Engame value of ko

Gérard TAILLE · Post by **Gérard TAILLE** » Wed Dec 09, 2020 8:20 am

Bill Spight wrote:
$$Wc
$$ -----------------
$$ | 1 X . X . . . |
$$ | X O X X . . . |
$$ | O O O X . . . |
$$ | . . . . . . . |
$$ | . . . . . . . |
$$ | . . . . . . . |
$$ | . . . . . . . |
$$ -----------------
Click Here To Show Diagram Code
[go]$$Wc $$ ----------------- $$ | 1 X . X . . . | $$ | X O X X . . . | $$ | O O O X . . . | $$ | . . . . . . . | $$ | . . . . . . . | $$ | . . . . . . . | $$ | . . . . . . . | $$ -----------------[/go]
Do you disagree that the count of this position (given no ko threats) (Edit: after , OC) is -1 ⁴/₉?

I suppose that we could set up 9 such corners and see if White has 13 points. But that would be tedious and possibly unclear, since kos do not add and subtract like combinatorial games.

I not only agree that the count after

is -1 ⁴/₉ but that was exactly my calculation.

Click Here To Show Diagram Code: [go]$$Bc $$ ----------------- $$ | . X 1 X . . . | $$ | X O X X . . . | $$ | O O O X . . . | $$ | . . . . . . . | $$ | . . . . . . . | $$ | . . . . . . . | $$ | . . . . . . . | $$ -----------------[/go]

In addition, after

the count of the position is -¹/₃

That was my first calculation and my first conclusion : if white plays first he will reach a position counted -1 ⁴/₉ and if black plays first she will reach a position counted -¹/₃ then the original position should be counted -⁸/₉ with a miai value of ⁵/₉.

In fact this result is not correct because we have now to take into account gote/sente situation.
When white plays first it seems he gains ⁵/₉ but now a following move will gain ⁷/₉. That means that white

is sente.
In the other hand when black plays first it seems she gains ⁵/₉ and a following move will gain ¹/₃. That means that

is gote.
Finally I count the initial position -²/₃ with a black reverse sente move equal to ¹/₃.
OC, the exchange

gains nothing because it is a "normal" sente move which can act as a ko threat if temperature in the environment is between ¹/₃ and ⁷/₉.
Do you agree Bill?

Bill Spight · Post by **Bill Spight** » Wed Dec 09, 2020 8:57 am

Gérard TAILLE wrote:
Bill Spight wrote:
$$Wc
$$ -----------------
$$ | 1 X . X . . . |
$$ | X O X X . . . |
$$ | O O O X . . . |
$$ | . . . . . . . |
$$ | . . . . . . . |
$$ | . . . . . . . |
$$ | . . . . . . . |
$$ -----------------
Click Here To Show Diagram Code
[go]$$Wc $$ ----------------- $$ | 1 X . X . . . | $$ | X O X X . . . | $$ | O O O X . . . | $$ | . . . . . . . | $$ | . . . . . . . | $$ | . . . . . . . | $$ | . . . . . . . | $$ -----------------[/go]
Do you disagree that the count of this position (given no ko threats) (Edit: after , OC) is -1 ⁴/₉?

I suppose that we could set up 9 such corners and see if White has 13 points. But that would be tedious and possibly unclear, since kos do not add and subtract like combinatorial games.
I not only agree that the count after is -1 ⁴/₉ but that was exactly my calculation.
$$Bc
$$ -----------------
$$ | . X 1 X . . . |
$$ | X O X X . . . |
$$ | O O O X . . . |
$$ | . . . . . . . |
$$ | . . . . . . . |
$$ | . . . . . . . |
$$ | . . . . . . . |
$$ -----------------
Click Here To Show Diagram Code
[go]$$Bc $$ ----------------- $$ | . X 1 X . . . | $$ | X O X X . . . | $$ | O O O X . . . | $$ | . . . . . . . | $$ | . . . . . . . | $$ | . . . . . . . | $$ | . . . . . . . | $$ -----------------[/go]
In addition, after the count of the position is -¹/₃

That was my first calculation and my first conclusion : if white plays first he will reach a position counted -1 ⁴/₉ and if black plays first she will reach a position counted -¹/₃ then the original position should be counted -⁸/₉ with a miai value of ⁵/₉.

