What is your definition of your 'ideal environment' and why / how does it avoid the odd parity problem of not exactly t/2?



Of course. Siegel lets it converge to 0 :)

What is your definition of your 'ideal environment' and why / how does it avoid the odd parity problem of not exactly t/2?

By definition, an environment of simple gote is ideal if playing in it alone yields a result of t/2 no matter who plays first. That is the basic idea of button go.

Also, for whatever position you are considering, the environment should contain a simple gote of the same size as that position and all subpositions.

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

OK Bill let's try to generalise.
Consider the environment of pure gote areas g1 >= g2 >= g3 >= g4 ....

Black plays immediatly in the corner:
score1 = -g1 + g2 + g3 + g4 - g5 + g6 - g7 ...
Black waits one move before playing in the corner:
score2 = g1 - g2 - g3 + g4 + g5 + g6 - g7 ...

score2 - score1 = 2 (g1 - g2 + g3 - g5)
score2 >= score1 <=> g1 - g2 >= g3 - g5
In my example I have g1 - g2 = ½ and g3 - g5 = 1 => black cannot wait
In your example you have g1 - g2 = ½ and g3 - g5 = ½ => black can play immediatly or can wait
In you consider a regular serie of gote moves then you have g1 - g2 > g3 - g5 and black cannot wait one move.

In you take the environment 4½, 3, 2½, 2, 1½, 1, ½
you can see that g1 - g2 > g3 - g5 and you must wait one move before playing in the corner which is just common sense when you see the big gap between g1 and g2.

Here is an SGF file for a chilled go environment with chilled temperature 1½. OC, when the chilled temperature is less than 0, neither player will play. The ½ point territory is marked by ∆.



All roads lead to Rome.

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

yes Bill with g1 - g2 = g3 - g5 = 0 it is indifferent to play immediatly in the corner or to wait one move.
We can see clearly the problem with the definition of an "ideal" environment. If the temperature do not drop then black can always wait before playing in the corner. As soon as the temperature may drop (t becomes t-0.01 or whatever you want) you have to compare g1 - g2 to g3 - g5 in order to know if you have to play immediatly in the corner.

The question of when to play sente is not an easy one, and I am not sure that the bots know, either.

The sente that you found in this corner with no ko threats is brilliant. First, it does not really threaten to win the ko, but to win points elsewhere in the ko exchange. Second, even though it threatens to win points elsewhere on the board, it allows White to take the largest play. A truly remarkable concept!

I do suspect, however, that there is an ideal environment at chilled go where either the sente or playing in the environment will be preferable.

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

Yes. At temperature 7 one option is preferable, because of tunneling. The threshold should be temperature 6½, as I indicated before.


Of course, Black does not have to accept the offer.

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

The example here after should answer your question

Click Here To Show Diagram Code

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

According to my previous analyse, because we have g1-g2 < g3-g5, the best way to play the ko variant is to provoque the ko immediatly.
But that doesn't mean that the ko variant is better than the sente variant. Here the sente variant is better than the ko variant and black must either play immediatly in the environment or play in sente in the corner.

Note :

Click Here To Show Diagram Code

[go]$$B
$$ ---------------------------------------
$$ | . 2 . b X X X X X X O . . . . . . . . |
$$ | b 1 O O X X . O O . O . . . . . . . . |
$$ | a . O X X X X X X X O . . . . . . . . |
$$ | . O O X . . X O O . O . . . . . . . . |
$$ | b O X X . . X X X X O . . . . . . . . |
$$ | . X X . . . X . O . O . . . . . . . . |
$$ | . X . . . . X X X X O . . . . . . . . |
$$ | X X . . . . X X O . O . . . . . . . . |
$$ | . . . . . . X X X X O . . . . . . . . |
$$ | . . . , . . X X . . O . . . . , . . . |
$$ | . . . . . . X X X X O . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . , . . . . . , . . . . . , . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ ---------------------------------------[/go]

Black may begin by the exchange but after this exchange black should either play in the environment or play a sente move in the corner at one of the "b" points. If I am not wrong, a black move at "a" to provoque a ko is bad.

As you can see, though a lot of progress have been made, we do not have yet all the information saying when black has to play in the corner and which sequence black should use, depending of the environment.

