Very good.

I think you are on to something. Let me get back to you on this.

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

Visualize whirled peas.

Everything with love. Stay safe.

Let’s set up an ideal environment for these two lines of play.

Click Here To Show Diagram Code

[go]$$B Sente
$$ ---------------------
$$ | . 4 3 1 X O 5 O . |
$$ | . 2 O O X X X O . |
$$ | . . O X X O 6 O . |
$$ | 8 O O X X X X O . |
$$ | 7 O X X X . 0 O . |
$$ | 9 X . . X X X O . |
$$ | . X . . . . . . . |
$$ | X X . . . . . . . |
$$ | . . . . . . . . . |
$$ ---------------------[/go]

Net result: 1 point for White

Click Here To Show Diagram Code

[go]$$B Environment
$$ ---------------------
$$ | . . 4 3 X O 1 O . |
$$ | 2 . O O X X X O . |
$$ | . . O X X O 8 O . |
$$ | 6 O O X X X X O . |
$$ | 5 O X X X . 9 O . |
$$ | 7 X . . X X X O . |
$$ | . X . . . . . . . |
$$ | X X . . . . . . . |
$$ | . . . . . . . . . |
$$ ---------------------[/go]

Net result: 1 point for White

Black is indifferent between these two plays. QED.

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

Visualize whirled peas.

Everything with love. Stay safe.

OC white has to answer black 1. In that sense black is sente but I do not like this wording:

Click Here To Show Diagram Code

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

The point is the following : the exchange black "a" white "b" is not a gain for black but rather a gain for white because now black cannot provoque a ko.
Let's change the order: if white plays first "b" then white is happy if black answer with "a" because white "b" becomes sente.

In such situation I call the sequence black "a" white "b" a thank-you sequence and I call the sequence white "b" black "a" a sente sequence.

Sure you will understand why I am reluctant to say that black "a" is sente. In the other hand you may say that black "a" is a ko threat but that is quite different.

I never said that Black a gains anything. In fact, sente gains nothing on average.



sets up a delayed sente for Black, which is sometimes advantageous for White.

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

Visualize whirled peas.

Everything with love. Stay safe.

Many thanks, Gérard. You are always an inspiration.

I have studied this position a bit more, and I believe that I understand it better. As always, I follow Berlekamp's dictum that to understand a position, start with its thermograph. The above position has the corner plus three simple gote with local temperatures greater than the temperatures of the ideal environment.

To understand the corner better, I made the three simple sente identical, with a temperature equal to that of the ideal environment, t = 4. IOW, we have the corner plus an ideal environment with 6½ ≥ t > 1.

In the sgf file below I compared these two lines of play with Black playing first.

1) Black plays first in the environment and then White plays in the corner, and then Black plays in the environment again.

2) Black starts the ko, which is sente. In the fight, with neither side having a ko threat, White plays once in the environment and Black plays twice, and then White plays in the corner to win the ko, and then Black plays in the environment again.

These two lines of play should produce the same net result. As the sgf file indicates, they do.



The three simple gote give leeway by adjusting their miai values for starting the ko to be the best play. What is intriguing is that values that create an environment that is very close to ideal does that.

Also, if it is White to play before the global temperature drops to 1, White will almost surely play in the environment. So Black will have many chances to gain from starting the ko at her turn. Well spotted, Gérard!

_________________
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]$$B
$$ ---------------------------------------
$$ | . . . . X X X X X X O . . . . . . . . |
$$ | a . O O X O O O O 1 O . . . . . . . . |
$$ | . . O X X X X X X X O . . . . . . . . |
$$ | . O O X X O O O O v O . . . . . . . . |
$$ | . O X X X X X X X X O . . . . . . . . |
$$ | . X X . X O O O O w O . . . . . . . . |
$$ | . X . . X X X X X X O . . . . . . . . |
$$ | X X . . . . . . . . . t . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . , . . . . . , . . . . . , . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . , . . . . . , . . . . . , . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ ---------------------------------------[/go]