Based upon the assumption that the position is gote.

Gérard TAILLE wrote:In fact this result is not correct because we have now to take into account gote/sente situation.
When white plays first it seems he gains ⁵/₉ but now a following move will gain ⁷/₉. That means that white is sente.

would gain ⁵/₉ if it were gote, but it is sente-like, so it gains ⁷/₉ = -⅔ + 1 ⁴/₉. (It is not sente because the sente sequence,

-

, does not lower the original temperature.)

Gérard TAILLE wrote:In the other hand when black plays first it seems she gains ⁵/₉ and a following move will gain ¹/₃. That means that is gote.

And it gains only ⅓ = -⅓ + ⅔.

Gérard TAILLE wrote:Finally I count the initial position -²/₃ with a black reverse sente move equal to ¹/₃.
OC, the exchange gains nothing because it is a "normal" sente move which can act as a ko threat if temperature in the environment is between ¹/₃ and ⁷/₉.
Do you agree Bill?

Like a normal sente,

-

gains nothing in terms of points, and

can be a ko threat.

Gérard TAILLE · Post by **Gérard TAILLE** » Wed Dec 09, 2020 10:35 am

Click Here To Show Diagram Code: [go]$$Wc $$ ----------------- $$ | . X . X . . . | $$ | X O X X . . . | $$ | O O O X . . . | $$ | . . . . . . . | $$ | . . . . . . . | $$ | . . . . . . . | $$ | . . . . . . . | $$ -----------------[/go]

We saw that if temperature of the environment is greater than ⅓ then black cannot play in the area and white can play in sente if temperature is not greater than ⁷/₉.
What if temperature is lower or equal to ⅓ ?

Click Here To Show Diagram Code: [go]$$Wc White to play $$ ----------------- $$ | 1 X 2 X . . . | $$ | X O X X . . . | $$ | O O O X . . . | $$ | . . . . . . . | $$ | . . . . . . . | $$ | . . . . . . . | $$ | . . . . . . . | $$ -----------------[/go]

connect

in the environment
white reaches in gote a position counted -1.

Click Here To Show Diagram Code: [go]$$Bc Black to play $$ ----------------- $$ | 2 X 1 X . . . | $$ | X O X X . . . | $$ | O O O X . . . | $$ | . . . . . . . | $$ | . . . . . . . | $$ | . . . . . . . | $$ | . . . . . . . | $$ -----------------[/go]

in the environment

connect

in the environment

Comparing the two results the difference is roughly the temperature of the environment. What does that mean? It appears that white interest is to wait as far as possible before playing in the area. Consequently the expecting sequence is the last diagram with black playing first in the area.
This is an unexpected result : white must avoid the exchange

(except for using this sequence as a ko threat) and black must play the exchange

in sente as soon as the temperature of the environment drops to ⅓.
Here is an example:

Click Here To Show Diagram Code: [go]$$Wc white to play $$ ----------------- $$ | b X . X . X . | $$ | X O X X X X X | $$ | O O O O O O O | $$ | . . O O O . . | $$ | O O O O O O O | $$ | O X X X X X X | $$ | a O X . X X . | $$ -----------------[/go]

white to play must play "a" to win. Playing "b" in sente will be a mistake.

It is a rather strange result : the white sente

exchange is bad unless it is used as a ko threat. The expected play is the sente exchange

as soon as temperature drops to ⅓.
Very interesting indeed.

