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 Post subject: Informal global endgame theory #1 Posted: Fri Mar 26, 2021 11:18 am
 Tengen

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What informal / verbal / traditional endgame theory do we have for global move decisions? I have collected the following. What have I missed? I ask for theory with a general scope but neither ask for details, local-only theory nor formal / mathematical / modern theory. Although the list is a bit longer than I expected, it is still shockingly short.

- Consider the local endgames with the largest move values.
- Play gote in order of decreasing move values. 1)
- Play but delay sente to preserve ko threats, options or aji. 2)
- Play double sente early. 3)
- Sente or reverse sente are worth twice as much as gote.
- Seek the last move before a large value drop (tedomari). 1)
- Ignore equal options.
- Consider the alternative strategy of mutual reduction.

(1) This often taught theory is wrong. Although these work as rough guidelines, countless exceptions exist.
(2) This includes sentes that are privileges.
(3) This is useful only as an informal guideline. In formal theory, local double sente does not exist.

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 Post subject: Re: Informal global endgame theory #2 Posted: Fri Mar 26, 2021 12:22 pm
 Oza

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O Meien in his much discussed endgame book shows us how to think about the endgame informally in several amateur games, six of hos own games, and in one by Yi Ch'ang-ho. He stresses the informality ("don't worry about precise values for counting").

The factors he mentions seem barely to touch on any item in your list. In fact the major difference may be that he doesn't make a list. The starting points vary according to the actual position (basically, is it a "hard" or an "easy" endgame ) and the various factors thereafter are integrated as and when needed.

Hard/easy is assessed on the basic of a count, of course (to "sense" who's ahead?), relative thickness and whether there are too many unresolved areas that are not easy to count precisely.

Along with relative thickness, he stresses eliminating one's own bad aji rather than preserving the opponent's. I.e. boosting one's own thickness and reducing uncertainty.

He includes the idea of looking for large-scale trades (for which reason he is less enamoured of sente than an amateur might be). This includes trades the opponent may initiate. Playing thickly can eliminate unexpected trades.

He makes great play (as he does in the rest of the book) of his concept of the "margin of error". This allows one to cut off all the dangly bits in one's thoughts and so creates certainty.

To look wider, the one word that comes up in almost every pro commentary on the endgame is "who is thicker?" Almost never "who is ahead?" (and then that question is broached it always designed for an amateur audience, or it comes at the every end to explain why a player resigned).

In a nutshell, the pro global strategy seems to centre on reducing uncertainty if you sense you are ahead. And if you are unsure who is ahead, "count" the thickness. If you are behind, look for trades.

Only then does move order according to size kick in, but (except in an "easy" game) it's still not entirely according size. Thickness still has to be factored in.

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 Post subject: Re: Informal global endgame theory #3 Posted: Fri Mar 26, 2021 2:19 pm
 Tengen

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I very much appreciate your input and refreshing view on the topic.

That said, much of what you say or attribute to O Meien affects topics I would consider to belong to the middle game, transition from it to the early endgame or positional judgement before the endgame. Until then, there is good use for imprecision or strategic concepts of the middle game, such as thickness, or for a combination of precise and imprecise aspects.

When concepts of the middle game become almost irrelevant, the preferably precise tools of the endgame are essential for avoiding losing points. Informal endgame theory still contains imprecision while formal endgame theory strives towards precision to gain more points.

EDIT: During the opening or middle game, there are decisions for which the precise endgame tools are already the most relevent. Other decisions involving imprecise and precise aspects profit from greatest precision of the latter.

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 Post subject: Re: Informal global endgame theory #4 Posted: Fri Mar 26, 2021 3:29 pm
 Gosei

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Robert, is formal endgame theory practical enough to apply under time constraints? I think traditional endgame theory aims at making decisions under time pressure, assuming that the imprecisions average out or a pro will realize the situation is out of the ordinary.

I see a possible application of assembling an encyclopedea of (late) endgame moves, for quicker decision making. For more intricate decision making, I'm curious, not sceptical per se.

I'm only a mediocre player of cours but I find myself making no calculations at all in the endgame. I'm happy if I can keep track of the big plays and grab more of them than my opponent.

The simple heuristics I apply:
- play big sente first then the biggest gote
- a big move in the vicinity of a group that's not yet clearly alive is even bigger, even if it's not immediately sente (the follow up probably is)
- reverse sente of similar size as gote is probably better, but sente of smaller size is still better
- except very small sente: these are ko threats

With 40s byo-yomi that's already challenging. Perhaps I should play slower games.

