This thread discusses the paper Principles from Basic Endgame Trees of Daniel Hu.

https://dhu163go.files.wordpress.com/20 ... ciples.pdf

Currently, I refer to its version on 2021-04-17. So far, I have only looked briefly into the paper and it may be a long time before I find time to read and understand it to its end. Nevertheless, I start discussion now because I have some questions whose answers might ease my further reading. Citation from page 1 with sentence numbers added by me:

"[1] we calculate the miai value of a node as the average of the miai value of its two children. [2] The miai value of a leaf node is simply the score there. [3] This recursively defines miai value for every node. [4] We continue by defining the miai value of a move as the difference between the miai values of the final node and starting node. [5] Note that miai values of sibling moves are equal. [...] [6] we will assume that in all binary trees, follow up moves have a lower miai value than preceding MOVES"

[1][2] define "miai value OF A NODE" as the average of the children's miai values and for a leaf as the score. This suggests that 'miai value OF A NODE' shall mean 'count'.

The recursion in [3] only makes sense if we already presume [6]. Since [6] appears later in the text, I wonder whether this is indeed the assumption for [3].

Is [6] a general presupposition for the entire paper? If yes, then why does the paper claim to introduce miai counting when it would only consider game trees with decreasing(-or-constant) miai values OF MOVES?

[4] defines "miai value OF A MOVE" as the difference between the miai values of the final NODE and starting NODE. I suppose "starting node" means the position left by the move and "final position" means the position created by the move. As a consequence of the definition of a miai value OF A NODE and its relying on [6], this definition also appears to presume [6].

The definition of miai value OF A MOVE is not the usual definition of move value of miai counting and is not the definition of gain. For Black's move, the definition of miai value OF A MOVE equals the definition of Black's gain. For White's move, the definition of miai value OF A MOVE equals the definition of the negation of White's gain, that is, a negative value. Is the definition as intended or has it been careless?

Supposing the miai value OF A MOVE was intended to be defined as gain, then the author's claim of it being part of miai counting again seems to presume [6]. Since [6] appears later in the text, I wonder whether this is indeed the assumption for [4].

If this is the intention indeed, then move value equals Black's gain and equals White's gain (*) so that miai value OF A MOVE can be defined as gain AS LONG AS WE PRESUME [6]. As soon as we drop [6], miai value OF A MOVE should be defined as move value.

[5] states "miai values of sibling MOVES are equal". I assume this refers to a node's moves to the black child and white child. Then, [5] expresses (*).

EDIT

Thanks.

It's been a few years since I checked Sensei's library for miai counting, but I learnt it there (which probably means indirectly from Bill). [6] comes from the trick of contracting sente (threat-response) edges, which I probably learnt here: https://senseis.xmp.net/?MiaiCountingWithTrees



I thought that [6] was the standard assumption. Without [6] it would still be interesting, and the recursion in [3] still makes sense, though you might not want to call it 'miai value' anymore. If a follow up move in a tree has a greater value than the preceding move, we normally assume it is sente. This doesn't always make sense globally, so I place in the pile of "issues with sente/gote in miai counting" that I mentioned before. [6] is a necessary assumption for all my calculations to work later, so yes it is assumed throughout



My focus was on the calculation. This just a rapid prelim to explain the setup.

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hmm. you understand what I mean correctly. It was convenient for me to take all variables as positive, though obviously not necessary. I guess I am saying White gains x or Black gains x rather than Black gains -x when White plays. I guess I started learning from this exchange with Bill https://www.lifein19x19.com/viewtopic.p ... 10#p219310 but perhaps I misunderstood the zero point.

I'm not so used to using the terms "gain" and "count" but I think that they correspond to what I mean by miai value of a "move" and "node", excepting the above point. Also, my tree diagrams only show "gains" rather than what seems to be the normal convention of "count". This was because gains alone are sufficient for strategy and it was quicker to draw diagrams.
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Yes. This is implied by the way the miai value of a node is the average of the miai values of its two children.
If by (*) you mean the CGT star, then no because the children might have children of their own.

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As a disclaimer, I did a quick skim through of Bill's This 'n' That over the last few days. My calculations appear to be the most basic generalisation of Bill's post https://www.lifein19x19.com/viewtopic.p ... 23#p243523
I have yet to check if my results are consistent with Bill's yet. If not, then that would be the quickest way to check if my formulae are wrong. Hopefully I will check tomorrow after work.

(edit: it was pretty hard to believe they are equivalent as the form was so different. However, they seem to be the same for 3 gotes and for now I can believe they are the same in general.
The information content of my formula over Bill's is almost zero after a translation by what I called c, though mine also has the pre-emptive formula and history of i which Bill essentially sets to zero.
I wonder how Bill came to his formula philosophically. Mine seems more based on strategy whereas his series formula looks more like successive improvements on an approximation.
I have also remembered why I started thinking about this problem in the first place. Bill mentioned his work on it to me nearly a year ago. There were some other related problems he hinted at as being difficult which I have yet to think about seriously.)