Bill Spight · Post by **Bill Spight** » Wed Dec 09, 2020 11:23 am

Gérard TAILLE wrote:
$$Wc
$$ -----------------
$$ | . X . X . . . |
$$ | X O X X . . . |
$$ | O O O X . . . |
$$ | . . . . . . . |
$$ | . . . . . . . |
$$ | . . . . . . . |
$$ | . . . . . . . |
$$ -----------------
Click Here To Show Diagram Code
[go]$$Wc $$ ----------------- $$ | . X . X . . . | $$ | X O X X . . . | $$ | O O O X . . . | $$ | . . . . . . . | $$ | . . . . . . . | $$ | . . . . . . . | $$ | . . . . . . . | $$ -----------------[/go]
We saw that if temperature of the environment is greater than ⅓ then black cannot play in the area and white can play in sente if temperature is not greater than ⁷/₉.
What if temperature is lower or equal to ⅓ ?

$$Wc White to play
$$ -----------------
$$ | 1 X 2 X . . . |
$$ | X O X X . . . |
$$ | O O O X . . . |
$$ | . . . . . . . |
$$ | . . . . . . . |
$$ | . . . . . . . |
$$ | . . . . . . . |
$$ -----------------
Click Here To Show Diagram Code
[go]$$Wc White to play $$ ----------------- $$ | 1 X 2 X . . . | $$ | X O X X . . . | $$ | O O O X . . . | $$ | . . . . . . . | $$ | . . . . . . . | $$ | . . . . . . . | $$ | . . . . . . . | $$ -----------------[/go]
connect
in the environment
white reaches in gote a position counted -1.

$$Bc Black to play
$$ -----------------
$$ | 2 X 1 X . . . |
$$ | X O X X . . . |
$$ | O O O X . . . |
$$ | . . . . . . . |
$$ | . . . . . . . |
$$ | . . . . . . . |
$$ | . . . . . . . |
$$ -----------------
Click Here To Show Diagram Code
[go]$$Bc Black to play $$ ----------------- $$ | 2 X 1 X . . . | $$ | X O X X . . . | $$ | O O O X . . . | $$ | . . . . . . . | $$ | . . . . . . . | $$ | . . . . . . . | $$ | . . . . . . . | $$ -----------------[/go]
in the environment
connect
in the environment

Comparing the two results the difference is roughly the temperature of the environment. What does that mean?

It means that you compared sequences with a different number of moves.

Let a play in the environment gain t points. Then the result after 4 plays in the first diagram is -1 + t. And the result after 4 plays in the second diagram is also -1 + t. All same same.

As I indicated above, the thermograph of this ko position with no ko threats is the line v = -1 + t at or below temperature ⅓.

Gérard TAILLE wrote:It appears that white interest is to wait as far as possible before playing in the area.

Above temperature ⅓ there is the ko threat matter which you have discussed. Below temperature ⅓ the inclined mast indicates that White should wait until the ambient temperature reaches 0, if possible, and Black should make the play as early as possible. (In real life there are no plays on the go board between temperature ⅓ and temperature 0 -- although I have constructed one or two -- so the question is moot.)

Gérard TAILLE wrote:Consequently the expecting sequence is the last diagram with black playing first in the area.
This is an unexpected result : white must avoid the exchange (except for using this sequence as a ko threat) and black must play the exchange in sente as soon as the temperature of the environment drops to ⅓.

Actually, it is more general than that. As a rule it is better to fill a ko at a certain temperature than to take a ko of the same size.

Gérard TAILLE wrote:Here is an example:
$$Wc white to play
$$ -----------------
$$ | b X . X . X . |
$$ | X O X X X X X |
$$ | O O O O O O O |
$$ | . . O O O . . |
$$ | O O O O O O O |
$$ | O X X X X X X |
$$ | a O X . X X . |
$$ -----------------
Click Here To Show Diagram Code
[go]$$Wc white to play $$ ----------------- $$ | b X . X . X . | $$ | X O X X X X X | $$ | O O O O O O O | $$ | . . O O O . . | $$ | O O O O O O O | $$ | O X X X X X X | $$ | a O X . X X . | $$ -----------------[/go]
white to play must play "a" to win. Playing "b" in sente will be a mistake.

