Basic Endgame Theory/ Discussion

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Sub-page of BasicEndgameTheory

Play double sente points first

Dieter: As the basic concepts of combinatorial game theory are not known to a wide audience, not even within the go community, I suggest that we stick to explanations using basic go terminology, such as sente, gote, and the value of plays. I think SL is a wonderful place to proliferate the CGT concepts, but as long as most people are more familiar with the theories outlined in James Davies' and Bozulich' endgame books, it is probably best not to put them on the basic pages.

Bill: The basics of traditional go theory are the same as CGT. One CGT term, temperature, has been borrowed by Western go players, as evidenced by posts on over several years. The primary meaning of temperature in CGT is the same as that of miai value in traditional go theory. However, the related concept of ambient temperature, which applies to the whole board, is the one that has caught on with go players. Properly speaking, double sente is best explained by reference to the ambient temperature. Otherwise, it makes no sense, as the discussion in the Ogawa and Davies book shows. Kano also struggles with it.

Rakshasa: It seems the text assumes the reader knows that the value of the threat is larger than any other moves on the board.

Captures which are rarely worth taking

The simplest capture is a stone in a ko which does not threaten the group - this stone has no value but one half a point and is not worth taking.

A white stone string (B3 fills up)  

aLegendWai: Let me explain more. The above diagram refers to the capture of a stone in a ko which does not threaten the group - this stone has no value but 0.5 point (deiri value) (d). (Note: notice the highlighted words! Important! Otherwise it is not simply worth 0.5 d-point)

It is said to be 0.5 d-point because it gets 1 d-point prisoner for the first move. But it is not the end. The player has to fill the hole up. It is the second move. So the calcuation is 1 / 2moves = 0.5/move.

0.5 d-point is the smallest value in Go world (?). If you ever make such move, say, in the middle game; you should do self-examination or heart-searching (joking). :D

Bill: This ko is commonly called a half point ko. However, that is not its theoretical value. Its miai value is 1/3. aLegendWai's derivation is incorrect.

I suppose that it is called a half point ko because, in a ko fight, if you do not know who will win, it is a 50-50 proposition whether Black will win one point or not.

Rakshasa: Using a lot of words does not make the explanation clearer. The 0.5 point does not have anything to do with 50-50, 1 / 2 moves = 0.5/move or similar. If you are in the endgame and there are an odd number of half point ko's on the board, then both players will fill up equal many and then fight for the last one. Thus it's not 50-50 but rather who has the most ko threats. Calling it a half point (or sometimes a 3/2 point ko) is just simpler than saying 0 points if I lose the ko or 1 is I win it.

On a side note; the explanation on the main page seems misleading to me. The title should be "Captures which are rarely worth taking early". And the text for the half point ko;

The simplest capture is a stone in a ko which does not threaten the group - this stone has no value but the possible capture of a stone. The player starting the ko gains either 0 or 1 points.

I have always had a problem with this page. To start off with 'gote = half sente' assumes a hell of a lot really: certainly it presumes both some idea of miai counting, and the sophistication to understand why miai counting might be useful, even though it frequently gives wrong answers in practical play.

Anyway, I think 'basic' is a long way off base.


Possibly true. What might be more useful is to highlight the importance of sente over gote in yose. Also Mutual Damage is an important principle for beginners to learn in the endgame. The page does seem a little too packed with confusing buzzwords and phrases though. It all depends on what level you are pitching to of course. Are we describing basic as sub 20kyu, sub 10kyu or sub dan?

I suppose you would want to sneak in the importance of good shape as well ;-). IanDavis

Naughty Ian. Charles

Bill: I contributed to this page, but it was here before me, and, I think, much more confused and confusing. If anyone introduced the idea of miai counting to this page, it was I. However, people were trying to use deiri counting like miai counting, and that was a mess. So, yes, I think it could use more than a WME.

Well what level do you want to pitch toward? For real beginners you could be highlighting

  • sente
  • mutual damage
  • eyes (not making 3 etc)
  • counting skills
  • a discussion of capturing or ko fighting

If you want an introduction to endgame theory maybe this is totally different.


Bill: Well, there is introductory material, then beginner, intermediate, etc. Before sente there is gote, no? Sente is already a sophisticated term. Even strong players can be unsure about it. And once we are talking about theory, I think that we are already at intermediate level.

