned as miai counting, or, if you will, Berlekamp-Spight-Mueller-Takagawa-Nakamura-Jasiek-et-al. miai counting.



No shite. (Pardon my French.)



Or, as our new President might say, "Follow me and make the endgame great again. We'll win bigly. I can't tell you how much we'll win!"

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

Well, you can consider taking a sente as a free move, but it also gains no points, so it's a wash.

Edit:

Or maybe you are saying something different. Suppose, for example, that you have a choice among 2 simple gote and one simple reverse sente. The larger gote gives the player who takes it 4 points of territory, and the smaller gote gives the player who takes it 2 points of territory. The reverse sente gives you 3 points of territory, while if your opponent plays it with sente, she gets 0 points.

You are saying that if you take the reverse sente, then your opponent takes the larger gote and then you take the smaller one, but if you take the larger gote, your opponent takes the sente and then takes the smaller gote, getting it for free because you cannot now take the reverse sente. Is that right?

Well, let's compare results.

First line of play: You take the reverse sente, then your opponent takes the larger gote and you take the smaller one. You get 3 points for the reverse sente and 2 points for the smaller gote, while your opponent gets 4 points for the larger gote. Net result for you: 3 + 2 - 4 = 1 point.

Second line of play: You take the larger gote, then your opponent plays the sente and you reply, then your opponent takes the smaller gote. You get 4 points for the larger gote, while you opponent gets 0 points for the sente and 2 points for the smaller gote. Net result for you: 4 - 0 - 2 = 2 points.

The second line of play is better for you.

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

Sneaky spambot. It copied topazg's post (2nd in the thread, on page 1) and thereby seemed to be contributing something relevant!

Working towards:

Let there be the environment T >= T1 >=...> 0 (or T = 0 if the environment is empty).
Let the starting player have, if any, the local sentes M0|F0, M1|F1,... (Mi are the sente move values, Fi are the gote follow-up move values, Mi < Fi is the local sente condition) with M0 >= M1 >=... (Note: this says nothing about the relative order of F0, F1,...)
Let the opponent have, if any, the local sentes N0|G0, N1|G1,... with N0 >= N1 >=...

These special cases with low temperature F0, F1,..., G0, G1,... >= T, I have proven:
- Only the starting player has local sentes: he can, e.g., play all his local sentes in sente in the order M0|F0, M1|F1,...
- Only the opponent has local sentes: the starting player's start is determined by the comparison N0 to twice the alternating sum T - T1 +...

However, apart from esoteric other special cases (or M0 = M1 =... = N0 = N1 =... = 1, see Mathematical Go Endgames), I am having difficulties to prove anything for more general cases including high (non-low) temperature (or further hot move values in the environment) and / or local sentes of both players (mutual reduction warning!). At first, I went for an explicit condition for all values - a dream. Then I became more modest and wanted simplifying reading guidelines, such as:

- For the opponent continuing with a local sente of his, he chooses in decreasing order N0 >= N1 >=...
- For the starting player continuing with a local sente of his, he chooses in decreasing order M0 >= M1 >=...
- For the starting player continuing with a local sente of his, he chooses in decreasing order of F0, F1,... ordered decreasingly.
- For the starting player continuing with a local sente of his, he chooses in decreasing order either among M0 >= M1 >=... or among F0, F1,...
- For the starting player playing reverse sente, he chooses in decreasing order N0 >= N1 >=...

But... can any of this proven in general, and how?

A similar question to the one you pose. It is little wonder that you are having difficulties with general cases.



An admirable goal. But even that is not so easy. For instance, on a board with no kos, now or later, given two independent positions with Black (Left) to play, a gote, {2g | 0} and a reverse sente, {r || 0 | -2s}, where g >= 0 and s > r >= 0, without knowing the rest of the board we can say little about which Black should play first. By size we can compare r and g, but we can only say for certain that Black should prefer the reverse sente if r > 2g, and we can only say for certain that Black should prefer the gote if 2g > r + 2s.