Some of your more recent exchanges with Bill on that thread also seem related, but I have yet to more than glance at it, but I plan to work through them, perhaps on the weekend.

My section 5 and corollary 8: I did calculations for this a few years back, but only for a special case. I don't remember it well enough to easily reproduce the proof.

My last conjecture is not that related to the rest of my paper, which also came from the same investigation a few years back. Another thing I learnt from the skim through of This 'n' That was that using 2^n difference games might render my conjecture ""obviously" true", but I have yet to think clearly enough about it. (edit: I don't think this helps. I still wouldn't be surprised if my conjecture is false)

Also, I hope to finish writing another paper on "constructing endgame trees on the board" in the coming weeks, to be put on my newly created wordpress site. It will pretty much just be reasoning out loud, since I haven't thought much about the topic until the last month or so.

Standard assumption for what? Not for miai counting in general.



But you do call it miai value of a node. So you need to explain how, without [6], the recursion in [3] still makes sense. If, however, you study purpose is limited, declare so. Put [6] first and prominently.



This is so for short alternating sequences of at most 2 moves. For longer sequences, analysis must be generalised by, e.g., my method of making a hypothesis checking for how long successive alternating play can occur. An initial local endgame can be a long gote or long sente.



I see.



Saying White gains x and meaning a (usually) positive number is using the concept of gain. Therefore, apparently you must clarify your definition in line of Berlekamps incentives and my gains so that, for White, it is count of the position before the move minus count of the position after the move.



For preceding count C, count B after Black's move and count W after White's move, the gain
- of Black's move = B - C,
- of White's move = C - W.

Gote count = average = like your miai value of a node.
Sente count = count inherited from the sente follower.
Calculate counts iteratively, where longer traversal sequences take precedence.

Gote move value = half the difference value.
Sente move value = the difference value.
Derive from the right followers, where longer traversal sequences take precedence.

For simple gote of short sequences: gote move value = Black's gain = White's gain.



Hm, will see.



Ok, since you presume [6].



LOL. No. It is the footnote I explain further below!



Please explain what you mean. What do you mean Bill to state there? What generalisation? Why the most basic?



Quickest? Depends on what you mean as before.



What is your formula? What do you call Bill's formula?



So will it it with caution:)



I am not there yet:)



You should read https://www.lifein19x19.com/viewtopic.p ... 23#p264723

I am feeling better these days, so I took some time to skim Daniel's paper.



I have skimmed the paper, and have some criticisms, which will wait. I gather that [6] is this;

Principle 6. You may have the right to delay playing a sente until the last possible moment, especially if the opponent has no good move to prevent the sente.

As a principle, it is a bit strong. The standard assumption is that one player has the privilege of playing the sente, which means that he will play the sente with sente before the opponent can afford to play the reverse sente. In practice, players and bots do not always wait until the last moment to play their sente. In the mid 20th century Takagawa preferred to play sente late, while Sakata preferred to play sente early. As they were rivals, there are many games of theirs where you can compare their different styles in that regard.

As far as miai values are concerned, however, the so-called n point sente is actually a sente for which the reverse sente gains n points. So for calculating the miai value, you want to figure out how late you can wait to play the sente.



In the go literature, miai value refers to the estimated gain of a gote or reverse sente sequence. It was taught in books when I was learning go, but was plainly not the traditional way of calculating move values, deiri counting. Somehow it was associated with getting the last play, by most authors. In the 1970s I discovered that a surprising number of players in the West were treating deiri values as though they were miai values, which clearly leads to incorrect decisions. On rec.games.go in the 1990s I went on a crusade to introduce miai values to the West. I partially succeeded, but, to my dismay, I seemed to have created more confusion. As a result, I now avoid the term in favor of gain. O Meien rebranded miai values as absolute counting™ a few years ago. His book seems popular. I don't know how much of an effect he has had.

In the go literature territory is used to refer to the local score or estimated territory. That's fine for non-ko positions, but Berlekamp wanted a word to apply to ko and superko positions, and chose count to cover everything.

Counts come first. From them you can calculate miai values (gains).



Way back when I realized that a set of simple gote could serve as a reasonable approximation of a go board. The best play is the largest one, which typically is the best play on a real board. But what about a position of the form, {2s | 0 || -r} (in CGT slash notation, which I did not know back then, and which may not have been invented yet), where numbers s,r > 0? If it is sente its miai value is r; if it is gote its miai value is (r+s)/2. I added that position to a set of gote of the form, {g|-g}, with miai value g > 0. The set of gote comprise an environment for {2s|0||-r}, although I did not use the term back then.

