Thank you so much to everyone ^^

Isnt the second statement immediately a counterexample to the first one if t<T/2?

True, I am being careless, just going off memory. Since I'm used to using deiri swings too.

I guess it should be less than 2T. I'll edit accordingly.

Now I think about it, the proof works for all endgames where gote/sente is clearly defined, not just simple gotes. But the bound is 2T-t.

In fact, the proof for the simple gotes bound of 2(T-t) is a more difficult proof.

First I had to learn traditional endgame theory. It was hard because there was little theory, little explanation and bad explanation verbally or written. Later I heard of modern endgame theory, of which there was more with more explanation but explanation was more often for CGT theorists than players. I had to write my own explanations to understand modern endgame theory well but this was my breakthrough to it so profoundly that I could also expand the theory a lot. Understanding modern endgame theory also enabled me to understand traditional endgame theory were it exists for some subconsciously but is well hidden and not described openly. Including this, there simply is very little traditional theory of endgame evaluation, and part of it is wrong. The amount of traditional endgame evaluation theory is a tiny fraction of the amount of modern endgame theory.

Supposing you are right that most pros do not or hardly use modern endgame theory, you ask why. I think some reasons are:

- They learn go early and spend much time on go study so that their mathematical skills are often under-developed. (I know there are exceptions.)
- Although there are expert explanations for expert theory, presumably there are still too few non-English practically useful explanations of modern endgame theory beyond its bare basics.
- Their tactical reading is so good that they can compensate their suboptimal knowledge of endgame theory to some extent by much use of the method of reading and counting.
- Most research in modern endgame theory is in English and does not reach most of them fast enough, especially if they under-estimate the volume of available knowledge.

If the pros had more insight (beyond O's book) in endgame evaluation theory, they would teach it, but they don't. We do not live in the Edo period with secrets kept.

Pros teach endgame as tactical reading (which is as important as evaluation), tesujis (hardly relevant except for a few moves per game) and, as a footnote only, the hint to calculate values. They know that evaluation is important but relatively neglect it in their teaching. Sure, teaching tesujis is 1000 times easier than teaching calculations, which takes time and easily made mistakes endanger reputation. (Yes, again, there are exceptions. Three?)

Just a dumb question: how practical is it to count precisely the values of moves during a live game? What time controls would be needed for a player with a lot of training?

Ke Jie has started live streaming lately and he sometimes counts out loud though most of it seems to be dissing his own moves, saying how badly he is playing and how he will lose, only to turn the game around. There is good Go theory he describes as well, when explaining his moves or his judgement of what is going on.

My guess is positions with depth 3 or more take more than 1 second.

He also seemed to take around 5 seconds to count the size of a 38 point seki and judge it was worth saving despite the lack of liberties.

I'd accept that all the reasons you list come into the reckoning, but would have reservations about the last: most modern research is in English and doesn't filter through.

First I think the term "modern" is misnomer, as I explained above ("post-modern" could be better!). But that's by the by. More important is that I think you are underestimating what has been going on for decades in Japan. Have you actually read Shimada? Have you actually read O Meien, apart from the bits I've posted here? And Berlekamp's work is certainly known in the Far East (though obviously only to the same sort of small audience that applies over here). It may well be, of course, that research is currently more active in the west. I haven't seen a mathematical discourse in Japanese for a while (or in Korean or Chinese, but I don't keep tabs on them as much).

I don't rate secrecy as a factor worthy of more than a milli-second of consideration. Recall that Genan's deiri classic Igo Shukairoku appeared in Edo times (1844 - see New In Go) and didn't attract any complaints from the other families. Josekis might be kept under wraps for important castle games, but that's just temporary gamesmanship. Deaths may also may kept secret temporarily to preserve stipends. Game records may be kept hidden (or destroyed) to save face - ever wonder why Dosaku won so many of his surviving games? But keeping boundary-plays secret? There's not a tittle of opening-preparation/cash/face advantage in that. If anything, people would laugh at a top pro who gave the impression he could only win with secret boundary-play counting techniques.

What I think may better explain lack of pro explanations is that Japanese amateur tournaments have traditionally been sudden death. Under those conditions there is little call for boundary-play techniques, especially those that are complex, which leads to little demand for pro instruction, which leads to pros not wasting their time giving it. Simple as that, though things might be changing in the era of fancy clocks. In similar fashion you don't get books on kiai. Even on things that matter to amateurs such as sabaki or tewari, books are like hen's teeth. It's all supply and demand in other words. And as in the real world there is always high demand for drugs: josekis and tsumego.

I understand that "modern" may not be the best word from a historian's perspective but I use a functional perspective in the context of usage by development of relative popularity.