We are talking about two different things. What you are showing is the corner plus three simple gote in an ideal environment at t = 1 in regular go, t = 0 in chilled go. I am talking about the corner alone in an ideal environment at chilled go, since I do not know of, say, an ideal environment at temperature 6½ in regular go.

To sum up, it appears that, at regular go in an ideal environment, above temperature 6½ a play in the environment is preferable to starting or winning the ko, and below temperature 1 a play in the environment is not good.

The point is, to find the temperature of indifference, any temperature drop must occur in the game, not in the environment. I am sorry if I gave a different impression or made a misstatement. Edit: As I said, I have never written academically about ideal environments and have no plans to do so.

Edit2: In my defense, 2½ - 2 + 1½ - 1 + ½ = 1½ > 1¼. so, by definition, this cannot be an ideal environment.

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

So let’s talk about this position. Since the idea of an ideal environment seems to be non-productive, let’s not use it.

Click Here To Show Diagram Code

[go]$$B
$$ ---------------------
$$ | . O . . X . . . . |
$$ | . X O O X . . . . |
$$ | . . O X X . . . . |
$$ | . O O X . . . . . |
$$ | . O X X . . . . . |
$$ | . X . . . . . . . |
$$ | . X . . . . . . . |
$$ | X X . . . . . . . |
$$ | . . . . . . . . . |
$$ ---------------------[/go]

White to play can live.

Click Here To Show Diagram Code

[go]$$W
$$ ---------------------
$$ | . O . . X . . . . |
$$ | 1 X O O X . . . . |
$$ | . . O X X . . . . |
$$ | . O O X . . . . . |
$$ | . O X X . . . . . |
$$ | . X . . . . . . . |
$$ | . X . . . . . . . |
$$ | X X . . . . . . . |
$$ | . . . . . . . . . |
$$ ---------------------[/go]

Now Black to play continues this way.

Click Here To Show Diagram Code

[go]$$W
$$ ---------------------
$$ | . O 4 2 X . . . . |
$$ | 1 X O O X . . . . |
$$ | . 5 O X X . . . . |
$$ | . O O X . . . . . |
$$ | . O X X . . . . . |
$$ | 3 X . . . . . . . |
$$ | . X . . . . . . . |
$$ | X X . . . . . . . |
$$ | . . . . . . . . . |
$$ ---------------------[/go]

Which leaves a net local score of -5 with White playing last.

There are other sequences that produce the same result.

So we know that lives for an average territory of -5. This is the same average territory we get if White had simply played the jump to A-18 first from the original corner position.

What if Black plays first?

Click Here To Show Diagram Code

[go]$$B
$$ ---------------------
$$ | 8 O 5 3 X . . . . |
$$ | 1 X O O X . . . . |
$$ | 2 6 O X X . . . . |
$$ | . O O X . . . . . |
$$ | 4 O X X . . . . . |
$$ | 7 X . . . . . . . |
$$ | . X . . . . . . . |
$$ | X X . . . . . . . |
$$ | . . . . . . . . . |
$$ ---------------------[/go]

The result is a local score of -4, which is perforce the maximum average value of this position.

Click Here To Show Diagram Code

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

This way results in a local score of -3 with Black playing last. That is consistent with an average local score of -4 at or above temperature 1.

Click Here To Show Diagram Code

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

Again, we get a local score of -3 with Black playing last.

The only hope for Black to do better is to play A-17.

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

Click Here To Show Diagram Code

[go]$$B Black to play
$$ ---------------------
$$ | . O . . X . . . . |
$$ | s X O O X . . . . |
$$ | k . O X X . . . . |
$$ | . O O X . . . . . |
$$ | . O X X . . . . . |
$$ | . X . . . . . . . |
$$ | . X . . . . . . . |
$$ | X X . . . . . . . |
$$ | . . . . . . . . . |
$$ ---------------------[/go]

Let's try to answer the following question: when black will choose the sente sequence "s" rather than the ko sequence "k".
Let's take the environment g1 >= g2 >= g3 >= g4 ...

Black chooses the ko sequence
ScoreKo = -5 - g1 + g2 + g3 + g4 - g5 + g6 ...
Black chooses the sente sequence
ScoreSente = -4 + g1 - g2 + g3 - g4 + g5 - g6 ...