The concept of an environment is peculiar. If you can read the whole board out you do not need theory, except perhaps as an aid for reading. As a practical matter, the environment is what is left after you stop reading. All you know about it is the miai value of its top play, and that may be a guess.

How much do you gain from playing first in the environment of simple gote? Following Chebyshev I decided to minimize the maximum error for my estimation. The minimum gain is, OC, 0, and the maximum gain is the miai value of its top play, t. We can minimize the maximum error to t/2 if that is the estimated gain.

The basic philosophy is that of strategy, but it is the lack of a theory for determining the environment that gives the sense of successive approximations.

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

Visualize whirled peas.

Everything with love. Stay safe.

Traditional? Hmmm. Japanese, yes. But O Meien in his Japanese book says,

"In Japan, it seems that very few people, of which I am one, are using Absolute Counting, but I have heard that in Korea and China many people use it.
I commend Absolute Counting to you, but if you can understand that a gote boundary play in Relative Counting (de-iri counting) is a set of two moves, it does not really matter if you do use Relative Counting. It is only that in the present Japanese go world this idea of a “two move set” has been forgotten and confusion has arisen over the criteria for units of a point, so that it is necessary, I believe, to sort that out."

References to the endgame are very frequent in old Chinese commentaries, but never in the sense of counting (except for a vague description of a move as big). However, there are also many books that include endgame problems, where the aim, invariably, is to get the maximum gain (to use your word, but that is supported by the typical use of 得 in describing the results - though usually in the form of gain an advantage, gain a wining position, and not a gain of X points). In the absence of even a single reference to deiri type counting, I'm inclined to think that that indicates the usual way of looking at endgame plays in China was miai-based, as O has found among modern Chinese players.

So, given that, it seems to me that the first and most important task of any writer on the endgame for the general go-playing public is to unravel why some people use one method and some another. In other words, the first task is about psychology or culture rather than decimal points.

My first guess is that Japan went down the deiri path because of their book-keeping culture. Both deiri and miai are accountancy terms. But why did they choose deiri? Accountancy is notoriously difficult. In that, it is very similar to the endgame in go. It looks as if everything should be very simple, but, as everyone who has had to fill in a tax return knows, it s very, very hard. Not only can even simple returns require a trained accountant to file, good accountants can even pull the wool over the eyes of the government's own accountants, i.e. the tax inspectors. In fact, I'd go so far as to say that the typical endgame writer is like the accountant who says, "If you follow my advice I can save you £200. That will be £500 for my fee, thank you very much."

I further guess that the Chinese may have gone down the miai route simply because they did so many endgame problems of the "gain as much as you can" type, and it became normal for them to look at every situation like that.

The poor player in the west, it seems to me, has suffered because they have taken it as an article of faith that the Japanese approach must be right (in all aspects of the game, not just boundary plays), but no-one has ever explained to them why (and therefore how) at a [u]psychological[/u] level deiri should be used (or even not used). Introducing miai into the conversation probably did caused confusion, but that's nothing compared with the confusion created by arguments over decimal points, definitions of words like count, privileges, trees and the price of fish. This is predominantly a western thing and, dare I say it, predominantly a numbers-guy thing.

The approach needed, I think, is like owning a car. The basic psychology, for most people, is I want to get from A to B. I want to know how to get from A to B quickly but safely. I don't need to know how the differential works, and I don't care too much about miles per gallon. And so on and so forth.

But if you examine the typical endgame book aimed at beginners, what you get first (O excepted) is an intricate demonstration that a move or a position is worth X points because if A has sente he can do this, but if B has sente he can do that, and so we average it out. O similar garbage, which if you think about, is like trying to tell the owner of a new car how the differential works. O Meien's approach is the only one I've seen that even approximates to the idea of point the car in the direction you want to go, press pedal A to go and pedal B to stop.

I can still remember, from maybe 50 years ago, the buzz at the London Go Centre when a new book arrived (I think it must have been Kageyama's) and it taught us in brief and simple words the principle of mutual damage. Even the many mathematicians in the centre were gobsmacked. It was a ponnuki moment.

The endgame as we are talking about here is somewhat trickier, but it still needs a Feynman much more than it needs new theorems.

Deire versus miai counting is not first of all a matter of psychology but about the ease of the second step. The first step is to count a settled position, that is to determine its 'count'. The second step compares the counts of two settled positions. There are different ways of doing this:

- Direct comparison: Which is the larger of the two numbers or are they equal?
- Deire counting: calculate the difference of the two numbers. This is the traditional move value.
- Miai counting: calculate half the difference of the two numbers. This is the modern move value.