I have not read Shimada's endgame texts; given your earlier hints and his rules texts (of which I saw some in English), he might have described some basics or maybe a bit maths. I have read O Meien's book (where "read" means diagrams, decrypting a few Japanese text symbols by positional text context, understanding the number calculations and finding that the contents must be very close to what you had described earlier about the book). I have had "read" some other Asian endgame books but as to calculations they were dull, such as stating the guesstimate 15 points, leaving the actual calculation of the move value as an "exercise".

Most research is a) combinatorial game theory by mathematicians (other than Berlekamp / Wolfe), b) my research and c) Bill's research. So far I see no indication of Asian professionals taking much notice of that.

Your reason about sudden death in Asia and related demand is valid.

"how practical is it to count precisely the values of moves during a live game?"

It depends on how complicated the calculation needs to be. how accurate it must be, whether verification of methodical correctness is done, the time limits, each player's general calculation skill and his skill needed for every particular problem. Except very great differences between players and different problems. In particular, expect many professionals to calculate (approximately) very much faster than almost all strong amateurs, like pros are very much faster on average in tactical reading.

Endgame evaluation is a skill that can be trained like other effortful go skill, such as tactics. A life and death beginner's tactical reading is extremely slow. A dan player is much faster. A pro yet much faster. Similarly, calculation speeds differ dramatically depending on training and talent. We should practise thousands of LD problems - we should also practise thousands of endgame evaluation problems (with correct calculations or only tiny approximation mistakes, verified due to solutions with trustworthy correctness or verified by one's own understanding).

In practise, calculations must be precise enough to make the relevant move decisions. "Roughly 7 is larger than roughly 5" can be good enough but it can be necessary to distinguish 4 1/3 from 4 1/4.

I understand from this that a trained and talented player is able to calculate the value of moves much faster and in many more situations than the average trained amateur, but often lacks time during a live game to make precise calculations so needs to make estimates?

Yes. Where the experienced, trained player's approximations are 1/4 or smaller whenever necessary.

I find it hard to recognize iterative deepening in your description of your method. In iterative deepening we do a sequence of depth first search that are limited at increasing but fixed depths. For example a chess program using iterative deepening and alpha-beta pruning might search depths 13-20 , first completing a depth first search to depth 13 then restarting and doing a new search to depth 14, and so on. This is very counter intuitive, especially when oversimplified. On the other hand, in your description you seem to pick some "best" sequences of different lengths that don't have tenuki, then add some sequences with tenuki. Your example is just too small for me to guess what the exact method is but I don't recognize iterative deepening in the example.



What I meant was that there are a plenty of lessons out there that teach how to compute endgame values, ko values (as in the size of the ko threats needed in principle) or some generic exchanges by taking into account the number of moves being played -- without invoking miai counting. I'd link some online lessons but it takes time to find and it is never really presented with any pretense of mathematical rigor. I remember some lectures of Guo Juan on big endgames, I have also learned a lot from her associates(?) Mingjiu and Jiuju because the three used to (and maybe still) give free lectures on Sundays that sometimes ventured into ko fights from real games. Michael Redmond also ventures into move values all the time but I have seen him mention miai counting, not sure if that is replies to the audience or if he is a (not so) secret practitioner. Because I mentioned ko I felt need for some example involving ko but I might just have left it at "you can easily find very good lessons on calculating move values that never invoke the miai method under any name".

I find such lessons to be very similar to your example (as understood by me): you read the likely lines for black and white, then update for possibility of tenuki which means accounting for the average of two options at some depths. The value stated at the top-level always seems to use the convention of "x in sente / gote". Many endgames over 10-12 pts tends to be too complicated to do exactly (if they involve multiple tenuki), and in that case is often more about timing than the exact value. These are never calculated exactly and the same applies for anything that is clearly biggest or clearly sente, it is not calculated because it is just played.

I have never found good systematic studies on ko fights. It sometimes seem like pros have a new way to fight a ko every now and then. With the latest bots we can easily confirm that famous players were mistaken at times.

Bots do not always play ko correctly. You have not found good systematic studies on ko fights because I think there still are none. Individual aspects are described but a systematic study is missing. (It is one of those things I want to do some time.)

"very good [online] lessons on calculating move values": ask yourself what they do not teach you. E.g., if they teach local evaluation, do they also teach global decisions better than just comparing local values? If they teach calculation of gote or sente, do they also verify by values that the type and therefore the values are correct? Do they actually show the iterative calculations of follow-ups? If they teach calculation of any local double sente's traditional difference value, this would indicate wrong teaching. Do they teach for how long play must be local, when it must be interrupted and do they verify the timing of interruption by values? Do their iterative calculations rely on sequences whose local alternation is interrupted at the right moments? Do they teach the value of positions aka count and the values of gains at all? I have yet to see teaching of traditional endgame values in more than basic shapes that would be correct and good.

It is also my experience that this is true. In doing my new Go Seigen book I came across something I had never seen mentioned about ko before. Utterly trivial and easy to understand once pointed out, but it had never occurred to me, and apparently to no-one else in the many books I have read.