Oops I do not understand your variation 2. After in the environment white cannot play in the corner when the temperature of the environment is as high as t = 4. White must answer in the environment waiting for a drop of this temperature. Basically white's line of play is not quite difficult. Black's problem is more difficult. She has to decide:
1) when to play in the corner
2) which move to play in the corner

Berlekamp's komaster theory stipulates that for ko positions there are enough plays at the temperature of the environment that the ko may be played out before the temperature drops. In that case White's reply in the environment is simply treading water; the temperature is never going to drop. Remember, CGT does not take into account whose turn it is. I omitted White's play for the corner when White plays first because we know that it is the jump to A-18.

By contrast with the case that the three simple gote are different, then we don't know that it is and von Neumann game theory applies. That's how we determined that White's play in the environment is better when the global temperature is greater than 1.

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

Visualize whirled peas.

Everything with love. Stay safe.

For any white go player, playing in the corner when t = 4 is a very bad move (it looks like a 1 point reverse sente move). You mentionned "Berlekamp's komaster theory" and I cannot believe that such theory stipulates that such white play in the corner might be good. Obviously there is a misunderstanding somewhere. Maybe here is two important points:
1) Neither black nor white have ko threats => "Berlekamp's komaster theory" does not apply (?)
2) For white point of view the ko does not exist. The ko can exist only if black plays first and only if black choose the ko variant. If black provoques really this ko then white OC will fight in the corner without waiting temperature drop.

The problem is me and Berlekamp. Berlekamp developed his komaster theory before I came along. In 1994 I attended his talk in which he presented the theory, but he apparently had developed it before 1990. He defined it in terms of tax, like the rest of thermography at that time. However, komaster theory obviously did not apply to multiple kos and superkos. In 1998 I redefined and expanded thermography to cover multiple kos and superkos by basing it on play in an ideal environment. When you define it that way you accommodate komaster theory by requiring all ko plays to be made before the temperature of the ideal environment drops. That is why made all three plays the same size and said that if Black starts by taking one of the three plays it does no good for White to take one, as well. There will always be enough plays of the same size for White to take one back. In Berlekamp's original formulation, there are no plays outside the corner, and you just apply the tax. Thermography is usually an approximation of the whole board game. If you can solve the whole board you do not need thermography.

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

Visualize whirled peas.

Everything with love. Stay safe.

Do you mean an arbitrarily dense "rich environment"?

Click Here To Show Diagram Code

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

Let me give you my analyse of black strategy by provoquing the ko, in a ideal environment without any ko threat.

1) Black starts the ko, white plays once in the environment and Black plays twice, and then White plays in the corner to win the ko, and then Black plays in the environment again
score1 = -t + t + t + t/2 - 5 = 3t/2 - 5
2) Black play in the corner and white give up the corner by playing in the environment
score2 = -t - t/2 + 22 = -3t/2 + 22

White chooses to give up the corner if :
score2 <= score1 <=> -3t/2 + 22 <= 3t/2 - 5 <=> t >= 9

3) The initial temperature being equal to t1 <= 9 assume black wait until temperature drops to t2 before starting the ko
score3 = (t1 - t2)/2 + 3t2/2 - 5 = t1/2 + t2 - 5
In order to reach the best score black must choose t2 to increase score3 as far as possible. As a consequence black chooses t2 = t1 and reaches the score1.

what conclusion in an ideal enviroment without ko threat? As soon as temperature drops under t= 9 Black should provoque the ko as soon as possible.

Note : it is quite interesting to study some non ideal environment u >= v >= t, with two gote points u et v and an ideal environment t. Depending of the values u,v,t it may be better for black to delay her move in the corner.

Robert : in your theory, did you take into consideration such local situation? If yes can you tell us what was your results?

In my paper, I used Berlekamp's terminology. I have since refined the concept to that of an ideal enivornment, because playing first in a dense environment does not always produce a result of t/2. The slight change is not worth a paper in itself.