Or let's compare two reverse sente, A = {a || 0 | -2b} and R = {r || 0 | -2s}. By size we can compare a and r. Suppose that r > a. We still cannot say for certain (with no kos, now or later), that Black should prefer R to A unless r + 2s >= a + 2b.

Now for the rules for comparing two sente. Let A = {2b | 0 || -a} and S = {2s | 0 || -r} be two sente which are not equal. Black should prefer A to S, for certain,

1) if a > r and 2b >= 2s or
2) if a = r and 2b > 2s or
3) if a < r but 2b > r + 2s .

_________________
The Adkins Principle:
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— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

In an environment with move values Tpq, suppose only the starting player has local sentes with follow-up move values Fij occurring in blocks so that T01 >= T02 >=... > F01 >= F02 >=... >= T11 >= T12 >=... > F11 >= F12 >=... >=...

Can we at least prove that playing any Fkl cannot be better than playing some still available Fij with k > i? That is, can the local sentes be played in order of their blocks? Or which counter-example can we construct?

We can also pose similar questions for only the opponent's local sentes.

IIUC, I think that this is a counterexample with F01 = 5, T01 = 4.5, F02 = 4.

T = {9 | 0}
S01 = {10 | 0 || -1}
S02 = {8 | 0 || -3}
Black to play

1) Black plays S02, without sente. Then
1a) White takes T, Black plays S01 with sente and then plays the threat of S02. Result: +8
1b) White takes S01 in reverse sente, Black plays T and then White plays the threat of S02. Result: 9 - 1 = 8.

2) Black plays S01, White takes S02 in reverse sente, then Black takes the threat of S01 and White takes T. Result: 10 - 3 = 7.

3) Black takes T, White takes S02 in reverse sente, then Black takes S01 in sente. Result: 9 - 3 = 6.

Black should choose option 1) and play S02.

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

Indices simplified, your example in game tree annotation is

T = {9|0}, S1 = {10|0||-1}, S2 = {8|0||-3}

or in move value annotation

T = 4.5, S1 = M1|F1 = 1|5, S2 = M2|F2 = 3|4

with T and follow-up move values

F1 = 5, T = 4.5, F2 = 4

and therefore two blocks of local sentes (each block consisting of one local sente) separated by the high temperature.


I complete the missing sequences and calculate resulting counts (from stated local endgames in which a move occurs) and net profits (stating the profit values of each move), of which either can be used. Sente sequences are ignored in the number value calculations. Rx stands for reverse sente in the local endgame Sx. * means relevant mistake.


1a)

S2 T S1 S1 S2 = 8 - 0 = 8.

F2 - T + F1 - F1 + F2 = 4 - 4.5 + 4 = 3.5.


1b)

S2 R1 T S2 = 0 - 1 + 9 = 8.

F2 - M1 + T - F2 = 4 - 1 + 4.5 - 4 = 3.5.

On move 2, White chooses (1a) or (1b).


2a)

S1* R2 S1 T = 10 - 3 - 0 = 7.

F1* - M2 + F1 - T = 5 - 3 + 5 - 4.5 = 2.5.


2b)

S1 S1* S2 T = 8 - 0 = 8.

F1 - F1* + F2 - T + F2 = 4 - 4.5 + 4 = 3.5.


2c)

S1 S1 T R2 = 9 - 3 = 6.

F1 - F1 + T* - M2 = 4.5 - 3 = 1.5.

On move 3, Black chooses (2b) and discards (2c).


2d)

S1 T S1 R2 = 10 - 0 - 3 = 7.

F1 - T + F1* - M2 = 5 - 4.5 + 5 - 3 = 2.5.


2e)

S1 T S2 S1 S2 = 0 - 0 + 8 = 8.

F1 - T* + F2 - F1 + F2 = 5 - 4.5 + 4 - 5 + 4 = 3.5.

On move 3, Black chooses (2e) and discards (2d).

On move 2, White chooses (2a) and discards (2b) + (2e).


3a)

T R2 S1 S1 = 9 - 3 = 6.

T* - M2 + F1 - F1 = 4.5 - 3 = 1.5.


3b)

T R1 S2 S2 = 9 - 1 = 8.

