Sente gains nothing, part 2: Fair exhanges

When estimating the score of an unfinished game, it is useful to know how to value unfinished territories fairly. Many amateur players prefer not to go through the effort, and instead spend their time in locating the largest move on the board. I would however argue that without evaluating the board position at all, the largest move will be challenging to find.

Dia 2
Dia 1

Dia 1 shows a common endgame position. Assuming that black and white stones are unconditionally alive, how many points do Black and White have?

If Black were to maximise his territory and minimise that of White’s, he would play 1-3 as in Dia 2. After black 3, Black and White would both have five points of territory (marked with △). “Five points each” is not a fair estimate of Dia 1, however—note that in Dia 2, Black has played one more stone than White!


In Dia 1, it could be just as well that White had the turn to play first, after which Dia 3 would follow. Now Black has four points and White six, and White has spent one more stone.

Dia 4
Dia 3

In Dia 1, both players have equally large endgame moves available. As the reader might have guessed by now, for a fair estimate of territories we should then imagine an even number of stones on the board—namely, the stones marked with △ in Dia 4. Thus in Dia 1, Black expects five and White expects six points of territory.

Let us return to the homework from Part 1, shown in Dia 5. How many points should Black expect there?

With the introduction above, many readers may now want to try an “even exchange of stones” as in Dia 6, thus getting the result of one black point of territory. This is a good try, but the 1-2 exchange is in fact not fair!

Dia 6
Dia 5

For a proper analysis, we should duplicate the position in Dia 5, as I have done in Dia 7. There, white 1 and black 2 are of equal size. After black 2, Black has secured two points of territory, and as we remember from Part 1, Black expects half a point more at △. Thus, in Dia 7, Black expects 2½ points. In Dia 5, Black logically expects half the territory of Dia 7, in other words 1¼ points.

Dia 7

Continuing from Black’s expected territory of 1¼, let us count the size of a move at A in Dia 8. If Black played at A, he would increase his territory from 1¼ to 2, making black A a ¾-point move. White A on the other hand would reduce Black’s territory from 1¼ to ½ points, being similarly worth ¾ points.

Dia 8

When counting expected territories or the size of moves such as A, it is in fact not necessary at all to go through the complicated shape duplication process of Dia 7! In an unfinished endgame situation, the following two formulae are all you need:

  • The expected territory is the average resulting score between Black or White getting to play the endgame move first. In Dia 8, we have (2+½)/2 = 1¼.
  • The value of a single endgame move is half the difference in score between Black or White getting to play the endgame move first. In Dia 8, we have (2-½)/2=¾.

Make sure you do not confuse the two!

Dia 6 (repeated)

The reason why Dia 6 is not fair is that white 1 is a ¾-point move while black 2 is a ½-point move. When Black answers white 1 with black 2, Black takes a ¼-point loss in the exchange.

Black 2 merely secures a half-point expected territory into one point of secure territory. White 1 does the same to the black potential territory formerly under black 2, while also preparing a follow-up move at 2 if Black does not answer. This follow-up possibility is the difference in value between white 1 and black 2.

For your next homework, I would challenge you to estimate the black and white territories in Dia 9. You may again assume that both groups are unconditionally alive. This time I have not explained all the theory necessary for the correct solution, but if you take your time, you might get within ⅓ points of the actual score!

Dia 9

18 thoughts on “Sente gains nothing, part 2: Fair exhanges”

  1. value *of

    Doesn’t the “expected territory is average of first-move territories” somehow assume there are no sente on the board? Is “endgame” same as all gote?

    1. Indeed, there is a lot more content yet to cover; as of my current plan, I will introduce “sente” in Part 4. The content that was finally split in parts 1-3 used to be a single text in my head, however, so I wouldn’t be surprised if another redesign occurred!

  2. Black starts with decend and get 9 points. White has 5 if responding, but follow up is 1/2 and its followup is 1/4 thus white expects 4 1/4.
    White starts with hane and get 5 points. Black would get 7 with decending, but white second have follow up would gain 1/3 due Black starts ko when responding. If not then white gain full point so expecting gain 2/3.
    Thus Black expects 6 1/3.

    Black expects (9+6 1/3) = 7 2/3
    White expects (4 1/4 + 5)/2 = 4 5/8

  3. I find that 12 copies of the position magically creates a situation where the optimal outcome is independent of who plays first… and the exact numbers I get are 7+5/6 points of expected black territory and 4+1/2 points of expected white territory. A play here should be worth 1+2/3 points to whoever gets there first.

  4. My try in prose:
    I think the right moves for both would be hane. Now i’ve got to build a tree to compare the territories when black hanes and white does so.