Later: Charles, you remark than miai counting frequently gives wrong answers in practical play. That is true, in terms of outcomes and choice of play, if you always play the move with the largest value. But you aren't suggesting that deiri counting is better in that regard, I trust.

Charles It doesn't much matter what conventions you use, in considering small examples; one can verify the results by hand. So in 'theorising' one is trying to counter the problem of scale, really. That's one point; I thought I'd post some examples this morning, but one indeed does need a clear head.

It seems that pages with 'basic' in the title generally have problems.

OK, this is the sort of thing I had in mind. I'm trying to connect with the kind of models I regard as truly basic. Apologies if I have misplaced a factor of two, yet again. The wacky terminology traditional here has a lot to answer for, In my view.

The basic question, as Bill knows, is to know when to play gyaku yose?. That is, the decision is of the type, play reverse sente, or play the largest gote, in a given position (i.e. whole board position). Let's look at a concrete example.

Suppose there is an 4-point sente play for White, and it's Black's turn, in a situation where all the other plays are gote. Yes, let's use deiri values for those: say they have values 8, 6, 5, 3, 2, 1. If we take as baseline the mean position, then for example the 8 point play yields +4 for Black if Black plays there, or –4 if White plays there

Black has essentially two ways to play here. Black can play reverse sente, White will take the 8 point gote, Black the 6 point gote and so on. Black's result will be then

(- 8 + 6 - 5 + 3 - 2 + 1)/2 or –2.5 relative to the mean.

On the other hand Black can take the 8 point gote, White will play out the 4 point sente and then take the 6 point gote, etc. Black's result is

(8 - 6 + 5 - 3 + 2 - 1)/2 – 4 = -1.5.

Therefore this is a case where the reverse sente is not correct play. It is easy to see the structure of the algebra here: we should compare

-(a – b + c – d + e - …)/2 with
(a – b + c – d + e - …)/2 – s;

or equally

(a – b + c – d + e - …) with s.

Now in cases where there are few endgame plays we may be able to see what the sum is. If not, we can guess that it comes out like a/2. In which case we are comparing a/2 with s; or a with 2s. That is the primary meaning, I say, of regarding gote plays as counted as half of sente plays: it takes us directly here.

It doesn’t however, recognise that the modelling assumption is quite likely to break down, if for example a is as large as 20. There is likely to be a sparse set of gote plays of value 19, 18, …, down to about a dozen. This would cause a bias in a real game.

Bill: Thank you, Charles. :-)

To add a tiny bit, let's say that instead of making an approximation with a/2, we decide to use a - b + c/2. Doing so is equivalent to taking c/2 as the ambient temperature. If we could read everything out, we would. In any given real game our choice of temperature is pragmatic, based largely on our limits of reading.

Gronk: I add this comment only with the motivation that it may help the stronger contributors (Bill, Charles) MWE in a more accessible way (to strong kyus, say). I don't understand and can't really follow most of the endgame or counting theory available here. I barely know what miai counting is, for example. But, for a while now it has been more-or-less clear to me that the "sente worth twice gote" or "reverse sente worth twice gote" type advice is a more-or-less continuum approximation. That is, it only applies when the possible moves have slowly decaying value (e.g., 10, 9, 8, ...). I think this is what Charles has said above. Based on this idea, I only bother to look and count what I believe are the 3 largest available endgame plays. In doing this I end up evaluating more than three because I can't tell at a glance if A is bigger than B etc. But typically I can see what are the five or so likely candidates. This is good enough. Then, I can apply rules like, "well if I let him have that big gote play but I can get the next biggest sente and then the next gote I come out ahead." Or, more simply, I can allow my opponent to take the gote worth X if I can find a sente + gote worth at least X,etc. Embeded in this thinking must be some kind of rudimentary theory that probably isn't correct but is WAY better than not counting. I've won a lot of close games with this heuristic method. It doesn't deserve to be called theory but it may be the best a mid-level amateur can do and, thus, might be worth trying to explain.

Bill: Gronk, what you are doing sounds good. Let me repeat it back just to make sure I understand you.

You identify and calculate or estimate the sizes of what look like the five largest plays or so and then try to find the best line of play for them. Is that right?