It is a rather strange result : the white sente exchange is bad unless it is used as a ko threat. The expected play is the sente exchange as soon as temperature drops to ⅓.
Very interesting indeed.

Yes, it is very interesting.

But it is not unusual.

Gérard TAILLE · Post by **Gérard TAILLE** » Wed Dec 09, 2020 11:53 am

It seems we have now a common understanding.

Click Here To Show Diagram Code: [go]$$Wc $$ ----------------- $$ | a X b X . . . | $$ | X O X X . . . | $$ | O O O X . . . | $$ | . . . . . . . | $$ | . . . . . . . | $$ | . . . . . . . | $$ | . . . . . . . | $$ -----------------[/go]

A white move at "a" not only gains nothing but may be a mistake and must be avoid unless white can use this move as a ko threat between temperature 1/3 and 7/9. The expected local sequence shoud be the black sente exchange black b white a which must occur as soon as temperature drops to 1/3.

Bill Spight · Post by **Bill Spight** » Wed Dec 09, 2020 12:50 pm

Gérard TAILLE wrote:It seems we have now a common understanding.
$$Wc
$$ -----------------
$$ | a X b X . . . |
$$ | X O X X . . . |
$$ | O O O X . . . |
$$ | . . . . . . . |
$$ | . . . . . . . |
$$ | . . . . . . . |
$$ | . . . . . . . |
$$ -----------------
Click Here To Show Diagram Code
[go]$$Wc $$ ----------------- $$ | a X b X . . . | $$ | X O X X . . . | $$ | O O O X . . . | $$ | . . . . . . . | $$ | . . . . . . . | $$ | . . . . . . . | $$ | . . . . . . . | $$ -----------------[/go]
A white move at "a" not only gains nothing but may be a mistake and must be avoid unless white can use this move as a ko threat between temperature 1/3 and 7/9. The expected local sequence shoud be the black sente exchange black b white a which must occur as soon as temperature drops to 1/3.

Well, in the Fujisawa-Son game Son as Black filled a ko in the other corner instead of filling the one with this shape. It didn't matter in that game, but I think that that is generally best, because Black may have the chance to win this ko.

Gérard TAILLE · Post by **Gérard TAILLE** » Wed Dec 09, 2020 2:18 pm

Bill Spight wrote: Above temperature ⅓ there is the ko threat matter which you have discussed. Below temperature ⅓ the inclined mast indicates that White should wait until the ambient temperature reaches 0, if possible, and Black should make the play as early as possible. (In real life there are no plays on the go board between temperature ⅓ and temperature 0 -- although I have constructed one or two -- so the question is moot.)

I never saw an area with a miai value less than 1/3 and I will be interested to analyse some of these.
Can you show us such position Bill?
Each time I tried to find such position I finally failed due to sente/gote consideration which modify the intitial evaluation like our last example where the miai value 5/9 was not correct.

Here is another example

Click Here To Show Diagram Code: [go]$$Bc $$ ----------------- $$ | O 1 . X . . . | $$ | . O X X . . . | $$ | O O O X . . . | $$ | . . . . . . . | $$ | . . . . . . . | $$ | . . . . . . . | $$ | . . . . . . . | $$ -----------------[/go]

looks like gaining 1/9 but it is sente and OC white will answer by taking the ko. In the absence of ko threat it is loss for black and

is a mistake.

The other solution would be to assume black is komaster.
In this case the black move

looks like the first move in a corridor where black can follow by taking the other ko. That means that black

value is ¾.
As you see I failed here to build a 1/9 miai value.