    And i cant get the concept through, as i gpt problems with pragmatics i want to apply. If i would take the concept down the road complete i would have to imagine possiblities where black plays all the moves and white tenukies 3 times so that white got no territory and black does get all it could have (9) and then build the tree up with all possibilieties. and average down to the stem. But i really don’t want to do that. I don’t think it would be valid also. If black hanes thats a move id estimate at around 1 point value, If white thereafter went down and black tenukied, the next move would be of greater value than one because after white gets the hane stone and black defends, there is some fraction of a point to be gained in the ko over the single stone. So i dont want to consider variations in which black play a move with a certain value and then doesnt play a move that has a greater value. So ill make hard the fact that black will answer a white atari on his hane stone. And as white also thinks my way, he will descend. So the first part of the equation where black play has only one outcome i consider: 9points for black and 4 points for white. I know that im reaching far inteo the planned schedule here, i tried to do it considering all possible variations, but that just felt wrong.

    The other prune white playing first doesnt work that gentle.
    The first variant is black tenukiing three times: both have 5 points then. And black wont tenuki any more cause h can atari 4 white stones, a thing white would answer. If black hadnt tenukied the last time it gets complicated, cause there is the one stone corner ko. I forgot the right fraction that has to be applied here, if it was a half or a third i think a third. Also i need that short cut here because i can’t evaluate ko and tenuki until Final Justice. Ok with black tenukiing 2 times i have 5|5 and 6_¹/³|5¹/³ prunes averaging one 5_2/3 | 5_1/6.

    So now to the variation in that black tenukies only one time. Taht situation looks even more complicated to me: The easy variant is the answering of white connecting its hane stone, its the same variation as before and the territories are 6_¹/³|5¹/³
    White tenukiing results in a ko. I have no idea how to evaluate the ko situation. White could just fight the ko, and this situation can’t be resolved without convention how to evaluate ko fights. I can’t decuce from teh value of the moves that there are no thereats around cause somebody of the two could have a dead group ko factory on the board.So i have two variants here black winning ko fight: white answers, white wining ko fight –> the situation before the tenuki. Goddamnit. an unbounded recursion. It sure is bounded on board but dependent on the unknown moves that white would spend before tenukiing. Ill ignore that stuff right now. It is a fraction cause it can’t be grater than the difference bewtween the move white tenukied for and the connection of the hane stone.
    So white has connected and we have the exact same situation like in the first prune plus some unknown fraction to be possibly gained by white white with tenukiing: 6_¹/³|5¹/³ and 6_¹/³|5¹/³ averages on …

    So i have both prunes for black tenukiing one time: 5_2/3 | 5_1/6 and 6_¹/³|5¹/³
    averaging on 6 | 5_¼

    If black doesnt tenuki at all it is easy he will descend to 1-2 point there is no move thereafter that gains anything, as white must connect in the end and the territories are 7 | 5

    So the white first branch gives 6.5 | 5.125

    Overall we have 7.75 for black | 4.625 for white, though white still could have some points tenukiing and fighting a ko in the one complicated variation.

  5. oxxxxx
    o. .oox

    In this ascii-diagram I would in some sense argue that black’s (x’s) expected territory is 4 rather than the (5+2)/2 = 3+1/2 points that the formula presented in the article yields. A side effect of evaluating this way though, is that a move here will be worth 2 points for white and only one for black. I haven’t got any hard proof for my viewpoint, but it seems somewhat consistent with what I think is the practical meaning behind these fraction numbers.

    1. Morkus, indeed you are right! I will be covering this part of endgame theory in the coming Sente gains nothing essays.

      In a nutshell, the defending side’s expected territory can never be lower than what they’d end up with if they just answered the opponent’s move (thus, rather than accept the 3.5 points, they’ll answer and have 4). Your board position is an example of a “privileged white endgame”.

  6. Value of move = 3.33 points
    White expected territory = 4.5 points
    Black expected territory = 7 and 5/6 points.
    If black plays first black wouldn’t descend, but hane.
    if white plays, the hane is gote, then hane answer exchange, later connect capture ko is also an exchange. So, black has 2/3 of the ko completed for a value of 6 + 2/3 points if white plays first.

  7. For the value of a move in the homework, 3.33 is technically right if you compare the difference between Black or White playing a move. The current score of the game, however, doesn’t change by 3.33 by that single move, but only by the half of that, ie. 1.67.

    In Dia 6 in this post, if you call black 2 a 1-point move, then 3.33 for the homework is correct. If you instead call black 2 a 0.5-point move, then 1.67 for the homework is correct.

  8. I get 1+3/8 for the value of the move.
    The expected territory difference is 3+5/8
    Black can bring it to 5
    White can bring it to 2,5
    My variation tree has 8 nodes

    As predicted, my answer is within 1/3 of Ten’s answer (which I assume is correct :) )

    1. Your value for after white’s play is the source of the problem! White’s move does not bring the position to 2.5, but to the average of 2 and 1.333, i.e., 1.667.

  9. Hi Ten,

    Thanks for pulling together these explanations. I’ve just stumbled upon them, and they have been catalytic to breakthroughs in my understanding of endgame.


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