ScoreSente >= ScoreKo <=>
-4 + g1 - g2 + g3 - g4 + g5 - g6 ... >= -5 - g1 + g2 + g3 + g4 - g5 + g6 ... <=>
2(g1 - g2 - g4 + g5 - g6 ...) >= 1 <=>
2(g1 - g2 + g3 - g4 + g5 - g6 ...) >= 1 + 2g3
Asuming 2(g1 - g2 + g3 - g4 + g5 - g6 ...) = g1 then
ScoreSente >= ScoreKo <=> g1 >= 1 + 2g3

If now you assume a regular set of gote points with g_i = g_i-1 + ½ then you have g1 = 1 + 2g3 with the environment
3, 2½, 2, 1½, 1, ½

What conclusion? In general, if temperature > 3 then black chooses the ko variant and if temperature < 3 then black chooses the sente variant.
In addition when you choose the ko variant you have in general to play immediatly in the corner. If you choose the sente variant then, in general you can wait befor playing in the corner.
OC, with a real environment and a temperature near 3, you have to read all the yose to decide correctly.

By inspection we note that the final alternating sequence for each sequence starts with g4. Therefore we may take g4 as our first candidate for t, depending on how far we can read the sequences out.

Since we are trying to gain by the ko sequence, we subtract the sente sequence from the ko sequence to see if it is positive.

ScoreKo - ScoreSente = -1 - 2g1 + 2g2 + 2(g4 - ...) > 0

For a first approximation we set the temperature to g4 and get

g4 > 1 + 2g1 - 2g2

g2 ≥ g4 , so

3g2 - 1 > 2g1

If g2 = 2 then 2½ > g1

If g2 = 4 then 5½ > g1

If g2 = 6 then 8½ > g1

But we also know that 13 - g2 > g1, so

If g2 = 6 then 7 > g1

If g2 = 5 then 7 > g1

That gives us the upper limits,

7 > g1

and

2 > g1 - g2

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

Addendum

If, as you suggest, Gérard, g1 = 2g1 - 2g2 + 2g3 -2g4 + ..., then

And if

ScoreKo - ScoreSente = -1 - 2g1 + 2g2 + 2(g4 - ...) > 0

then

ScoreKo - ScoreSente + g1 = -1 + 2g3 > g1

This gives us another upper constraint on g1, but also a lower constraint on g3.

g1/2 + ½ < g3

If g1 = 2 then 1½ < g3.

This constraint is not a problem for chilled go, but for simple gote with regular go, we cannot have a denominator greater than 2. That means that at regular go g3 = g1 = 2.

That does not work if there is a constant difference between successive miai values. However, the constraint, g1 = 2g1 - 2g2 + ..., is not as strict as that. It could be satisfied at regular go with g4 = 1½, g5 = 1, g6 = ½, with no further gote, for instance.

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

Good idea Bill to analyse such position.
let me continue your analyse by this play A-17

Click Here To Show Diagram Code

[go]$$W White to play
$$ ---------------------
$$ | . O . b X . . . . |
$$ | a X O O X . . . . |
$$ | X . O X X . . . . |
$$ | . O O X . . . . . |
$$ | . O X X . . . . . |
$$ | . X . . . . . . . |
$$ | . X . . . . . . . |
$$ | X X . . . . . . . |
$$ | . . . . . . . . . |
$$ ---------------------[/go]

Now the point is to compare my new move "a" to the well known "b".
The ko we can reach with "a" and "b" are quite different and I will take again the environment g1 >= g2 >= g3 >= g4 ....

white plays at "a":

Click Here To Show Diagram Code

[go]$$W white to play
$$ ---------------------
$$ | 2 O 4 5 X . . . . |
$$ | 1 X O O X . . . . |
$$ | X . O X X . . . . |
$$ | . O O X . . . . . |
$$ | 6 O X X . . . . . |
$$ | . X . . . . . . . |
$$ | . X . . . . . . . |
$$ | X X . . . . . . . |
$$ | . . . . . . . . . |
$$ ---------------------[/go]

take g1

Click Here To Show Diagram Code

[go]$$Wcm7
$$ ---------------------
$$ | X 1 X O X . . . . |
$$ | 3 X O O X . . . . |
$$ | X 5 O X X . . . . |
$$ | . O O X . . . . . |
$$ | X O X X . . . . . |
$$ | . X . . . . . . . |
$$ | . X . . . . . . . |
$$ | X X . . . . . . . |
$$ | . . . . . . . . . |
$$ ---------------------[/go]

takes g2
takes g3
takse g4
and black score is ScoreA17 = -g1 + g2 + g3 + g4 - g5 ... -5

