Thanks for the exercise. Did I get it wrong?
Well B plays first with one more liberty so I expected them to win the shape unconditionally rather than getting to a ko even if it is 1 step ko. However, you are right that deciding between them is difficult.
- Click Here To Show Diagram Code
[go]$$W
$$ -----------------------------------------
$$ | . . O . 5 . 2 1 X 3 . 6 4 . . X . . . |
$$ | . . O O X X X X O O O O O X X X . . . |
$$ | . . . O . O O O X X X X X X . . . . . |
$$ | . . . O O O . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$[/go]
This is a 1 step ko in B's hand in gote, but B's forcing moves at
,
have already gained something for B. The swing is 26+kotake = 7 pts? (territory). So under NTE, the gain of the next move is 27/5 = 5 2/5. If we count B's win as zero and count W's gain relative to that, the count is 5 as long as T>=5.
Note that if
played elsewhere, then B could connect at
to win the semeai but with 1 less point (i.e. a count of 1), but might also risk playing at
themselves for 1 more local point but passing in the ko fight. So the move
commits to profiting from the ko since W has already lost almost 1 point from starting the ko. This is why they say prepare (manufacture) big ko threats before starting a ko (because starting it loses you points locally if you lose).
I think that
is sente on
and
on
because they are equal liberty points and W is threatening to move to a direct ko and a count of up to 2/3(27)=18 points. Note that
is not necessary and if not played, allows B to win the semeai to a count of zero. However, note that without
, B could have played to a count of 2 points at best.
The average is before
is a count of 3 1/5 with
having gain 2 1/5.
- Click Here To Show Diagram Code
[go]$$W
$$ -----------------------------------------
$$ | . . O . . . 6 1 X 5 . 3 2 4 . X . . . |
$$ | . . O O X X X X O O O O O X X X . . . |
$$ | . . . O . O O O X X X X X X . . . . . |
$$ | . . . O O O . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$[/go]
B gets a 2 step (B's favour) ko in W's hand in gote.
The swing is now increased to 28+1 = 29, but the zero point hasn't shifted.
At temperatures above the ko gain 29/8 = 3 5/8, this is close to but slightly worse for B than a 1 step ko in B's hand since 2K/8 > K/5. This is 2 factors, both worse for Black.
The count is 29/4 = 7 1/4
Furthermore, W has the option of not playing
, and just shorting liberties, moving to 2 directly, because B hasn't captured a stone to occupy
yet.
Under NTE, when T is the gain of the first and smallest ko involved, an n-step (your favour) ko in opp's hand has the same count as an (n-1)-step (your favour) ko in your hand. I don't know how this varies with temperature. But as T goes down, smaller ko threats are required to maintain the ko since they no longer need to balance the value of sente. Local profit matters more than sente and hence winning the ko is more important.
Is there any situation where the 2 step ko is better?
last check
- Click Here To Show Diagram Code
[go]$$W left of 4
$$ -----------------------------------------
$$ | . . O . 9 . 8 1 X 4 5 3 2 7 O X . . . |
$$ | . . O O X X X X O O O O O X X X . . . |
$$ | . . . O . O O O X X X X X X . . . . . |
$$ | . . . O O O . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$[/go]
This is like a golden chicken standing on one leg shape and doesn't look good for W initially. However, upon further inspection, B has gotten a 2 step ko (at
) in W's hand in gote. Although the zero point has been avoided since B occupies
in this variation, W has made more inroads into B's position, the swing has increased to 29 2/3, so this is probably marginally worse for B than the variation above, or at best, hard to distinguish.
I almost completely concede this one. The one-step ko is better for B in the large majority of cases.
One last try: when could the 2 step ko possibly be better? W has to expect more than 2pts locally. Also, the temperature has to be low enough to change the influence of sente-gote and prioritise local variations. However, I can't see any possible way my 2 step ko could be better. Since it seems worse whichever side is to play. So the 1 step ko dominate in this case?