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 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.
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!
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.
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.
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!
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!