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 Post subject: Re: Informal global endgame theory #5 Posted: Fri Mar 26, 2021 3:55 pm
 Honinbo

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John Fairbairn wrote:
The factors {O Meien} mentions seem barely to touch on any item in your list. In fact the major difference may be that he doesn't make a list. The starting points vary according to the actual position (basically, is it a "hard" or an "easy" endgame ) and the various factors thereafter are integrated as and when needed.

Hard/easy is assessed on the basic of a count, of course (to "sense" who's ahead?), relative thickness and whether there are too many unresolved areas that are not easy to count precisely.

Along with relative thickness, he stresses eliminating one's own bad aji rather than preserving the opponent's. I.e. boosting one's own thickness and reducing uncertainty.

Both humans and AI play safe by making otherwise inferior plays. But, AFAICT, they make different inferior plays.

John Fairbairn wrote:
He includes the idea of looking for large-scale trades (for which reason he is less enamoured of sente than an amateur might be).

If these large scale trades involve raising the global temperature, then normally it is important to get the last play (i.e., take gote) before the temperature drops back down.

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 Post subject: Re: Informal global endgame theory #6 Posted: Fri Mar 26, 2021 4:05 pm
 Honinbo

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Knotwilg wrote:
Robert, is formal endgame theory practical enough to apply under time constraints? I think traditional endgame theory aims at making decisions under time pressure, assuming that the imprecisions average out or a pro will realize the situation is out of the ordinary.

Formal, exact, endgame theory is, IMHO, usually best studied away from the table. But, like life and death knowledge, it can simplify reading at the table. Here is a simple example. Suppose that you are reading from a position with no ko caveat, and you look at making a large simple gote. Then you may assume that your opponent will make all plays that carry larger threats and that you will answer them. That does not mean that that sequence of play is best, but if your original play is correct, then so are the other plays. You can also apply that reasoning heuristically, in which case if the original play is correct, the other plays are very likely to be correct.

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 Post subject: Re: Informal global endgame theory #7 Posted: Fri Mar 26, 2021 4:18 pm
 Oza

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Quote:
I see a possible application of assembling an encyclopedea of (late) endgame moves, for quicker decision making.

I forget the precise figure but Rob van der Zeijst (apparently following professional advice) supposedly learned the values of about 1,000 boundary plays. Speaking to pros myself has confirmed that at last some of them use this method, though none could put a figure on it. They just kept absorbing instances rather than working through a list. We also see this approach in Genan's Igo Shukairoku (available in New In Go), where he gives 155 such counts.

Quote:
With 40s byo-yomi that's already challenging. Perhaps I should play slower games.

But pros have told me they can't use this encyclopaedic method very well in fast games, and in games such as simultaneous play they don't even try. They do wkat we do - play on instinct. Of course they have better instincts.

At the other extreme, there are pros who can play to win a game against an amateur by 1 point. But they have been specially brought up to practise how to do that. My impression is that the teacher gives this task to pupils he expects to become lesson pros, as a PR trick - a trick which will only work in a minority of cases, anyway. Those pupils he expects to thrive in tournaments don't have to waste time on this task and so can't manage it easily. In other words, trying to overcount in the endgame in fast games is seen as too hard and even counter-productive.

That is why, in Japan at least, they keep coming back to this endgame codeword atsui (thick, which has nothing to do with atsumi BTW; it's to do with atsusa). It is, after all, relying on probabilities, which humans are geared up to do, and which also characterises the (?older) soba 相場 style of middle-game play they are familiar with in Japan.

My own impression is that the average amateur can only memorise a small double handful of josekis and only an armpit or two full of tesujis. That would suggest to me that trying to fit a large number of boundary-play counts into one's crotch isn't going to work very well either - not necessarily through lack of ability but for lack of time (including refresher courses) and commitment.

Incidentally, it seems intuitively likely to me that by playing the endgame thickly (which ideally means starting in the late middle game) means, probabilistically, that when boundary plays are actually made in individual areas, the thicker player will emerge with a numerical advantage in each case, on average. I have never felt the temptation to think that through, but someone more number-oriented may be willing to - and share their insights?

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 Post subject: Re: Informal global endgame theory #8 Posted: Fri Mar 26, 2021 4:58 pm
 Tengen

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Formal theory, which might be status verification by tactical reading, endgame evaluation or other things, consumes time. How much time depends on the position, which more specific theory applies and how much skill / practice one has with applying it. There are easy and difficult life and death problems, and there are easy and difficult evaluation problems. I apply formal theory as far as my skill and available thinking time allows it.