If I've got that right, then there is a step you can take to improve your estimated results. Let's say that the miai value of the smallest play is V. Then you can add V/2 to your estimate if you have the move at the end of the sequence, or subtract it if your opponent has the move.


Estimating results  

Let's say that this is the sequence you are considering. The miai value of B1 is 5, and the miai value of W2 is 4. In addition to the count for the rest of the board, the local count is +11 (for Black). Since you have the move, you can improve your estimate by adding 4/2 = 2 points to your overall count.

Gronk: To sum up, in a way I understand it, this looks like a method for estimating the value of sente, which can be tacked on to whatever stopping point I choose. So, if I am evaluating the most profitable way to play out what seem to be the five biggest endgame moves, I should tack on half the value of the last of them to whichever side ends with sente. Sounds reasonable. But, actually, I don't really figure all the permutations (or some short-cut equivalent) of what seem to be the five biggest endgame plays. Really I try to locate the three biggest, but in doing so I sort-of catalog the five biggest (typically) 'cause I can't tell whether A is bigger than B if they're close so I end up evaluating more than I need (so to speak). Then I do a bit of greedy logic to tell whether I think it best to take the big gote or play the reverse sente. This also helps me evaluate whether move X is really sente ('cause of the size of the gote follow up, say). I usually stop when I feel like I'm making a pretty smart decision, but by no means am I always certain it is optimal, even with respect to the five plays I've cateloged. My point was that I know it is not theoretically sound, but it is quite good at my level and far better than guessing. Typically my opponents are drawn to what seem to be sente plays but, in fact, are not (because the follow up is gote and not as big as the next available gote move). So, I win a lot of close ones this way. The key seems to be to know how big the follow up to the "sente" move is relative to the available gote moves.

Jonathan Cano: underlying this discussion is an unspecified set of objectives. Two somewhat competing goals to consider are:

  1. correctness is most important
  2. real world applicability is most important.

Perhaps we might bifurcate the discussion along these lines.

Gronk: Jonathan Cano's suggestion sounds sensible to me. I'm not too interested in correct theory beyond what I can reasonably apply in a game setting. Another thing I meant to add (along the lines of practical application) is the following. When time permits (and usually it does), I count the score very frequently. I only work as hard as needed in the end game to make sure I win. Well, usually I try a bit harder because I want the practice. Or I try extra extra hard because I am angry at having to play it out :) :) But, at some point it might come down to working for five minutes through a bunch of detailed calculations and permutations or just getting on with it and finishing the game. Now what is typical is that I'm a bit ahead or a bit behind as the end game starts. I'm a bit better than typical (for my strength) at endgame so I catch up or pull further ahead during the early endgame. Then, at some point, I can relax and coast to an easy win.

Charles In the case of a club-level player, there is probably ten points to be made every game by upgrading your endgame to dan standard. Plus there will often be some tesuji, or chance to kill (or at least use life-and-death knowledge to good effect). But of course it is time-consuming; and as a practical matter playing more ambitiously in the middle game may be the way to improve, as a choice of where to use time.

Bill: Plus another 10 points or so for the average dan player.

Charles A theory worth the name ought to be concerned with the truth of things; which is not to say that heuristics and rules-of-thumb have no value - most of us play go that way.

Bill: Traditional go endgame theory is almost all concerned with estimation and heuristics. Tedomari, which is mainly concerned with correctness (as I think you mean it), gets only a nod in the textbooks.

HolIgor: This is what I think. You have to take the score of all gote moves. If it is zero then the gote part is reduced to miai and you may play reverse sente. Otherwise you just lose the value of the sente for nothing. Let there are gote plays with miai values of 5, 5, 1 and a reverse sente play with the value of 2. You cannot play 5 gote, because the opponent plays 2, then without losing tempo, 5, and you are left with 1. You have to play 2 point reverse sente and then let the opponent win 1. Change 5 to 20 here, if the remaining plays are 20, 20 and 4 then the value of the reverse sente has to be compared with 4.

The above rule of reverse sente about a half of the largest gote is true for the case when about the half of the largest gote is the balance of the gote game, which is quite often not true, because all miai gote plays have to be eliminated first.

I am not able to make the balance of gote part in my games for sure. But that does not reduce the correctness of the approach. Don't unbalance the gote part.