Bill Spight · Post by **Bill Spight** » Wed Dec 09, 2020 3:03 pm

Gérard TAILLE wrote:
Bill Spight wrote: Above temperature ⅓ there is the ko threat matter which you have discussed. Below temperature ⅓ the inclined mast indicates that White should wait until the ambient temperature reaches 0, if possible, and Black should make the play as early as possible. (In real life there are no plays on the go board between temperature ⅓ and temperature 0 -- although I have constructed one or two -- so the question is moot.)
I never saw an area with a miai value less than 1/3 and I will be interested to analyse some of these.
Can you show us such position Bill?
Each time I tried to find such position I finally failed due to sente/gote consideration which modify the intitial evaluation like our last example where the miai value 5/9 was not correct.

Here is another example
$$Bc
$$ -----------------
$$ | O 1 . X . . . |
$$ | . O X X . . . |
$$ | O O O X . . . |
$$ | . . . . . . . |
$$ | . . . . . . . |
$$ | . . . . . . . |
$$ | . . . . . . . |
$$ -----------------
Click Here To Show Diagram Code
[go]$$Bc $$ ----------------- $$ | O 1 . X . . . | $$ | . O X X . . . | $$ | O O O X . . . | $$ | . . . . . . . | $$ | . . . . . . . | $$ | . . . . . . . | $$ | . . . . . . . | $$ -----------------[/go]
looks like gaining 1/9 but it is sente and OC white will answer by taking the ko. In the absence of ko threat it is loss for black and is a mistake.

The other solution would be to assume black is komaster.
In this case the black move looks like the first move in a corridor where black can follow by taking the other ko. That means that black value is ¾.
As you see I failed here to build a 1/9 miai value.

If Black is komaster the miai value of

is ⁴/₉.

Click Here To Show Diagram Code: [go]$$Bc Black komaster $$ ----------------- $$ | O 1 . X . . . | $$ | 3 O X X . . . | $$ | O O O X . . . | $$ | . . . . . . . | $$ | . . . . . . . | $$ | . . . . . . . | $$ | . . . . . . . | $$ -----------------[/go]

In 2 moves Black moves to a position worth ⅓, with a temperature of ⅓.

Click Here To Show Diagram Code: [go]$$Wc Black komaster $$ ----------------- $$ | O 1 . X . . . | $$ | . O X X . . . | $$ | O O O X . . . | $$ | . . . . . . . | $$ | . . . . . . . | $$ | . . . . . . . | $$ | . . . . . . . | $$ -----------------[/go]

In 1 move White move to a position worth -1.

Each move gains on average 1⅓/3 = ⁴/₉.

Gérard TAILLE · Post by **Gérard TAILLE** » Wed Dec 09, 2020 4:09 pm

Bill Spight wrote:
Gérard TAILLE wrote:
Bill Spight wrote: Above temperature ⅓ there is the ko threat matter which you have discussed. Below temperature ⅓ the inclined mast indicates that White should wait until the ambient temperature reaches 0, if possible, and Black should make the play as early as possible. (In real life there are no plays on the go board between temperature ⅓ and temperature 0 -- although I have constructed one or two -- so the question is moot.)
I never saw an area with a miai value less than 1/3 and I will be interested to analyse some of these.
Can you show us such position Bill?
Each time I tried to find such position I finally failed due to sente/gote consideration which modify the intitial evaluation like our last example where the miai value 5/9 was not correct.

Here is another example
$$Bc
$$ -----------------
$$ | O 1 . X . . . |
$$ | . O X X . . . |
$$ | O O O X . . . |
$$ | . . . . . . . |
$$ | . . . . . . . |
$$ | . . . . . . . |
$$ | . . . . . . . |
$$ -----------------
Click Here To Show Diagram Code
[go]$$Bc $$ ----------------- $$ | O 1 . X . . . | $$ | . O X X . . . | $$ | O O O X . . . | $$ | . . . . . . . | $$ | . . . . . . . | $$ | . . . . . . . | $$ | . . . . . . . | $$ -----------------[/go]
looks like gaining 1/9 but it is sente and OC white will answer by taking the ko. In the absence of ko threat it is loss for black and is a mistake.

