Corner 1-2 connection value

HolIgor asks: As you know I don't like to defend. Quite often early in the yose I would leave this unplayed.

The marked stone is in danger  

Assume that White stones are othewise alive. Now, how much is a is worth?

The question is not so easy because it involves ko and thus depends on the whole board position. How is this counted?

Safe defense  

Ko defense  

I will try to invent something myself later, but does anybody know the answer?

Yes. :-) --BillSpight

HolIgor's count:

With a white move  

Local score 10 (marked). Move value 0.

With a black move (safe defence)  

The position is unfinished. Local score 4 2/3 (Black captured a stone). Move value 1/3.

So the connection value was (10 - 4 2/3)/2 = 2 2/3.

With the ko defence the value of the ko should be determined.

If it is all or nothing: 10/2 = 5. The value of ko is 5 points.

It seems that before fighting this ko White has to have enough five-point threats.

As the result White wins something compared to the safe variation if she is ready to win a ko at with a 2.5 point ko-threat by opponent (or less). Quite tough ko conditions.

PS. I did not play costly atari today :).

BillSpight: To evaluate this position we can use Prof. Berlekamp's concept of the komaster. The komaster has just enough primary ko threats to win the ko.

First, suppose that Black is komaster. Then White should avoid the ko, and play the 'safe defense'. As HolIgor points out, the play has a miai value of 2 2/3.

Next, suppose that White is komaster. White plays the 'ko defense'. The ko is big. How do we evaluate this position, then?

White will win the ko. Does that mean that Black (the koloser) should not start the ko?

Actually, it is just the opposite. Black should start the ko as early as he can afford to do so. When White fills the ko, Black gets compensation by playing a move elsewhere. The earlier Black starts the ko, the larger that play will be.

How early can Black afford to cut and let White make the ko?

Suppose that he does so when the size of the play he gets in compensation for the ko (the ambient temperature) is three points. If White wins the ko, she protects the cut in exchange for a play worth three points. It is better to play the safe defense, and, instead of taking Black's cutting stone, play a three-point move herself.

Safe defense (ii)  

If the ambient temperature is somewhat above 2 2/3, Black can play Black 1 with sente, even if White is komaster. Then at temperature 2 2/3 someone will take a stone here.

So White only sets up the ko when the temperature is 2 2/3 or less (as a rule). It is the 'safe defense' that determines the size of the play. That means that, even if White is komaster and makes ko, the play has a miai value of 2 2/3.

A similar problem

If the followup isn't so simple...  

Depends on komaster?  

cynewulf: If Black is komaster and White plays b, Black can make a big ko with a.