And in real games one can safely assume that the gote part is balanced in about half the cases. Or, actually, this is quite a simple math problem. Suppose you have 20 random numbers in the range from 1 to 20. What is the balance? You have to generate the numbers, sort them in decreasing order and then take plus, minus, plus, minus to the end. I am not sure the even statistically the balance would be half of the largest number. There is a lot of maths pros in the library. Is the solution known? Otherwise I am going to run a test.

Charles I believe the obvious mathematical result to support this is true, and not hard to prove. By removing miai, we can formulate the question like this: consider random sums formed, for example, with numbers 1, 2, ..., N , where each number is present or absent with probability 0.5. Form in each case the sum

S = N - a + b - c + ...

where we assume the largest value N is definitely present, and

a > b > c ...

represents the ordering by size of the other numbers. Then the expected value of S is N/2.

Actually, I wrote about this at stacks of coins; before getting fed up of the whole issue there because of the comment that it was 'trivial', page title was misleading - the whole picky bit that has infected SL from time to time.

ilanpi: Here is a proof that the expected value of a - b + c - ..., is N/2, where a > b > c > ... are taken with probability 1/2 among the integers 1,..., N. Since each sum is equally probable, the expected value is (S1 + S2 + ... )/K, where S1, S2,..., are all the various sums, and K is the total number of sums (= 2^N). The result then states that S1 + S2 + ... = N * K/2. This is proved as follows: For each sum S which does not include N, one associates the sum constructed by including N as an extra term in S. This new sum will have value N - S since including a new largest term N to the sequence in S will change the signs in the alternating sum coming from S. Adding S and N - S gives N, so the total sum of sums is N * K/2, since there are K/2 such pairs, the correspondence I constructed being a proof of this last fact.

This argument also generalizes to alternating sums taken from a sequence A > B > C > ..., showing that the expected value is A/2. It also follows that the expected value in the problem given by Charles above is (N+1)/2 = N - (N-1)/2, since you are subtracting from N the problem with summands N-1,...,1.

Charles Yes, I see that there is a slight glitch in my formulation. When one says 'a play of value N is definitely present', there might be 2 (or any even number). Which means that a could be N, rather than < N.

Bill: It's easy to show with only two numbers.

What is the expected value of the difference,

 S = n - m

where n and m are positive numbers and m is selected with equal probability among the numbers between 0 and n.

The expected value of m is obviously n/2, so

 E(S) = n - n/2 = n/2

It is also easy to show asymptotically.

 S = n - m + p - q + ...

As the number of numbers approaches infinity, the expected value of m approaches n, and the expected value of S approaches

 E(S) -> n - (n - a + b - ...)


 E(S) -> n - E(S)


 E(S) -> n/2

Actually, it was always n/2 when the number of numbers was even.

HolIgor: Great, now we know that the number is really a half of the value of the laargest move statistically. Of course, it can be either larger or smaller than that. At least, and this can be easily evaluated, we have to exclude miai at the highest level in the case when there is a gap in the distribution.

Note to beginners. As far as I know, the above statement is not true literally. Simply speaking, the value of sente is determined by the value of next endgame move.

See Discussion of the value of sente and gote plays for the behind meaning of this statement.

Bill: When this was on the parent page, I objected strongly to aLegendWai's authoritative presentation of his opinion. But, since the Discussion page is the place to express opinions, I withdraw my objection (and delete it and the ensuing discussion). If I do not express my disagreement with something he (or anyone else) says, that does not mean that I agree. ;-)

One point that is still worth making:

Discussion of the value of sente and gote plays is itself confused, and would be confusing to beginners. It may have some value to more advanced players, as a kind of example of classroom discussion.

3 point sente  

aLegendWai: Speaking in terms of deiri value(d), the first move is worth 3 d-points (10-7). Don't make it wrong that it is worth 10 d-points. Try to think if you don't respond it, how much d-points you will lose?

The second move is worth 7 d-points (7-0). And the above endgame is W sente.

If B has another endgame play which is worth more than 3 d-points, B will not play here.

So you should see the importance of calculating d-value of endgame accurately. It is also related to the calculation of the value of sente and cost of playing elsewhere.

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Basic Endgame Theory/ Discussion last edited by on January 27, 2015 - 05:27
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