T - M1* + F2 - F2 = 4.5 - 1 = 3.5.

On move 2, White chooses (3a) and discards (3b).

On move 1, Black chooses (1a) + (1b) and discards (2a) + (3a).


Black's correct start is in the local sente S2 taking the follow-up move value F2 in the second block of local sentes.

Many thanks for constructing this counter-example to the conjecture that blocks of local sentes would be played in order of follow-up move values!

Let me try to understand this.

The possible, relevant profit value sequences and their net profits are:

1) b - b + s - s = 0

2) b - r + b = 2b - r

3) s - s + b - b = 0

4) s - a + s = 2s - a

These comparisons occur for choices between the sequences:

(1) ? (2) <=> 0 ? 2b - r.

(3) ? (4) <=> 0 ? 2s - a.

(2) ? (4) <=> 2b - r ? 2s - a <=> a - r ? 2s - 2b.

The first two comparisons are also summarised in the third comparison. There is also the implicit interpretation of the count differences in A between 2b and -a and in S between 2s and -r.

Now, for starting in A, you write the conditions

1) if a > r and 2b >= 2s or
2) if a = r and 2b > 2s or
3) if a < r but 2b > r + 2s.

Let me try to explain this in terms of the comparison (2) ? (4) <=> a - r ? 2s - 2b.

Case a > r:

a - r > 0 so with 2b >= 2s we have 2s - 2b <= 0, (2) > (4) <=> a - r > 2s - 2b fulfilled and therefore Black can start in (2), which means starting in A.

Case a = r:

a - r = 0 so with 2b > 2s (equality excluded by your assumption of different A and S) we have 2s - 2b < 0, (2) > (4) <=> a - r > 2s - 2b fulfilled and therefore Black can start in (2), which means starting in A.

Case a < r:

If a is infinitesimal positive, a - r is, at the mimimum of a - r, slightly less than r smaller than 0. Therefore, adding r compensates the excess of how much a - r is smaller than 0. For a - r = 0, we have 2b > 2s so, with a - r < 0, also 2b > r + 2s is fulfilled. As a consequence and since a > 0, we have (2) > (4) <=> a - r > 2s - 2b <=> a + 2b > r + 2s fulfilled and therefore Black can start in (2), which means starting in A.

This confirms your conditions.

***

How to interpret all this?

Can there be further cases when starting in A is correct?

I think that starting in A being correct does not always exclude the alternative of starting in S also being correct, right?

I mentioned for an environment with low temperature that the sente player could play all local sentes in sente in order of decreasing sente move values. Do your conditions contradict this or are they just another study aspect?

That may not be so when A and S are not the only plays on the board. The above conditions hold when there are other plays on the board — sans ko, now or later.

If those were the only four sequences, then the comparison would be

a+2b >?< r+2s

The above conditions are more stringent, as there are cases where the play in A is correct but those conditions are not met. For instance, when the only two plays on the board are A and S and a+2b > r+2s but a > r and 2s > 2b.

Let B = {2b | 0} and A = {B | -a}, b > a > 0.
Let T = {2s | 0} and S = {T | -r}, s > r > 0.
Let G = A + S.
Let E = the environment of other plays, with no ko, now or later.

When Black (Left) plays first in G, the result is B + S + E when Black plays from A to B, or T + A + E when Black plays from S to T. Which, if either, does Black prefer? When we compare them, E drops out (because of no ko), so we only have to compare S + B and A + T. We can do that in the game, D = S - T - A + B, or {2s | 0 || -r} + {0 | -2s} + {a || 0 | -2b} + {2b | 0}. If Black wins or ties even when White plays first, for Black to play A is at least as good as for Black to play S.



Sure, depending on the rest of the board.



If the above conditions are met and there is no ko in the game tree, Black can play in A instead of S with no reading.



In all three conditions, 2b >= 2s, so no, they do not contradict playing the sente in descending order of threats. Note, however, that when a < r but r + 2s >= 2b > 2s, the conditions are not met, and the play in S may be correct, depending on the rest of the board.

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