If endgame theory ended at the second step, deire counting would be simpler than miai counting, which divides by two. Traditional endgame theory, as was used by Western players, indeed pretty much did end at the second step, except for calculating twice the difference for (reverse) sente.

When exploring endgame theory more deeply, the third step calculates the count of the position preceding 1) Black's play to a settled position and 2) White's play to a settled position. Knowledge of also the count of the preceding position enables the concept of gain, also known as "absolute counting".

Now, we can compare a) the count of the preceding position to the count of the position after Black's play or b) the count of the preceding position to the count of the position after White's play. For each comparison, we calculate a difference of two numbers. For (a), the count of the position after Black's play minus the count of the preceding position is called Black's gain. For (b), the count of the preceding position minus the count of the position after White's play is called White's gain.

From the third step on, almost always miai counting (or its aspect of absolute counting) is simpler than deire counting. To start with, in a simple gote endgame, the modern move value equals Black's gain and equals White's gain. We cannot say something so simple for deire counting, for which it becomes HALF the modern move value equals Black's gain and equals White's gain. That is, for deire counting to become consistent, first it must be converted to miai counting.

For advanced endgame theory, it becomes more and more apparent that miai counting is simpler for every task that is more demanding than comparing the move values of two local gote endgames.

***

Traditional go theory has flooded us with countless terms, of which almost all are superfluous, especially arcane shape terms. Every strategic concept (such as "efficiency";) ) needs some terms to work efficiently. Endgame theory works more efficiently by using some terms for it, instead of avoiding all terms.

To start with, each time it is efficient
- to speak of the "count" of a position instead of speaking of the "number calculated when counting / assessing" position or the "expected positional value of" the position,
- to speak of "Black's follower" instead of speaking of "the position resulting from a sequence started by Black",
- to speak of the "gain" instead of speaking of "in the case of Black's play, it is the count of the position after Black's play minus the count of the preceding position. In the case of White's play, it is the count of the preceding position minus the count of the position after White's play.".

Endgame teaching must not end at the third step while pretending the theory would be complete then. Endgame aspects dominate the second half of every finished game and affect the first part of each game. One does not become strong at endgame decisions by stopping study of endgame theory at its beginning. Serious endgame teachers teach very much more endgame theory because more than 95% of it lies beyond the third step while half of all endgame decisions require endgame theory much more than tactical reading / tesujis. Throughout the history of go, almost all teachers have neglected 95% of endgame theory. Criticise them instead of those few serious endgame teachers also exploring and teaching the 95%!

John, it is your passion to cast doubts on using numbers. I have studied endgame theory for half a dozen of years full time now and realised more and more that
- guesswork and rough guidelines (proverbs even) are often wrong,
- tactical reading is often too complex,
- even seemingly very simple positions have unexpected behaviour (like change a stone in a life and death problem and you change the outcome),
- numbers enable correct decisions,
- value conditions (supported by theorems) greatly accelerate decisions.

The Feynman approach to correct, fast endgame theory is simplified thermographs but I think that quickly constructing them in one's head while playing does require Feynman himself! My approach to making numbers applicable is to simplify value conditions as often as possible so that decision boil down to comparing two numbers. Such is sometimes possible but sometimes the theory is more advanced and demands two to four comparisons of one to four numbers, and further simplification cannot be possible.

To start with, a local endgame with follow-up has a move value, a follow-up move value and a value of playing in the environment. Such can require consideration of all three values. During the early endgame, such cannot be replaced by global tactical reading.

NUMBERS ARE OUR FRIEND!

No. I cast doubt on an obsession with numbers, which in turn leads to trying to describe everything in number terms, even when that is either unnecessary or undesirable.



I see that as an example of what I have just said. You see Feynman and think instantly of numberss. My allusion to Feynman was for his words - his reputation of being able to explain complex things simply.



Read "Weapons of Math Destruction" (written by a mathematician who can also explain things simply and without numbers, incidentally).

The problem is: when you mention doubts concerning endgame numbers and consider them undesirable, they are necessary. We don't compare numbers by comparing something that is not numbers, such as psychology. When we compare resulting settled positions, we do so by representing each result as a number and comparing these numbers. Whenever we compare earlier positions and informal approximations are not good enough, we compare the positions by (also) comparing numbers for them. We compare endgame positions that cannot be solved tactically by only comparing numbers and the turn.



Of Feynman I recall his participation in the Manhattan project, colourful private live, contributions to physics and Feynman diagrams. Why shouldn't I mention them? He is famous for them, they simplify as you have requested and represent some algebra, in other words: abstract NUMBERS and formulas, of particle collisions.

I got distracted by Feynman and forgot to reply to your implied suggestion of greater need for easily understood, introductory explanations than for new theorems. It depends on what you refer to by "The endgame as we are talking about here".