White plays at "b"

Click Here To Show Diagram Code

[go]$$W White to play
$$ ---------------------
$$ | 4 O . 1 X . . . . |
$$ | 5 X O O X . . . . |
$$ | X 7 O X X . . . . |
$$ | 3 O O X . . . . . |
$$ | 2 O X X . . . . . |
$$ | . X . . . . . . . |
$$ | . X . . . . . . . |
$$ | X X . . . . . . . |
$$ | . . . . . . . . . |
$$ ---------------------[/go]

takes g1
takes g2
and black score is ScoreD19 = g1 + g2 - g3 + g4 - g5 ... -7

White should prefer A17 if ScoreA17 < ScoreD19 that means
-g1 + g2 + g3 + g4 - g5 ... -5 < g1 + g2 - g3 + g4 - g5 ... -7 <=>
2(g1 - g3) > 2

That proves that the white move at "a" is in general better than "b" as soon as g1 - g3 > 1

Two examples:
with the environment 3½, 3, 2½, 2, 1½, 1, ½ (g1 - g3 = 1) the move "a" is better than "b" but
with the environment 3½, 3½, 3, 2½, 2, 1½, 1, ½ (g1 - g3 = ½) the move "b" is better than "a"

To confirm your results I tried the following difference game

Click Here To Show Diagram Code

[go]$$W White to play
$$ ---------------------
$$ | . O . . X . O O . X . |
$$ | X X O O X . O X X O . |
$$ | . . O X X . O O X . . |
$$ | . O O X . . . O X X . |
$$ | . O X X . . . O O X . |
$$ | . X . . . . . . O O . |
$$ | . X . . . . . . . O . |
$$ | X X . . . . . . . O O |
$$ | . . . . . . . . . . . |
$$ | . . . . . . . . . . . |
$$ | . . . . . . . . . . . |
$$ ---------------------[/go]

White to play, I expected a white win but it looks black can reach the draw

Click Here To Show Diagram Code

[go]$$W White to play
$$ ---------------------
$$ | 7 O 4 2 X . O O . X . |
$$ | X X O O X . O X X O . |
$$ | 1 5 O X X . O O X . . |
$$ | . O O X . . . O X X . |
$$ | 3 O X X . . . O O X . |
$$ | 6 X . . . . . . O O . |
$$ | . X . . . . . . . O . |
$$ | X X . . . . . . . O O |
$$ | . . . . . . . . . . . |
$$ | . . . . . . . . . . . |
$$ | . . . . . . . . . . . |
$$ ---------------------[/go]

Click Here To Show Diagram Code

[go]$$Bcm8
$$ ---------------------
$$ | O O X X X . O O . X 4 |
$$ | . 5 O O X . O X X O 3 |
$$ | O O O X X . O O X . 2 |
$$ | . O O X . . . O X X . |
$$ | O O X X . . . O O X 1 |
$$ | X X . . . . . . O O . |
$$ | . X . . . . . . . O . |
$$ | X X . . . . . . . O O |
$$ | . . . . . . . . . . . |
$$ | . . . . . . . . . . . |
$$ | . . . . . . . . . . . |
$$ ---------------------[/go]

The unexpected allows black to reach the draw by using the ko threat
We know that this ko fight invalidate the difference game and OC we cannot claim for a black move dominating the other. It was just for fun

b is well known? In general it doesn't look so good to me.{shrug}

Edit: I have to take that back. In general it does look better.



How about this?

Click Here To Show Diagram Code

[go]$$W White to play
$$ ---------------------
$$ | . O . 1 X . . . . |
$$ | . X O O X . . . . |
$$ | X . O X X . . . . |
$$ | 3 O O X . . . . . |
$$ | 2 O X X . . . . . |
$$ | 4 X . . . . . . . |
$$ | . X . . . . . . . |
$$ | X X . . . . . . . |
$$ | . . . . . . . . . |
$$ ---------------------[/go]

Assuming no ko threats, now or later, this may well become seki, with Black having 1 point by Japanese/Korean rules, effectively 0 by AGA rules. But it is known that this position favors Black in general, because Black has little to lose in the ko and much to gain. This is the main reason why, in general, D-19 does not look so good to me.

1) Suppose that White makes seki with .