The other solution would be to assume black is komaster.
In this case the black move looks like the first move in a corridor where black can follow by taking the other ko. That means that black value is ¾.
As you see I failed here to build a 1/9 miai value.
If Black is komaster the miai value of is ⁴/₉.

$$Bc Black komaster
$$ -----------------
$$ | O 1 . X . . . |
$$ | 3 O X X . . . |
$$ | O O O X . . . |
$$ | . . . . . . . |
$$ | . . . . . . . |
$$ | . . . . . . . |
$$ | . . . . . . . |
$$ -----------------
Click Here To Show Diagram Code
[go]$$Bc Black komaster $$ ----------------- $$ | O 1 . X . . . | $$ | 3 O X X . . . | $$ | O O O X . . . | $$ | . . . . . . . | $$ | . . . . . . . | $$ | . . . . . . . | $$ | . . . . . . . | $$ -----------------[/go]
In 2 moves Black moves to a position worth ⅓, with a temperature of ⅓.

$$Wc Black komaster
$$ -----------------
$$ | O 1 . X . . . |
$$ | . O X X . . . |
$$ | O O O X . . . |
$$ | . . . . . . . |
$$ | . . . . . . . |
$$ | . . . . . . . |
$$ | . . . . . . . |
$$ -----------------
Click Here To Show Diagram Code
[go]$$Wc Black komaster $$ ----------------- $$ | O 1 . X . . . | $$ | . O X X . . . | $$ | O O O X . . . | $$ | . . . . . . . | $$ | . . . . . . . | $$ | . . . . . . . | $$ | . . . . . . . | $$ -----------------[/go]
In 1 move White move to a position worth -1.

Each move gains on average 1⅓/3 = ⁴/₉.

Obvioulsy you assume here that black has only one ko threat and black is unable de delay resolving the ko while the temperature of the environment drops. OK in this case I agree on the figue 4/9 providing you play the correct moves for black:

Click Here To Show Diagram Code: [go]$$Bc Black komaster $$ ----------------- $$ | O 1 b X . . . | $$ | a O X X . . . | $$ | O O O X . . . | $$ | . . . . . . . | $$ | . . . . . . . | $$ | . . . . . . . | $$ | . . . . . . . | $$ -----------------[/go]

First of all white must first take the ko with

in order to force black to use her ko threat and retake the ko. Obviouly you play that way by assuming black is komaster so, let's forget about that.
Secondly, unless black has other ko threats, black

must not be played at "a" but at "b".

What about a position with a miai value less than 1/3. Did you find one?

Bill Spight · Post by **Bill Spight** » Wed Dec 09, 2020 5:24 pm

Gérard TAILLE wrote:What about a position with a miai value less than 1/3. Did you find one?

Click Here To Show Diagram Code: [go]$$Wc White komaster $$ . . . . . . . . . | $$ . . . . . . . . . | $$ . . . . . O O O O | $$ . . . X X X X O . | $$ . . X X O O O . . | $$ . . X . X . . . . | $$ ------------------[/go]

Gérard TAILLE · Post by **Gérard TAILLE** » Thu Dec 10, 2020 7:38 am

Bill Spight wrote:
Gérard TAILLE wrote:What about a position with a miai value less than 1/3. Did you find one?

$$Wc White komaster
$$ . . . . . . . . . |
$$ . . . . . . . . . |
$$ . . . . . O O O O |
$$ . . . X X X X O . |
$$ . . X X O O O . . |
$$ . . X . X . . . . |
$$ ------------------
Click Here To Show Diagram Code
[go]$$Wc White komaster $$ . . . . . . . . . | $$ . . . . . . . . . | $$ . . . . . O O O O | $$ . . . X X X X O . | $$ . . X X O O O . . | $$ . . X . X . . . . | $$ ------------------[/go]

If I analysed correctly, the position is sente for black and the score is -4. But the point is that white can play and gain 1/5 point in reverse sente provided white is komaster (5 ko threats).
You found here a beautiful position Bill.