If you refer to Daniel's paper, yes, exactly. Before understanding the theorems or studying the conjectures, understanding the introduction is necessary.

If, however, you refer to the endgame in a broader sense including miai counting, different theory and practice in China, Japan, Western countries etc., I oppose your drawn competition of urgencies between introductory explanations and theorems. Both can coexist together with all grey area in between. Introductions are needed for those new to certain endgame topics. Theorems are needed to solve things in general with absolute correctness. Theorems are read by mathematicians and first are translated into 100% correct principles accessible for players understanding general statements of value comparisons or conditions. Next, partial simplifications sacrifice some correctness while also providing access to other players. Besides, informal summaries of new theory are written as overviews on the topics without already enabling reasonable application. Application does require some diving into a theory. No benefit without effort, as always.

Numbers obsession:

It has been reported that in a French research project 40 years ago, 75% of maths students in four countries answered the following question by "manipulating the numbers" but gave a wide variety of numbers for the answer.

"If a ship has 26 sheep and 10 goats on board, how old is the ship's captain?"



Agreed. But writing about the endgame for ordinary players requires the effort of diving into pedagogy theory.

As far as culture is concerned, IMX Western go players tend to think of the values of plays as miai values. That may be a result of the scientific revolution in the West, I don't know. Unfortunately, many, if not most of them learned deiri values. If all you want to do is to compare the average values of two plays, deiri values are fine, and perhaps easier for most people to handle, as most people have difficulties with fractions. But Westerners tend to want to do arithmetic with move values, adding and subtracting gains and losses, and for that you need miai values.

As we know, playing the averages by choosing the theoretically largest play is typically correct, and not often wrong. If this is all that people want to know, then why complicate things by saying that some people use this method and other people use that method and there may be cultural reasons behind these choices and so on and so forth? Leave well enough alone.

Now, as taught when I was learning go, to find the mean value of a non-ko position you had to know whether it was a sente position or a gote position. Unfortunately, sente and gote were not well defined. Furthermore, positions were classified as double gote, double sente, or sente for one player only. Furthermore, most books simply declared positions to fall into one of these categories and made calculations accordingly. So the reader was pretty well left with by guess and by golly.

In practice, it is easy to find examples of each of these categories. However, by the 1970s players and writers realized that there was a theoretical problem with double sente. To find the miai value of a double sente you had to divide by zero. Tilt!

I'm going to blow my own horn, here, but I was perhaps one of the few people who understood the problem and saw the theoretical answer. Double sente depend upon what is on the rest of the board, the environment, as we now say. But it is possible to define sente and gote without regard to the rest of the board, intrinsic sente and gote. These technical, theoretical definitions are somewhat different from the traditional meanings of sente and gote, but easily defended as such. When I taught players in the 1970s in Los Alamos about intrinsic sente, they say that I should publish something about it, but I thought it was too obvious. Little did I know. In the early 1980s I tried to get Bozulich to publish an article in the Go World about how to solve the problem with double sente, but he did not reply. I should at least have tried Kido after that.

One of the Japanese professionals who did understand the problem with double sente and classified positions simply as sente or gote was Shimamura Toshihiro, then 8 dan, in 1954. I have written about Shimamura before. (See https://www.lifein19x19.com/viewtopic.p ... 35#p194535 and following.) The thing is, with technical matters you want to have the right concepts and the right way of thinking, culture be damned. The theory requires the concepts of intrinsic sente and gote.

Basic gote is simple enough. {g|-g}, where number g > 0. Each player to play can move to a local score with an average gain greater than zero.

Basic sente is not difficult, either. {2s|0||-r}, where numbers r,s > 0. If it is a Black sente, its miai value is r. If it is gote, its miai value is (r+s)/2. If r > s, then it is intrinsic gote, even though on some boards White will reply locally. If s > r, then it is intrinsic sente, even though on some boards White will reply elsewhere and Black can play to 2s. If s = r, it is ambiguous between sente and gote, a new classification I came up with.

Why do we classify this as sente when s > r? Because typically at some point Black will be able to play to {2s|0} when White's reply will be the largest play on the board, before White will be able to play the reverse sente. We say that Black has the privilege of playing locally with sente. Privilege is not a Western concept, BTW, it is part of the Japanese go literature. Privilege is why we assume, when estimating territory, that sente plays will be played with sente.

The theory is about playing the averages, and sometimes playing the averages is wrong. With correct play, sometimes White gets to play the reverse sente, and sometimes Black plays the intrinsic sente but White does not answer it, allowing Black to play the threat. Theory only provides heuristics.

For certain positions, difference games provide correct answers, not just heuristics. But that's another story.

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

Visualize whirled peas.

Everything with love. Stay safe.