AGASekiScore = g1 - g2 + g3 - g4 + ....

2) Suppose that takes g1 and makes the ko.

Click Here To Show Diagram Code

[go]$$W White to play
$$ ---------------------
$$ | 6 O . 1 X . . . . |
$$ | 7 X O O X . . . . |
$$ | X 9 O X X . . . . |
$$ | 3 O O X . . . . . |
$$ | 2 O X X . . . . . |
$$ | 4 X . . . . . . . |
$$ | . X . . . . . . . |
$$ | X X . . . . . . . |
$$ | . . . . . . . . . |
$$ ---------------------[/go]

= g1 = g2

Throw-inScore = -6 - g1 + g2 + g3 - g4 + ....

White will prefer to make seki in gote when

g1 - g2 > 3

Which is very rarely.

If neither player throws in then the AGA score will be

NoThrow-inScore = -g1 + g2 - g3 + g4 - ....

Black will prefer to throw in when

2g3 - 2g4 + ... > 6

Assuming, as a first approximation, that g4 = 2g4 - ..., we have

2g3 - g4 > 6

Normally, g3 ≅ g4, so we may write

g3 > 6,

So at lower temperatures, we may assume that normally the connection will result in this score.

AGAD19-SenteSekiScore = -g1 + g2 - g3 + g4 - ....

ScoreD19 = g1 + g2 - g3 + g4 - g5 ... -7

This is better for Black when

3½ > g1.

Now let's compare this sequence with White's throw-in instead of D-19.

ScoreA17 = -g1 + g2 + g3 + g4 - g5 ... -5

AGAD19-SenteSekiScore = -g1 + g2 - g3 + g4 - ....

White's throw-in is better for White when

2½ > g3.

These simple gote environments are not close approximations, and typically the largest play in the environment does not have a miai value 1 point larger than the third largest play. So it looks like once the miai value of the largest play drops below 3½, White D-19 would typically make seki with sente. I.e., White D-19 is not so good at low temperatures when there are no ko threats. OC, at higher temperatures the conditions when it is best may well arise.

_________________
The Adkins Principle:
At some point, doesn't thinking have to go on?
— Winona Adkins

Visualize whirled peas.

Everything with love. Stay safe.

Click Here To Show Diagram Code

[go]$$B
$$ ---------------------
$$ | . 2 . c X . . . . |
$$ | a 1 O O X . . . . |
$$ | . . O X X . . . . |
$$ | . O O X . . . . . |
$$ | b O X X . . . . . |
$$ | . X . . . . . . . |
$$ | . X . . . . . . . |
$$ | X X . . . . . . . |
$$ | . . . . . . . . . |
$$ ---------------------[/go]

Bill, studying the black moves a,b,c is good OC but my conclusion is the following. Playing a,b or c just after the exchange is in general tecnically bad moves because in this case the exchange is a pure lost of one ko threat.
In order to see this point see the following position:

Click Here To Show Diagram Code

[go]$$B
$$ ---------------------
$$ | . . . a X k X O . |
$$ | . c O O X X O O . |
$$ | . . O X X . . . . |
$$ | . O O X . . . . . |
$$ | b O X X . . . . . |
$$ | . X . . . . . . . |
$$ | . X . . . . . . . |
$$ | X X . . . . . . . |
$$ | . . . . . . . . . |
$$ ---------------------[/go]

In order to win the ko in "k" black must begin by a move at "a" or "b". By playing at "c" black will not be able to win this ko.
A black move at "c" is good when black wants to start a ko in the corner, otherwise this move appears a bad move.

Edit : OC, playing the exchange as a ko threat is very good because black does not lose the two options ko or sente.

Oh, yes. Generally speaking the point of Black starting on the 2-2 is to make the ko, not to generate ko threats.

Up until now we have focused on the play that you discovered where Black makes the ko, despite neither player having a ko threat, now or later.

Evaluation of kos, except for placid kos, depends upon what assumptions we make. Berlekamp's brilliant komaster analysis depends upon assumptions that are usually not met exactly on the go board, but which provide useful approximate limits. When neither side is komaster, he came up with the idea of a neutral threat environment (NTE), where each player has the exact opposite of the ko threats of the other, where the ko threats are sufficiently large. How good an approximation NTE produces in practice is another question, but let's give it a shot.