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

Visualize whirled peas.

Everything with love. Stay safe.

Since we agree that a White play in the corner gains only 1 point there in regular go, why shouldn't Black wait to play in the corner? Assuming a temperature drop between the top plays in the rest of the board, why not play elsewhere and pick up the difference?

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

Visualize whirled peas.

Everything with love. Stay safe.

Look at my previous example Bill:

Click Here To Show Diagram Code

[go]$$B
$$ ---------------------------------------
$$ | . . . . X X X X X X O . . . . . . . . |
$$ | . a O O X O O O O u O . . . . . . . . |
$$ | . . O X X X X X X X O . . . . . . . . |
$$ | . O O X X . O O O v O . . . . . . . . |
$$ | . O X X X X X X X X O . . . . . . . . |
$$ | . X X . X X O O O w O . . . . . . . . |
$$ | . X . . X X X X X X O . . . . . . . . |
$$ | X X . . . . . . . . . t . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . , . . . . . , . . . . . , . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . , . . . . . , . . . . . , . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ ---------------------------------------[/go]

the environment is the following:
Three gote points u, v, w with the values 4, 3½, 3 and I assume the remaining environment being an ideal environment at temperature t = 2½.

Can you reach a better result for black than playing immediatly in the corner at "a"?

That does not answer the general question.

BTW, I do not know of an ideal environment in regular go at t = 2½. It is possible to construct an ideal environment in chilled go at chilled temperature 1½, however. If there is no ko fight aside from this one, which we know that White is going to win, anyway, correct play in chilled go should be the same as correct play in regular go. Since we can construct an actual ideal environment for chilled go, why not use it?

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

Visualize whirled peas.

Everything with love. Stay safe.

For the problem where the rich environment is not ideal, Berlekamp used one with a delta of 0.01 point, such that, as a practical matter a ko fight would hardly ever result in a result different from that in an ideal environment. If necessary he would resort to an even smaller delta.

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

Visualize whirled peas.

Everything with love. Stay safe.

Or, we can construct an ideal environment with temperature 2 in regular go, so why no try that, with appropriate changes to your example, to avoid a sizable temperature drop? (We will start at temperature 3½ instead of 4. )

Click Here To Show Diagram Code

[go]$$B Corner first
$$ ---------------------------------------
$$ | 5 2 7 8 X X X X X X O . . . . . . . . |
$$ | 4 1 O O X . O O O 6 O . . . . . . . . |
$$ | 3 . O X X X X X X X O . . . . . . . . |
$$ | . O O X X X O O O v O . . . . . . . . |
$$ | 9 O X X X X X X X X O . . . . . . . . |
$$ | . X X . X X . O O w O . . . . . . . . |
$$ | . X . . X X X X X X O . . . . . . . . |
$$ | X X . . . . X O O . O . . . . . . . . |
$$ | . . . . . . X X X X O . . . . . . . . |
$$ | . . . , . . X O O . O . . . . , . . . |
$$ | . . . . . . X X X X O . . . . . . . . |
$$ | . . . . . . X O O . O . . . . . . . . |
$$ | . . . . . . X X X X O . . . . . . . . |
$$ | . . . . . . . X O . O . . . . . . . . |
$$ | . . . . . . . X X X X . . . . . . . . |
$$ | . . . , . . . . . , . . . . . , . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ ---------------------------------------[/go]