If they do try to play an endgame properly, I suspect most people this way most of the time - perhaps even pros, too. Calculating the numbers is indeed easy enough. What is not easy is to form a view on how much it matters (on average) when using these values turns out to be wrong. Such a view by an endgame expert is one of the things I have in mind when I argue in favour of words and not purely numbers in the exposition of boundary play. If, for example, it turns out that at low dan level the most you can lose on average in a typical endgame against a player of your level (who makes the same sort of mistakes) is just 5 points, it would be very useful to know that. MUCH MUCH more useful than being told how to calculate fractions of a move value, and MUCH MUCH more imbibable then the usual Humpty Dumpty approach of "When I use a word it means just what I choose it to mean - neither more nor less."

I can understand that endgame books are usually written by people of mathematical bent and such people might feel uncomfortable with inelegant approximations and fuzzy definitions. But I say they have to be less concerned with what other mathematicians might think of them and more concerned with how they can help the typical reader.



I'm sure I'm part of a chorus here: what are they?



That doesn't seem to square with Edo players being so good at the endgame. But, leaving that aside I'll make a couple of other points. One is that you say in the thread you referenced that O Meien didn't talk about double sente. He did, twice, and on one occasion said he felt smug about playing it. He also talks about privilege, which is not absent from other Japanese literature but is not very common. But it is pretty common in old Chinese commentaries. They may not have seen things through the prism of division by zero, but they and their Edo counterparts seemed to know to put the ballin the back of the net.

Switching back to words over numbers, one of the most noticeable features of pro commentaries towards the end of a game is that they VERY rarely mention numbers (for move sizes) but they BVERY often describe a move or a position as thick, and therefore advantageous. Now it seems to be that if it is worth harping on about gaining a point or a fraction in the micro-endgame (not that I think it is, but there's an 'if' in there), then it must be worth a humungous amount more to study the earlier and bigger stage when thick moves or positions become an issue. mIt would be immensely valuable, therefore, to have an authoritative endgame writer write a book or a chapter on how and when to play such thick moves, how to recognise andr appreciate them, and so on. There's vast number of examples in the literature. This exposition would have to be done mostly in words, of course, but that underlies my point: words are more useful than numbers for many people. Instead of improving by a point two you can look forward to improving by a grade or two.

How many endgame points do players lose at what rank?

Too many at all ranks!

I and others have estimated amounts and those are large amounts. So large indeed that then cast is doubt that those numbers could not possibly be correct. Of course, they are approximations but let me reaffirm: their order of magnitude is correct. Too many points!

Half of the game is in the endgame phase but do we study the endgame half of our time? Most don't and not even close. This is another strong indication of how very weak most players' endgame is. By far too weak!

***

Are larger mistakes more important than smaller mistakes?

A single large mistake is compared to a single small mistake. However, beyond beginner level, we make large mistakes only every couple of moves. At dan level, much less frequently.

We make small mistakes all the time, often without even noticing them.

Suppose one plays 100 endgame-related moves. John's persistent claim is that fractions would not matter. The contrary is true: even with an average loss of only 1/8 per endgame move, already this amounts to 1 lost rank. FRACTIONS MATTER!

I can understand time management though. If one can spend the same amount of time on avoiding a 1 point mistake as on a 1/8 point mistake, of course one should spend the time on the larger.

***

Endgame books are usually written by maths people?

Let us say that this might be so for those half a dozen endgame books that explain values and have almost no calculation mistakes. (Not the countless books with evaluation mistakes,) Writing while avoiding calculation mistakes requires discipline and the basic maths skill to recognise one's own mistakes. Explaining endgame calculations requires firm understanding of their theory. Therefore, such books are and have to be written by people with solid maths thinking.

***

Do maths-inclined writers feel uncomfortable with inelegant approximations and fuzzy definitions?

The question is too one-sided.

Some write only for mathematicians and their texts look accordingly. Berlekamp is a soft example. Conway and Siegel are not the hardest examples.

Some (me!) write different texts for different readers. From informal beginner texts to ivory tower texts (doing live research in drafting proofs) meant to be understood by at most a few readers.

When writing theorems and their proofs, less than 100% correctness is unacceptable. Only thereby are things solved in general. Making a tiny mistake in a theorem means that each person makes it in each application so the total amount of all mistakes made becomes great.

When translating theorems to principles, the writer must still approach correctness as well as anyhow possible and explain whenever approximation is used intentionally.

Feeling uncomfortable is not the best description for bending truths to weak approximations. It is about responsibility. Teaching must not mislead learners.

A typical reader should not expect easy to swallow entertainment with wrong contents but should expect (almost) correct contents, even if there is an entertaining touch.

***

Pros rarely teaching endgame numbers in game commentaries:

This is a great weakness of their teaching! Why does it occur nevertheless?