Click Here To Show Diagram Code

[go]$$W White first
$$ ---------------------
$$ | C 5 4 2 X . . . . |
$$ | 1 C O O X . . . . |
$$ | C C O X X . . . . |
$$ | C O O X . . . . . |
$$ | . O X X . . . . . |
$$ | 3 X . . . . . . . |
$$ | . X . . . . . . . |
$$ | X X . . . . . . . |
$$ | . . . . . . . . . |
$$ ---------------------[/go]

gets a local score of -5 in one net play.

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[go]$$B Black first, no ko
$$ ---------------------
$$ | . 4 3 1 X . . . . |
$$ | . 2 O O X . . . . |
$$ | . . O X X . . . . |
$$ | 6 O O X . . . . . |
$$ | 5 O X X . . . . . |
$$ | 7 X . . . . . . . |
$$ | . X . . . . . . . |
$$ | X X . . . . . . . |
$$ | . . . . . . . . . |
$$ ---------------------[/go]

gets a net local score of -3 in one net play.

Click Here To Show Diagram Code

[go]$$B Black first, ko, NTE
$$ ---------------------
$$ | 7 2 . 4 X . . . . |
$$ | . 1 O O X . . . . |
$$ | 3 . O X X . . . . |
$$ | 6 O O X . . . . . |
$$ | 5 O X X . . . . . |
$$ | . X . . . . . . . |
$$ | . X . . . . . . . |
$$ | X X . . . . . . . |
$$ | . . . . . . . . . |
$$ ---------------------[/go]

Now we have reached a position after one net move by Black such that either player, playing first could win the ko in two local plays by ignoring the opponent's threat.

Click Here To Show Diagram Code

[go]$$W White first, NTE
$$ ---------------------
$$ | X O . O X . . . . |
$$ | 1 X O O X . . . . |
$$ | X 3 O X X . . . . |
$$ | O O O X . . . . . |
$$ | X O X X . . . . . |
$$ | . X . . . . . . . |
$$ | . X . . . . . . . |
$$ | X X . . . . . . . |
$$ | . . . . . . . . . |
$$ ---------------------[/go]

= ko threat, completes threat

After there is a local score of -7 plus a gote worth on average 0 when t > 1.

Click Here To Show Diagram Code

[go]$$B Black first, NTE
$$ ---------------------
$$ | X W 1 W X . . . . |
$$ | C X W W X . . . . |
$$ | X 3 W X X . . . . |
$$ | W W W X . . . . . |
$$ | X W X X . . . . . |
$$ | C X . . . . . . . |
$$ | C X . . . . . . . |
$$ | X X . . . . . . . |
$$ | . . . . . . . . . |
$$ ---------------------[/go]

= ko threat, completes threat

After there is a local score of +21.

By assumption, what each player gains by playing and completing their threat is the same, and therefore cancel out in the average, so the average value of the given position under NTE when t > 1 is the average of the value of these two politions. That value is (21 - 7)/2 = 7, 10 points better than -3.

So, under NTE conditions, the average value of the corner is (7 - 5)/2 = 1, and the miai value of a play is 6.

Edit: Suppose neither komaster not NTE conditions apply, but Black can sacrifice the ko for making and completing a ko threat. If the ko threat is a simple sente, then the result after Black completes her threat is -7 + 2θ, where θ is the miai value of the threat. As long as θ > 1½ that result is better on average than -4, which is the average when Black does not make the ko.

_________________
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Though I follow your calculation the conclusion does not fit with my understanding of the miai value basic concept. For me a miai value of say "6" means that each player has to play in the local area when the temperature drops under 6.

Click Here To Show Diagram Code

[go]$$W White first
$$ ---------------------
$$ | . . . . X . . . . |
$$ | . . O O X . . . . |
$$ | . . O X X . . . . |
$$ | . O O X . . . . . |
$$ | . O X X . . . . . |
$$ | . X . . . . . . . |
$$ | . X . . . . . . . |
$$ | X X . . . . . . . |
$$ | . . . . . . . . . |
$$ ---------------------[/go]

Without any ko threat in the environment I do not think white has to play in this local area when temperature drops under 6 do you?
For a simple direct ko I have no problem with miai values but here the ko in the corner is more complex and in the initial position black is the only one who can start the ko. In that case it appears quite difficult to calculate a miai value without changing its basic concept.
My feeling is that, by starting the ko, black increases the temperature but that does not mean that the intial miai value is that high.