This interesting variation came up, reviewing one of my games with KataGo:

Click Here To Show Diagram Code

[go]$$Wc19
$$ +---------------------------------------+
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . O O X . . . . . . . . . . . . . . |
$$ | . . O X X . . . X , . . . . . X . . . |
$$ | . . . O X . . . . . . . . . . . . . . |
$$ | . . . O . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . O . . . . . . . . . . . . . . . |
$$ | . . . , . . . . . , . . . . . , . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . X . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . O O O . . . . . , . . . . . . . . . |
$$ | . X X X O O O . . . . . . . . X . . . |
$$ | . . . . X X 1 . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ +---------------------------------------+[/go]

How should black respond to ?

Hint:


What KataGo showed me:

(And to save anyone else saying it: yes, black should have played at a fair bit earlier.)

Your question is very interesting and I have difficulty to find good arguments

Just wondering how you reached this position (which looks slightly odd in the lower left corner). Did the game proceed like this?

I had the same thought, but I note 'a' also creates a living shape, and has more impact on white's left side.

Not quite.

OK, I may as well share the full game in all its ugliness. Joseki mistakes were made in other corners too.
I'm still not sure if I enjoy these fast, messy online games. But opportunities to play over a board are a bit limited here. Looking forward to my US trip...

If black leaves out and white kills with a, there's an important difference from the killing sequence shown earlier on this page. But it's nothing to do with the local tally (the exchange - is one stone for each side). The difference is that black no longer has the threat of escaping at b, and so the point c is no longer a forcing move for black. So killing this way is more profitable for white, compared to the previous sequence.

Click Here To Show Diagram Code

[go]$$Wc19
$$ +---------------------------------------+
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . O O X . . . . . . . . 2 . . . . . |
$$ | . . O X X . . . X , . . . . . X . . . |
$$ | . . . O X . . . . . . . . . . . . . . |
$$ | . . . O . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . O . . . . . . . . . . . . . . . |
$$ | . . . , . . . . . , . . . . . , . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . X . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . O O O . . . . . , . . . . . . . . . |
$$ | . X X X O O O . . . . . . . . X . . . |
$$ | . . . . X X 1 . . . . . . . . . . . . |
$$ | . . . . . 3 . . . . . . . . . . . . . |
$$ +---------------------------------------+[/go]

May be I am wrong but if I understand correctly you claim that, according to katago, the above sequence is not bad for both.
Typically that means that the miai value of each o these three moves would be about 13 points.
Now my difficulty is the following: if is consider a bad move in the bottom left corner why this white can be considered a good move?
IOW why the arguments saying that is a bad move in the bottom left corner are not valid when analysing the value of the move in the same corner? If the corner is small for black, could it be big enough for white?

Sorry, I wasn't clear. In my first post, I said "If ... and if white goes on to kill the corner..." This was my way of hinting that KataGo does not think is the best move. It prefers to play in the bottom right. I just put on the board to see what KataGo thought of it, and then it "explained" to me that capturing the corner is not so good for white, because it leaves some aji for black.

I am trying to convinced me that defending the corner with black is not good at this stage of the game, but it is still difficult.

What is my reasonning?

Firstly I imagine that the status of the black has no impact on the outside as in the following diagram

Click Here To Show Diagram Code

[go]$$B
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | O O . . . . . . . . . . . . . . . . . |
$$ | . O . . . . . . . . . . . . . . . . . |
$$ | . O O O a . . . . , . . . . . . . . . |
$$ | . X X X O O O . . . . . . . . X . . . |
$$ | . . . . X X O O O . . . . . . . . . . |
$$ | . . . . . . . . O . . . . . . . . . . |
$$ +---------------------------------------+[/go]

If black lives in gote I imagine that the very final result could be something like (I do not look for a very accurate value):

Click Here To Show Diagram Code

[go]$$B
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | O O . . . . . . . . . . . . . . . . . |
$$ | . O . . . . . . . . . . . . . . . . . |
$$ | 6 O O O . . . . . , . . . . . . . . . |
$$ | 5 X X X O O O . . . . . . . . X . . . |
$$ | . . 1 . X X O O O . . . . . . . . . . |
$$ | . . 7 . 3 2 4 . O . . . . . . . . . . |
$$ +---------------------------------------+[/go]

and I see a swing value about equal to 28 points (miai value 14).

Secondly comes the crucial point. How to evaluate the impact of black group on the outside?
For me the position is quite difficult because three situations exist:
1) Black plays first in the corner in order to live
2) White plays first in the corner in order to kill black
3) Neither player play in the corner
In each case the influence of the corner on the outside is not easy to evaluate but general comment may be formulated:
In situation 1) I imagine the influence on the outside is in white favor because white may have outside moves renforcing white and threatening black corner
In situation 2) I imagine the influence on the outside is in black favor because black may have outside moves threatening to connect the killed black group
In siutation 3) I imagine the influence on the outside is still more in black favor due to the potential cut at "a": if black can cut at "a" threatening to attack one of the white groups, then white can hardly sacrify one of her groups because black stones in the corner will live without having added a defensive stone.

The conclusion is not easy is it?
At least I note a very general point. When you count the gain for living or for killing a group, and ignoring the influence on the outside, then you get always an overevaluation.

Here, prefering to play tenuki (about a move gaining 13 points) instead of gaining 14 points in the corner means that the influence on the outside must be more than 2 points : (14-13)*2