The most time-consuming endgame methods are pure tactical reading for more than a few local endgames or infinitesimals for large local regions. Both become inapplicable quickly. Intermediate time consumption occurs for calculation of a move value of a deeply iterative local endgame. Provided local values are known or reasonably estimated, the fastest formal endgame decisions are global.

Regardless of this description of average time consumption, every major aspect of formal endgame evaluation has a variety from little to much time consumption. For example, some global decisions are easy while others can consume much time (relatively much for global decisions, that is) if one wants to solve a position of the early stage of the late endgame as if it was a very late endgame decision.

For formal global endgame, summation of some values consumes relatively little time. It can be more time-consuming to do the right thing with calculated values: add them or subtract them? In one case, addition is right - in another case, subtraction is right. Was it the move value or the follow-up move value to be applied? Etc. One has to think carefully which applies. With much practice, such considerations should be(come) instant or at least fluent.

Heuristics are useful to some extent as long as one can be sure that their application is right but they only bring a player so far. Beyond them, many more points must be gained that one can only get by evaluation.

Last edited by RobertJasiek on Fri Mar 26, 2021 5:11 pm, edited 1 time in total.
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 Post subject: Re: Informal global endgame theory #9 Posted: Fri Mar 26, 2021 5:04 pm
 Tengen

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If Black has the thicker position and both players enter the late endgame with an equal number of safe points, the thick black shapes oppose thin white shapes in small, almost neutral regions. There, it is very likely that Black can make a few points more than White, e.g., by capturing one white stone or another.

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 Post subject: Re: Informal global endgame theory #10 Posted: Fri Mar 26, 2021 11:03 pm
 Tengen

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- Usually, play sente before gote.

reminds me to include my own earlier compilation of informal advice of common knowledge above weak kyu level, in particular:

- Avoid premature endgame.
- Maximise territory when defending your boundary stones.
- Minimise territory when attacking the opponent's boundary stones.
- Choose the locally best endgame.
- Distinguish the basic endgame types.
- Update the basic endgame types after every move.
- Imagine different sequences, count at the end of every sequence and play the first move of a sequence leading to the most favourable count.

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 Post subject: Re: Informal global endgame theory #11 Posted: Sat Mar 27, 2021 5:52 am
 Dies in gote

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On the 'informal side' of endgame theory, I think there's a psychology/practical element is missing...

I think there's often a thought process/thought pruning process at the outset of the endgame along the lines of below (at least in tournament games):

We're about to begin the endgame...Who is ahead...How close is the game? Where are the areas of the remaining aji/uncertainty? How much time is on each of our clocks?

If I am well ahead, what is the only way that my opponent can beat me? Where do i need particularly to watch out? Can I afford a move to eliminate any aji?

If they are well ahead - do I need to raise the stakes or complicate the game to have a chance of winning? What are the areas of aji or uncertainty where I can possibly turn the game around? What's the best way of doing this? Straightforward? Or would some preparation make it work better? Do i need to play before they eliminate the aji?

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 Post subject: Re: Informal global endgame theory #12 Posted: Sat Mar 27, 2021 7:37 am
 Oza

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The first references to the endgame are in ancient Chinese books. We may safely assume that the first book was the Guanzipu 官子谱. The version we have today was compiled in 1694 but it was based on a now lost older source. Either way, it has a preface in which a noted player, Wu Ruizheng, remarked that this was the first ever book on the endgame.

The Guanzipu itself is devoted mainly to what I call "encroachment problems" i.e. featuring endgame tesujis. There is nothing at all about counting.

However, in the books of commentaries that started appearing just before, there are references to the endgame. The Yo Mo of 1662, for example, ends many commentaries with the phrase: 官子俱细
- the entire endgame was played with precision. Later commentary books make similar if more long-winded remarks, and in general the micro-endgame stage is mentioned very frequently, but never with any reference to a count.

There are many references to a move being "big" but this is almost invariably in the sense of "big and important" rather than "numerically big."

But when it comes to the macro-endgame, there are even more references to boundary plays and they centre on the word shou 收. In classical games this refers to a "walling off" (including squeezes) to make the boundary plays. Again, there are no numbers involved, except that a particular walling-off may be described as "big", or more likely "well-timed.".

