British Go Journal No. 56.  June 1982. Page 20.
In this article I introduce the idea of "average result" which is a powerful tool for counting the game accurately and for calculating the value of yose plays.
Diagram 1 ||
Diagram 2 ||
Diagram 3 |
Consider the three positions shown in Dias. 1, 2 and 3. The white plays in these diagrams are all worth one point in the sense that the difference between black playing there and white doing so is one point. Some texts distinguish between them as "one point in gote", "one point in ko", and "one point in gyaku yose", but it can make things much easier to measure them all on the same scale. This is done by calculating the expected result before and after the move. One advantage of this is that, if you're counting the game properly, the value of the move is the amount by which it changes the count.
How to calculate the average result? The simplest way is to consider a game consisting of nothing except a huge number of positions equivalent to the one in question. Doing this to Dia 1 we achieve Dia 4. Playing the "game" out we see that Black gets one point in exactly half the positions. The "average result" is thus that Black gets half a point in each such position.
Diagram 4 |
The vital intellectual leap is to think of this, not as a position in which Black "might get a point", but as one in which he already has half a point. This way of thinking makes counting easy, and lets positions involving kos and other complications be included in the count without difficulty.
Diagram 5 |
White 3 fills below 1
White 7 fills below 5
White 11 fills below 9
Applying the same arguments to Dia 2 we get Dia 5, in which White gains one point in just 1/3 of the cases. Thus the value of the position before White 1 in Dia 2 is 1/3 of a point to White. After playing White 1, the value is 2/3 to White (or one prisoner to White and 1/3 to Black, which comes to the same thing).
Diagram 6 |
In Dia 3 we have an altogether different situation. If Black plays here white will have to answer. The calculation 'game" now looks like Dia 6. The expected result is that Black gets all such points, so white 1 in Dia 3 is worth a whole point, i.e. twice as much as in Dia 1, and three times as much as Dia 2.