Click Here To Show Diagram Code

[go]$$Wm10 Black has captured 2 stones in the ko fight
$$ ---------------------------------------
$$ | B 1 B O X X X X X X O . . . . . . . . |
$$ | 3 B O O X . O O O O O . . . . . . . . |
$$ | B 5 O X X X X X X X O . . . . . . . . |
$$ | . O O X X X W W W 2 O . . . . . . . . |
$$ | X O X X X X X X X X O . . . . . . . . |
$$ | . X X . X X C W W 4 O . . . . . . . . |
$$ | C X . . X X X X X X O . . . . . . . . |
$$ | X X . . . . X W W 6 O . . . . . . . . |
$$ | . . . . . . X X X X O . . . . . . . . |
$$ | . . . , . . X O O 7 O . . . . , . . . |
$$ | . . . . . . X X X X O . . . . . . . . |
$$ | . . . . . . X W W 8 O . . . . . . . . |
$$ | . . . . . . X X X X O . . . . . . . . |
$$ | . . . . . . . X O 9 O . . . . . . . . |
$$ | . . . . . . . X X X X . . . . . . . . |
$$ | . . . , . . . . . , . . . . . , . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ ---------------------------------------[/go]

If I have counted correctly, the net result is +14 for Black.

Now let Black play first in the largest play on the board.

Click Here To Show Diagram Code

[go]$$B Largest play first
$$ ---------------------------------------
$$ | 7 4 9 0 X X X X X X O . . . . . . . . |
$$ | 6 3 O O X C W W W 1 O . . . . . . . . |
$$ | 5 . O X X X X X X X O . . . . . . . . |
$$ | . O O X X X O O O 2 O . . . . . . . . |
$$ | . O X X X X X X X X O . . . . . . . . |
$$ | . X X . X X . O O 8 O . . . . . . . . |
$$ | . X . . X X X X X X O . . . . . . . . |
$$ | X X . . . . X O O . O . . . . . . . . |
$$ | . . . . . . X X X X O . . . . . . . . |
$$ | . . . , . . X O O . O . . . . , . . . |
$$ | . . . . . . X X X X O . . . . . . . . |
$$ | . . . . . . X O O . O . . . . . . . . |
$$ | . . . . . . X X X X O . . . . . . . . |
$$ | . . . . . . . X O . O . . . . . . . . |
$$ | . . . . . . . X X X X . . . . . . . . |
$$ | . . . , . . . . . , . . . . . , . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ ---------------------------------------[/go]

Click Here To Show Diagram Code

[go]$$Bcm11 Black has captured 2 stones in the ko fight
$$ ---------------------------------------
$$ | B 2 B O X X X X X X O . . . . . . . . |
$$ | 4 B O O X C W W W X O . . . . . . . . |
$$ | B 6 O X X X X X X X O . . . . . . . . |
$$ | . O O X X X O O O O O . . . . . . . . |
$$ | 1 O X X X X X X X X O . . . . . . . . |
$$ | . X X . X X . O O O O . . . . . . . . |
$$ | C X . . X X X X X X O . . . . . . . . |
$$ | X X . . . . X W W 3 O . . . . . . . . |
$$ | . . . . . . X X X X O . . . . . . . . |
$$ | . . . , . . X W W 5 O . . . . , . . . |
$$ | . . . . . . X X X X O . . . . . . . . |
$$ | . . . . . . X W W 7 O . . . . . . . . |
$$ | . . . . . . X X X X O . . . . . . . . |
$$ | . . . . . . . X O 8 O . . . . . . . . |
$$ | . . . . . . . X X X X . . . . . . . . |
$$ | . . . , . . . . . , . . . . . , . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ ---------------------------------------[/go]

Again, if I have counted correctly, Black is 14 points ahead, the same as when she starts in the corner. Note that in the second sequence Black played twice in the ideal environment before White won the ko.

I'm taking a break now. More later.

_________________
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 if ideal environment notion do not simplify the reasoning we had better to not use it.

Click Here To Show Diagram Code

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

Let's add the following (quasi ideal?) environment : the 8 following pure gote points : 4, 3½,3, 2½, 2, 1½, 1, ½.

Black plays immediatly in the corner:
Score1 = -4 + 3½ + 3 + 2½ - 2 + 1½ - 1 + ½ -5 = -1
Black waits one move before playing in the corner
Score2 = 4 - 3½ - 3 + 2½ + 2 + 1½ - 1 + ½ - 5 = -2

You see that black loses one point if she waits a move before playing in the corner

If you can read everything out, you do not need any theory.