Teaching endgame numbers requires time and the skill to evaluate and calculate correctly. Avoiding the topic avoids spending more time on teaching than they want. It also avoids displaying their possible weakness. (Very few pros can teach it but still do so little because of the time investment.)

Do not draw the wrong conclusion that because they teach it little we should study it little! See above why that would be wrong for us.

***

Thick and other early endgame topics:

Sure, relevant, too. Again, very many decisions do require numbers, that is, they can never be made correctly without numbers, and the long stage comes when all moves are decided by numbers while no moves still require consideration of thickness etc.

***

Words are more useful than numbers for many people?

You might also say "words are more useful than tactical reading for many players".

Nonsense!

Use words when words are useful. Use numbers when numbers are useful. Use tactical reading when tactical reading is useful.

Use all tools when appropriate!

Do not use words when they involve making significant mistakes - instead use more precise tools as necessary.

EDIT: typos.

We'll suppose nothing of the sort. This is close to dishonest. WORDS MATTER! A game may have 100 endgame moves, but each player only makes half of them. Many games don't have endgame m0ves at all, many don't have as many as 50 each, and in many of those that do, even duffers get the moves right in many cases. Furthermore, there are many cases where fractional differences do exist but don't matter. One, but not the only, case is the Monte Carlo syndrome where you are so far ahead, perfect play doesn't matter in the slightest. A 1-point victory is worth the same as a 100-point victory.



Examples of such useful "nonsense": The L group is dead. The monkey jump is worth 8 points; Play safe when ahead. Or anything else that serves up candidate moves and evaluations on a plate.

Tactical reading obviously underpins these statements, but not necessarily in each game. Once learned, it becomes part of our intuition.

But other kinds of reading are useful, too. Although forums are not the best environment, accurately reading the words that are written by someone else is an especially desirable skill. Consider the difference between "words are more useful than tactical reading for many players" and "words are more useful than tactical reading for all players". Then square the former with the putative killer counter-claim "Use all tools when appropriate". Lewis Carroll logic will suffice.

I'm bored with diving down this rabbit hole again. The floor is all yours.

As a weak player, I thought each player would play 30 endgame moves. As a strong player, I consider endgame aspects of all moves of a game and also apply endgame calculations for my ca. 100 last moves of every scored game.



1) It is trivial to find excuses for not improving one's endgame skill.

2) When I say that 1/8 loss for 100 of one's own moves equals roughly 1 lost rank, then this refers to the scored games or the games that might as well be scored instead of resigned. Ca. half of all games. Thus 1 lost rank per (might be) scored game amounts to 1/2 rank per any played game.



Recall that I have spoken of average loss. A player makes occasional 5-point mistakes, frequent 1/8-point mistakes, everything in between and some perfectly correct moves.

An average of 1/8 loss per endgame-like move indicates a strong player.

For amateurs, the average loss per endgame-like move tends to be larger and for kyus much larger.



I have said: Use words when words are useful. For these applications, words are useful. Even so, the first example is off the point because it does not really belong to endgame evaluation.



One of the worst endgame advices ever given and designed to keep players weak! This kind of proverb encourages an endgame mistake of up to 4 points per monkey jump. The right guideline in words is: A monkey jump is worth between about 4 and 8 points per excess play.

EDIT

Yes. We pretty much have to rely upon experience there. In a way it is a statistical question, but questions such as the universe of players arise. If your opponents do not punish your errors, making them isn't so bad.



Not having good numbers does not mean that words are better. Since experience matters, feelings and hunches may be best.



Based on my experience, starting from the mid-endgame, where plays gain on average around 5 points, low level dans chuck around 15 points. To anticipate the discussion below, if they play solidly they can probably cut that loss in half. If their opponent does not play solidly, the question then becomes how well they can punish their opponent's mistakes.



It is painfully obvious that most of them are not.



Although Mathematical Go sold pretty well, especially in Japan, I felt that it suffered from having a dual audience, one of mathematicians and one of go players. It was not easy to satisfy both.

Back in 2011 I decided to write an endgame book aimed at middle SDKs, and assembled a group of readers who would give me feedback. To my surprise, it wasn't the math that was the problem, it was the go.



The obvious one is the strategy of playing the largest play, AKA hotstrat. The main exception comes when there is a significant drop in the global temperature. It will often be worth playing to get the last play before that drop. This is something that go players noticed long ago, leading to the advice to get the last big play of the opening, the last play of the large yose, and the last play of the game. For the last play of the game, there are both absolute plays and heuristics. My research with the Elf files suggests that getting the last big play of the opening is almost worthless advice. There is in general no significant temperature drop between the opening and the middle game. I have, however, come up with the following last play advice, which is to occupy the last open corner, as a rule. I know that occupying any open corner is said to be good, but in the pre-AI era pros often left the last open corner unoccupied in order to fight elsewhere. That was usually an error. Getting the last large yose does seem to be good advice. More on that later.