I don't recall any modern commentator ever criticising the endgame play of the Chinese ancients, but in any case I'm certain any criticisms are rare - just as there is a general assumption that the endgame play of the Edo masters was close to perfect. There is nothing in Genan's Igo Shukairoku that suggests deiri counting was remotely new, and if we add in little details that there were influential players such as court astronomer Shibukawa Harumi (i.e. Yasui Santetsu in the time of Dosaku, which is also the time of Yi Mo etc) who were rather obsessed with numbers, it beggars belief to assume they did not know how to do deiri in some form or another in both China and Japan at a very early period.

Yet, just as today, the pros then did not normally talk in number terms. They were very conscious of timing and the order of moves and all the other aspects of technique, but the word they used most of all was shou (making walls) in Chinese and atsui (thick) in Japanese.

Before you shrug off that observation, note that in a frequently list of all the words used in the early Qing commentaries (I've made a corpus), 收 cones up in 24th place, behind words such as not, corner, good, at - and ahead of words such as one, can, is, also. In short, it is very safe to say that this was the primary way the old Chinese masters talked, and presumably thought, about the endgame.

Indeed, on the board, the boundary walls are a fairly distinctive feature of the game.

So think on this: walls represent thickness. It therefore seems safe to say that old Japanese and Chinese masters were thinking along the same lines, only using different words.

If they thought that way, why shouldn't we? You can always brush up your technique, of course, but that not much use if your overriding thought processes are not in control.

There was a good example of all of this in a video I watched last night. It was the divine Joyce DiDonato teaching a young singer who has some impeccable technique. When Joyce suggested a different point to take a breath, the singer was aghast. As Joyce astutely realised at once, the singer was breathing where she did because that's where she had seen Cecilia Bartoli or whoever once breathe. She was thinking only about technique (i.e. counting for us).

Joyce then explained that the primary thought process was to establish what emotions the person depicted by the role would have in the opera. Once the singer knew that and attempted to internalise these emotions, the sound that came out of her mouth would be driven by the emotions and she would automatically breathe where a real emoting person would breathe. I would suggest that internalising the emotions is equivalent to us playing thickly - and thinking about that seems to tell us quite a bit about how to play thickly, i.e. how to breathe at the right times in a game.

 This post by John Fairbairn was liked by 2 people: dhu163, ez4u
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 Post subject: Re: Informal global endgame theory #13 Posted: Thu Apr 08, 2021 3:56 pm
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JF:
Quote:
I forget the precise figure but Rob van der Zeijst (apparently following professional advice) supposedly learned the values of about 1,000 boundary plays.

Wow.

Quote:
the thicker player will emerge with a numerical advantage in each case, on average.

Hmm, perhaps. But perhaps when you are thick you may want to set up more groups further from the thickness to prevent overconcentration, and there is subtlety in how far away they can be while remaining safe. It is always nice to squeeze extra points in the endgame by capturing stones when your opponent has ventured slightly too far from their thickness as then you have the added bonus of nullifying their thickness in the process (which is often the biggest move anyway). Even one space jumps from a wall can be attached to and exploited. But I guess you mean using thickness to play safely enough that you can read all the follow ups and not get captured!

Perhaps my point at the end of this post is also relevant.

RJ:
Ignore equal options - i.e. pair off miai moves (e.g. to simplify reading, counting).

Well, there are always more complicated "principles" that you can generate in addition to the fundamentals you have mentioned. This may already be too detailed compared to what you are looking for, but I can't think of others that you haven't already mentioned at the moment.

For example, how about read the follow up and estimate the size of the first move via

"count half the value of the follow-up in the value of the first move"

(of course, miai counting requires you read all leaf nodes, but sometimes you can estimate/bound them)

Similarly, you can probably generate the principles for one move look ahead on standard principles. For example:

"if playing the largest gote move means your opponent gets tedomari, then consider playing a smaller reverse sente instead to fight for tedomari"

or

"sacrifice a small amount locally to add or remove a follow up move, in order to change the parity of moves of the next common size in order to fight for tedomari on them"

CGT:

I'm not familiar enough with CGT to be sure, but I think there is the principle that among corridors of otherwise equal value where you have a sente at the end,
"play in the corridor that ends in the smaller sente"
in order to save your larger sente as an intermezzo move in response to your opponent's smaller sente at the end of the game. There is probably a slight generalisation to larger endgames but it is beyond me at the present time.