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

Visualize whirled peas.

Everything with love. Stay safe.

Now that we have found an equilibrium position, we can fiddle with it.

Click Here To Show Diagram Code

[go]$$B Reduce the top play
$$ ---------------------------------------
$$ | . . . . X X X X X X O . . . . . . . . |
$$ | . . O O X B O O O u O . . . . . . . . |
$$ | . . O X X X X X X X O . . . . . . . . |
$$ | . O O X X X O O O v O . . . . . . . . |
$$ | . O X X X X X X X X O . . . . . . . . |
$$ | . X X . X X . O O w O . . . . . . . . |
$$ | . X . . X X X X X X O . . . . . . . . |
$$ | X X . . . . X O O . O . . . . . . . . |
$$ | . . . . . . X X X X O . . . . . . . . |
$$ | . . . , . . X O O . O . . . . , . . . |
$$ | . . . . . . X X X X O . . . . . . . . |
$$ | . . . . . . X O O . O . . . . . . . . |
$$ | . . . . . . X X X X O . . . . . . . . |
$$ | . . . . . . . X O . O . . . . . . . . |
$$ | . . . . . . . X X X X . . . . . . . . |
$$ | . . . , . . . . . , . . . . . , . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ ---------------------------------------[/go]

Now it is better to create the ko in the corner.

Click Here To Show Diagram Code

[go]$$B Reduce the second largest play
$$ ---------------------------------------
$$ | . . . . X X X X X X O . . . . . . . . |
$$ | . . O O X . O O O u O . . . . . . . . |
$$ | . . O X X X X X X X O . . . . . . . . |
$$ | . O O X X X C O O v O . . . . . . . . |
$$ | . O X X X X X X X X O . . . . . . . . |
$$ | . X X . X X . O O w O . . . . . . . . |
$$ | . X . . X X X X X X O . . . . . . . . |
$$ | X X . . . . X O O . O . . . . . . . . |
$$ | . . . . . . X X X X O . . . . . . . . |
$$ | . . . , . . X O O . O . . . . , . . . |
$$ | . . . . . . X X X X O . . . . . . . . |
$$ | . . . . . . X O O . O . . . . . . . . |
$$ | . . . . . . X X X X O . . . . . . . . |
$$ | . . . . . . . X O . O . . . . . . . . |
$$ | . . . . . . . X X X X . . . . . . . . |
$$ | . . . , . . . . . , . . . . . , . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ ---------------------------------------[/go]

Now it is better to take the largest play first.

Click Here To Show Diagram Code

[go]$$B All same same
$$ ---------------------------------------
$$ | . . . . X X X X X X O . . . . . . . . |
$$ | . . O O X X X O O u O . . . . . . . . |
$$ | . . O X X X X X X X O . . . . . . . . |
$$ | . O O X X X X O O v O . . . . . . . . |
$$ | . O X X X X X X X X O . . . . . . . . |
$$ | . X X . X X X O O w O . . . . . . . . |
$$ | . X . . X X X X X X O . . . . . . . . |
$$ | X X . . . . X O O . O . . . . . . . . |
$$ | . . . . . . X X X X O . . . . . . . . |
$$ | . . . , . . X O O . O . . . . , . . . |
$$ | . . . . . . X X X X O . . . . . . . . |
$$ | . . . . . . X O O . O . . . . . . . . |
$$ | . . . . . . X X X X O . . . . . . . . |
$$ | . . . . . . . X O . O . . . . . . . . |
$$ | . . . . . . . X X X X . . . . . . . . |
$$ | . . . , . . . . . , . . . . . , . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
$$ | . . . . . . . . . . . . . . . . . . . |
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And in this particular ideal environment, playing in the environment is better, right?

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

Visualize whirled peas.

Everything with love. Stay safe.