traditional analysis of small endgame
Charles Matthews This is really a 'toy' analysis, but worth writing out.
Suppose 'small' here amounts to: two-point and one-point gote plays^{[1]} and one-point sente plays for Black and for White^{[2]}. We take 'sente' to mean absolute sente: requiring an immediate answer. This takes all the subtlety out of the argument, but means it becomes very clear what should happen ^{[3]}.
We assume Black is to play. Black can play all his sente plays before stopping to think.
Firstly, we can remove from consideration pairs of miai amongst the gote plays, since with best play they will always be shared between the players. That means we need reckon only with at most one two-point play A, and at most one one-point play B.
Second, if there is no sente play for White, there is no difficulty at all: Black takes the larger gote play if there are two.
So, we assume there is a one-point sente play S for White, and Black must decide whether to play reverse sente at S or start on the gote play(s).
Thus there are four cases:
- S, A, B
- S, A
- S, B
- S
In cases 3 and 4, the absence of a two-point play means that Black might as well play S anyway^{[4]}. So it is just cases 1 and 2 to analyse.
In case 2 Black actually loses a point by playing reverse sente; that's because White can do nothing useful with sente in that case. In case 1 it makes no difference: either way of playing for Black, starting with A or with S, makes two points for Black versus two points for White.
See CGT Analysis of Same Small Endgame
[1] I.e. plays of miai count 1 and 0.5.
[2] We can leave out double sente since the first player, Black in this case, will have no hesitation in playing any such plays.
[3] The non-traditional method based on atomic weight copes with open-ended plays gaining one point per turn, that aren't treated as sente.
[4] In case 4 Black can lose a point, like this. Remember that we have eliminated pairs C, C' that are miai. If Black forgets about that piece of reasoning and plays in C of a position S, C, C', then White should clearly play S in sente and then C', gaining one point. Therefore if Black employs a 'random strategy' (actually, assumes that the parities of numbers of two-point and one-point gote plays in the game are random variables determined by tossing a fair coin) of playing reverse sente in S if there is such a one-point sente for White present, this should lose and gain a point equally often. It is in a sense as good a strategy as always playing a two-point play if you can find one. But it is evidently a worse strategy than counting and getting the parities accurately; which is the only guarantee of getting tedomari, whether that is a reverse sente (case 4) or a two-point play (case 2) in the particular case.