This is somewhat related to mutual reduction as well. Often mutual reduction means that you still have a big move to play after the opponent plays, whereas if you make the mistake of defending your own areas first (all else being equal), then the final plays are in the opponent's sphere of influence, leading to the opponent to get tedomari when they respond to your attack.

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 Post subject: Re: Informal global endgame theory #14 Posted: Thu Apr 08, 2021 10:05 pm
 Tengen

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dhu163 wrote:
"if playing the largest gote move means your opponent gets tedomari, then consider playing a smaller reverse sente instead to fight for tedomari"

or

"sacrifice a small amount locally to add or remove a follow up move, in order to change the parity of moves of the next common size in order to fight for tedomari on them"

One can make many such guesses of allegedly correct guidelines. At closer inspection, one finds that they are wrong. Do you think that your two statements have been part of informal endgame theory or have you just made up them? I have not heard them before. If you have just made up them, do you think we should include every ad hoc guess?

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 Post subject: Re: Informal global endgame theory #15 Posted: Sun Apr 11, 2021 2:42 am
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Huh, clicking on the link led to "L19 is down", but removing "f=15" from the URL worked.

Well they certainly need more detail to correctly apply them. Did I make them up? You caught me ! The first, probably yes, just now. The second, I think I have rephrased something I've heard in pro commentary. Have I used them in a real game? I don't remember doing so consciously, only in post game analysis. Have I used the principles you first listed? Yes often.

variations 1-3 are not directly relevant, but something else I was thinking about
variation 4 is my impromptu construction of my second principle
variation 5 is a discussion of mutual reduction

I could generate more "principles" like those two but if expressed in words they would get very clumsy and would essentially just be CGT (which I plan to learn more about).

adhoc: not sure what you mean. Probably they aren't relevant enough to be included in your book, no.

I note that all your principles are about the structure of endgame trees like in CGT. Shapes on the board such as thickness, tesuji are not mentioned, which is probably what pros talk more about. For example Gopro yeonwoo made an endgame video, and it made me wonder about when it is right to play from inside the territory (i.e. cut) vs from outside. It is probably somewhat related to inside/outside in semeais. And might be worth investigation.

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 Post subject: Re: Informal global endgame theory #16 Posted: Sun Apr 11, 2021 8:06 am
 Tengen

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dhu163 wrote:
Probably they aren't relevant enough to be included

Of course, there can also be flawed ad hoc theory. Without such, we might not always be able to find correct, permanent theory.

Quote:
I note that all your principles are about the structure of endgame trees like in CGT. Shapes on the board such as thickness, tesuji are not mentioned, which is probably what pros talk more about. For example Gopro yeonwoo made an endgame video, and it made me wonder about when it is right to play from inside the territory (i.e. cut) vs from outside. It is probably somewhat related to inside/outside in semeais. And might be worth investigation.

Currently, I study the purely territorial aspects of the endgame. Already they are very much richer than one might expect at first. Bill will tell you to have spent some 5 decades on them and my related study has at least reached a couple of years now.

One can also study non-territorial aspects (such as those you mention) but most of that belongs to the middle game.

Before we can hope to develop a systematic understanding of the combination of territorial and non-territorial aspects, we must first establish it for territorial aspects alone. It was done for the basic microendgames. (Bill started to do it and) I am doing it for the large territorial endgame.

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 Post subject: Re: Informal global endgame theory #17 Posted: Mon Apr 12, 2021 2:26 am
 Tengen

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You suggest mutual reduction if there are more local endgames with the opponent's follow-ups than local endgames with the player's follow-ups. Please explain why you think so and why it should be independent of local move values and follow-up move values!

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 Post subject: Re: Informal global endgame theory #18 Posted: Sat Apr 17, 2021 1:32 pm
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I will be quite busy in the foreseeable future, and I wanted to reply properly, so it took me a while.

I'm not sure what you classify as "informal endgame theory".
Personally, my ideas are mostly based around miai counting (which I learnt from Sensei's library) because I think the mean value theorem makes non-trivial progress. But miai counting does seem to have some flaws around distinguishing gote/sente and hence infinitesimals. As far as I can tell, your first post is pretty much a list of the concepts that miai counting is based on (in the sense that "play gote in order of decreasing move values" isn't as good advice if you use deiri counting)

For sure move values come first. Tedomari is subtle enough that it is as easy to lose as gain if you don't read far enough ahead, and even if you succeed in tedomari, you gain at most the largest move value (in the rest of the game) compared to the greedy algorithm of always taking the move of largest miai value.