Professor Berlekamp came up with a heuristic he called sentestrat, which is basically to answer your opponent's sente if it raises the global temperature. The idea is to limit your losses. Berlekamp was around 3 kyu, and I privately scoffed at the idea, dubbing it gotestrat. As a dan player I felt that I was duty bound to find a good intervening play, if I could. Sentestrat sounded like trying to make a virtue of followitis. Berlekamp was right about limiting your losses, OC.

But consider your question about how many points a low dan chucks in the endgame. Sentestrat is a way that they could limit their losses. It goes against the grain, but there we are. High dans and pros may interpose a play instead of meekly replying to an opponent's sente. OC, we cannot say that they are wrong to do so, but the result is often a fairly long term increase in the global temperature. When that occurs, it will frequently become important to get the last play before the temperature drops back down. This is why I say that getting the last big yose is often a good idea.

This kind of phenomenon cannot simply be observed by words. It is necessary to have the technical chops to do so.

Today's top bots do not often seem to play thickly, but play lightly and flexibly. That style of play may be difficult for humans to emulate, given our propensity for depth first search. However, especially at the large yose stage, it behooves us to be on the lookout for temperature drops and to consider tenuki when that happens.



The reputation of the Edo players is mostly intact. True, the bots may differ, but the large yose is one place where humans, given enough time, may play better than today's bots. By the 20th century, with 10 hours of time, I noted in the 1970s that top pros rarely made errors when the average gain dropped below 3 points. 3 points is in the middle yose range.



I acknowledged that at the time, didn't I? But O Meien did not include double sente as a classification for calculating the size of plays, which was, and is, my concern. Kano in the 70s and the Nihon Kiin, as late as 2000, tripped up badly in that regard. As I said above, it is not difficult to find examples of double sente in practice. The problem is that it is too easy to find false examples if you believe in double sente.



As you know, I gradually developed a thick style of play. The bots would probably regard it as heavy, what can I say? But one advantage of playing thickly in the large yose, or maybe earlier, is that it frees you play enterprisingly to take advantage of your opponent's misplays. Also, IMX, when the average gain drops to around 7 points is often when plays in the corner, often involving ko, become significant again. Thick play prepares the way for ko fights by reducing the number of possible ko threats by the opponent.



In vol. 6 of Sakata no Go, which is about endgame calculation, Sakata shows a good example of thick play in the endgame. That play allowed Sakata to continue enterprisingly. To show the advantage of the thick play, you cannot just rely upon words, you have to have the technical chops.

This is speculative, but it may well be that thick play tends to produce positions where humans, even strong amateurs, can play better than today's bots. I doubt if I well be able to research that question, but I think it is worth doing.

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

Visualize whirled peas.

Everything with love. Stay safe.

What was problematic about the go (theory)? That you had thought to have it ready but found that you did not have it ready yet?



As is KataGo's? However, it may simply be that nobody has studied their endgame play carefully enough to verify it against modern theory. I may do it some time if I find the necessary time.

No, my readers understood the math just fine. But they did not find the plays, which I wouldn't even call tesuji. {shrug}

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

Visualize whirled peas.

Everything with love. Stay safe.

Bill: But I think I learned more from your one post (thank you) than I've learned from all other endgame writers excluding O Meien. Why? Because of the words.

Technical nous matters of course, but I think it matters far more for the person framing the advice (as it should) than for the end user. To repeat my previous analogy, to drive a car I need some basic technical knowledge (gear shifts, how to fill up with petrol, etc), but I don't have to know how the differential works - or even where it is - or even if it's there or not.

KataGo does   N O T   care about points!

In this sense, it might be a useless tool for you (during "play" / "analyse"), if you ever wanted to use its assistance for the verification of some parts of your "modern theory".
If KataGo has never encountered an endgame position in question (nor anything comparable), it is quite likely that it will be unable to find the "correct" (your understanding) endgame sequence, no matter how powerful the machine is it runs at.

If you ever wanted KataGo to "solve" such kind of endgame positions, you would have to go back to the selfplay training of the net, and feed the training with appropriate sample material.
The underlying problem is absolutely the same as with solving Igo Hatsuyôron 120.

If you let KataGo play games, I am very sure that it will never reach any of the endgame positions, where your "modern theory" possibly matters (just as it would never reach the starting position of Igo Hatsuyôron 120).
And this might be also true for games of the old masters, which are said to have had a similar style as modern AI.

_________________
The really most difficult Go problem ever: https://igohatsuyoron120.de/index.htm
Igo Hatsuyōron #120 (really solved by KataGo)