KarlKnechtel: This is the original board position, we are asked to give the best option for black - a, b, or c. I will attempt to calculate the miai values of the moves.
is White's sente. and eventually (or the other way around) are forced and the local tally is 4 points.
Black threatens x with a net result of 6 points. This is thus sente, so white will play 4 at x to limit the result to 4 points.
Since an exchange by both players in either order leads to 4 points after , the tally is 4 points after .
Black has this option if white plays first starting at a. ( at 5 is of course inferior). and are sente so this is the limit of the analysis. The result is -2.
Alternately, black starts a ko. Black has had 2 stones captured, but has a solid point of territory and is about to begin the capturing. Black reaches this position in sente I believe.
Call this score y. If white plays next, the score is -1/3, or x, whichever is less (favourable to white). If black plays again...
If black plays yet again, and also connects, the score is 4 (3 white stones captured, plus 3 points territory, less 2 black stones captured). If black and white trade here, the score is z again.
Putting all that together: When white chooses to simplify the ko in all cases, I get a value of 28/9 for z, and hence -29/36 for x. When white keeps the kos complicated (does not play passively), I get -8/5 by solving the simultaneous equations in x,y and z (and assuming that sente doesn't apply to ko since you don't necessarily get to respond anyway. I suppose this ko could depend on who is komaster, too).
Putting all that together, black will want to choose the ko in response to at a (in the original diagram), and white will play the ko aggressively, yielding a score of -8/5 after -.
Finally, if white is allowed two moves in the starting diagram, we get this situation. Black at a is sente and leads to a score of -3. White at a ends up with a score of -4 points.
Putting everything together, the tree becomes consistent with a value at the top of (-7/2 + -8/5)/4 + 4/2 = .725, and thus a miai value of 3.275 points for a play at a. Playing here is not sente for either player.
Now then, let's try the miai value of ''b''?. :)
Bill: Nice try. :-) However,
If , White does better to avoid the ko with . could also be at a.
Karl Knechtel: I could have sworn I thought about a line of play like that, but didn't think to analyse it. :) So the tally is -2 here right? That would make the miai value 3 3/8 instead, with the original score being 5/8.
Bill: The tally here is -2. If the Black sagari is correct, then the tally after it is +4, as you figured. :-)