As for mutual reduction, I guess one would need to define or work out what counts as a defender's endgame tree vs attacker's.

All else being equal, my point was that mutual reduction only works if you have some sort of advantage that breaks the symmetry, generally being "thicker" on the board, or having more follow ups from going further into your opponent's area (the opposite of how you phrased my point). I now see that what I said about more follow ups may be misleading, at the least it is non-linear. It seems not so easy to describe what the conditions are for mutual reduction to work except in the simplest cases.

You could view this in terms of miai values: if your follow ups are larger/sente this increases the miai values of moves you play attacking your opponent's territory rather than defending your own.

You could also view this in terms of the structure of the game tree (my point still can hold if miai values are the same). For example you could say that in the defender's tree, there are more places for them to retreat to and often just one move of theirs will settle the position, whereas for the attacker to enter further, they need several moves in a row. This becomes related to the UPs and DOWNs of CGT. In my sgf there was the most simple scenario of two corridors that end in TINY/MINY and that you have the opportunity to play a mutual reduction if the corridor lengths are equal and you have a larger sente at the end.

If the corridor lengths are not equal or there are multiple, I'm not sure what happens. But in the simple case of two corridors ending in TINY/MINY, it seems your phrasing works (sort of). If the corridor in your opponent's territory is shorter, you have an advantage because you can reach the sente faster before coming back to defend your own. (it is a bit like moves closer to the root of the game tree count with a higher weight than those further down)

I wrote up some somewhat related ideas, nothing to do with mutual reduction, but just some calculations of elementary endgame scenarios over the last few days here: https://dhu163go.files.wordpress.com/20 ... ciples.pdf

I see there has been a lot of discussion on the endgame on L19 over the last few years and I hope to read some of it.

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 Post subject: Re: Informal global endgame theory #19 Posted: Sat Apr 17, 2021 4:01 pm
 Honinbo

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Global endgame theory is definitely not traditional go endgame theory, except for getting the last play, which is not at all developed. Traditional go endgame theory is mainly about estimating local territory and the size of local plays. It makes use of sente and gote, but they are not formally defined. Combinatorial game theory (CGT) is a more formal theory of local positions and plays, in which it is possible to formally define local sente, gote, and ambiguous positions. In both traditional go theory and CGT whose turn it is is not part of the definition of a game.

There is another type of game theory in which whose turn it is is part of the definition of the game. Starting from node A in the game tree with perfect play to leaf node Z, the value of A is equal to the value of Z. It is possible to evaluate a game node as win, loss, or draw, or as a score. There is only one game, so this is indeed a global game theory. With alternating play, sente and gote have no global meaning. Without knowing correct play, today's neural net bots can evaluate a node in terms of the estimated probability of winning, given certain assumptions. These estimates provide an informal global endgame theory.

There are problems with such a theory for human play. First, humans cannot make these evaluations. A theory of how humans might evaluate positions and plays has not yet been developed, nor even how to make relative evaluations. Second, humans play differently from bots, and the assumptions behind these evaluations are typically that bots are playing. Third, humans play better than bots in some ways. A good global theory should take these human plays into account.

Not to be particularly discouraging, because I do think that better endgame theories for humans can be developed with the help of neural networks, but not very quickly. Meanwhile, the proof is in the pudding. How well do certain rules of thumb work, and under what conditions?

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 Post subject: Re: Informal global endgame theory #20 Posted: Sat Apr 17, 2021 8:39 pm
 Tengen

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dhu163, I read your paper a bit later. BTW, except for CGT people, the number of people digging into formal endgame theory has recently increased from 2 to 5. The future is bright! :)

Formal endgame theory is established as maths. Informal endgame theory is not. Of course, there is a grey area of principles and methods containing cores that we can hope to be proved by maths later but whose presuppositions might need to be worked out carefully and possibly corrected.

Classification of local endgames as gote / sente etc. is not flawed per se but does not claim to always predict global play in global environments. A local gote / sente need not be a global gote / sente.

Not move values come first but scores do, then counts.

Mutual reduction does not only occur as a consequence of thickness but also occurs in the microendgame. Right, solving mutual reduction in general is hard.

Bill, traditional endgame theory is not only about local sizes but also about tactical reading and the occasionally useful, but heavily over-emphasised tesujis.

Some theories of positional judgement have been developed (see, e.g., my texts) but they float in the air without being based on and derived from low level maths. Such theories are very useful in practice but unproved in maths.

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