We talk about getting compensation for a ko fight. You lose the ko but you then you pick up something somewhere else to compensate. But in this game Go's opponent taught the young Go a thing or two. He got his compensation first, well before starting the ko, and then lost the ko. But he also got a free boundary play from the aji left over, AND got some more compensation several moves later, on the other side of the board. It all sounds like a conjuring trick, but it was beautifully simple in practice. I'm sure it must have happened many times before and since, but it was new to me, and I've never seen it covered in a book on kos. The overriding impression I was left with was: another of the many things pros know and amateurs don't even suspect. It wasn't a new technique or deep calculation. It was just a matter of a different mindset. Too hard to teach because it's so hard to change people's mindsets?

I'd recommend a bit of caution with your nuances here. You will recall that Emanual Lasker, in the throes of planning to go to Japan, said “The Japanese haven’t as yet produced a mathematician who compares with the best we can muster. I am convinced that we can, ultimately, beat them at go, the ideal game for a mathematical mind.” His go equal, Dueball (also a mathematician), did go to Japan and had to play Shusai on 8 stones.

The Japanese player who was most instrumental in promoting go in Germany was Fujita Goro. He was later to meet Einstein at Princeton. This visit was arranged by guess who, the Princeton-based Japanese mathematician Yano Kentaro. One of those Lasker was ignorant of. I do hope the phrase he was only a "professor at Princeton" will not join "he is only a Japanese 9-dan" in the go archives I can't comment on anyone at such stratospheric levels in maths or go, but our own Charles Matthews, who taught at Cambridge and Princeton, may be able to say something about how good Japanese mathematicians were.

kvasir: I think my intended method is based on iterative deepening as you know it, though I am not familiar with i.d.

My main knowledge of miai counting was from Bill Spight's writings on SL and L19 so I was only changing the order in which move values are calculated, but the formula and method are the same.

Deiri counting requires comparing two board positions. For mathematical rigor that requires two settled positions where you know the score. So the plan is to compute all nodes to depth 1, then for each node play it out until the end of the game (alternating play is preferred but the method will converge whichever sequence of tenukis you choose, as long as you choose locally optimal play). Use that as an estimate for that node. Then compute to depth 2 and so on for increasing precision (Compare MCTS).

Another reason it may look unfamiliar was my use of the shortcut of assuming sente responses, a sort of lookahead. I didn't prove the moves were sente but in the first diagram, technically a full systematic method would require a proof.

If you want more details I suggest RJ's books.

Ko: I certainly cover compensation in my model, but only for simple kos. It sounds like you are talking about an approach ko or thousand year ko, though the model can be extended to cover that. Assuming I didn't make a mistake, you can look forward to it. Ko has been on my mind for many years since someone at the club complained they didn't understand ko and there was a debate between whether you should play large ko threats first or small threats. My intuition had a conclusion which I think I have been able to affirm.

I assume an ideal environment, so my setup is somewhat different from Berlekamp and Bill's whose papers on ko thermographs I still can't make sense of. However an RJ style of explicit simple gote environments is included in the model, though I admit I haven't managed to do so cleanly.

I do not doubt that there have been good Japanese mathematicians (e.g. Teigo Nakamura https://www.dumbo.ai.kyutech.ac.jp/~teigo/GoResearch/ studied some advanced corridors as combinatorial games), physicists etc. I doubt that those studying endgame theory during the first half of the 20th century made more than a) basic contributions (regardless of how important they might have been in themselves) or b) theory more useful for go players than researchers in game theory.

The question of when Japanese mathematicians joined the mainstream of European/American professional mathematics has a fairly definite answer, I think. Takagi Teiji (高木 貞治) 1875–1960 did pioneering work in algebraic number theory (my field) after a period at Göttingen before WWI. After the work of Kunihiko Kodaira in algebraic geometry, Lasker's remark could be disregarded.

Lasker of course didn't have it right about go. I would regard attitudes to the traditional "mind games" in the European context, up to around 1920 and the rise of Soviet chess, as a sort of cultural history that might repay study. A Japanese visitor to Cambridge once asked me for help the use of "game" in some writings of T. E. Hulme, the poet. The usage of ideas like "formal game" and mathematics as one, and "language game" in Wittgenstein, really are odd to those who spend considerable time on chess and go, etc. When they cross over from pastimes and entertainment to another level of "seriousness", it is easy to get confused as to what is going on. British and Central European attitudes, for example, appear to me to have been quite different. But that's a long story.

Go and the various chess-like games can be considered as examples of Formal systems as in mathematical logic. A formal system consists of well formed formulas, axioms, and rules of inference. In the case of go the well formed formulas would be the legal positions in go, the only axiom is the empty position, and the rules of inference are the rules describing legal moves. The theorems in this formal system would be legal positions that arise from sequences of moves from an empty board.