Gérard TAILLE · Post by **Gérard TAILLE** » Fri Dec 11, 2020 5:06 am

Bill Spight wrote:
Gérard TAILLE wrote: Eventually this white "a" black "b" exchange looks like a reversible play and we can verify that point by the following difference game:

$$W
$$ -----------------
$$ | . X . X . O O O X |
$$ | X O X X . O O X . |
$$ | O O O X . O X X X |
$$ | . . . . . . . . . |
$$ | . . . . . . . . . |
$$ | . . . . . . . . . |
$$ | . . . . . . . . . |
$$ | . . . . . . . . . |
$$ -----------------
Click Here To Show Diagram Code
[go]$$W $$ ----------------- $$ | . X . X . O O O X | $$ | X O X X . O O X . | $$ | O O O X . O X X X | $$ | . . . . . . . . . | $$ | . . . . . . . . . | $$ | . . . . . . . . . | $$ | . . . . . . . . . | $$ | . . . . . . . . . | $$ -----------------[/go]
the two positions are equivalent and we can conclude that the white "a" black "b" exchange reverses for both players.
Kos do not add and subtract, although what Berlekamp dubbed placid kos typically do so in terms of average counts. We cannot say that these two kos sum to 0, even though their mast values do so and they have the same temperature.

Bill, I do not understand in which cases I can use or not difference game when ko are implied.
In many occasions you use yourself such difference game (see for example viewtopic.php?p=260154#p260154) so it is not clear for me what is correct or not.

Bill Spight · Post by **Bill Spight** » Fri Dec 11, 2020 6:41 am

Gérard TAILLE wrote:Bill, I do not understand in which cases I can use or not difference game when ko are implied.
In many occasions you use yourself such difference game (see for example viewtopic.php?p=260154#p260154) so it is not clear for me what is correct or not.

First, if there is no ko fight there is no problem. For instance:

Click Here To Show Diagram Code: [go]$$Wc Kosumi vs. Monkey jump 1, White first $$ -------------------------------- $$ | . . B 1 . . O | X B W 2 W . . | $$ | X X 4 W . . O | X . B 5 3 O O | $$ | . X O O O O O | X X X X X O . | $$ | . X X X O O . | . X X O O O . | $$ | . . . X O O O | X X X O . . . | $$ | . . . X X X X | O O O O . . . | $$ | . . . . . . . | . . . . . . . | $$ --------------------------------[/go]

fills ko.

Also, in general, G - G = 0, even if G is not a combinatorial game. But G - G must be finite. Obviously this lies outside the scope of CGT, but mirror play will lead to 0 if G - G is finite.

Also, 3 copies of a simple ⅓ point ko add to 1 or -1. And 3 copies of any simple ko likewise add to an integer.

We may assume any ko fight away. For instance, the final evaluation of Three Points without Capturing assumes no ko fight, although a ko fight may be possible.

Another idea, which I think Martin Mueller came up with, is to assume that one player is not only komaster, but komonster, so that ko fights are futile. For instance, if Black is komonster White can win a ko but cannot take one. And if Black takes a ko she wins it. This generally does not apply in real life, but it offers limits. For instance, if Black is komonster but White wins the difference game, the difference game is good for White. OC, if the komonster wins the difference game that does not mean much.

Even though ko fights were possible in the examples you linked to, I believe that I avoided them. Maybe I goofed, and avoiding them was not always reasonable, but I thought so. This was early on in our discussions, and I wanted to avoid complicated explanations. I would not have chosen your example to start out with, but there we were.

As for this combination:

Click Here To Show Diagram Code: [go]$$Wc $$ ----------------- $$ | . X . X . O O O X | $$ | X O X X . O O X . | $$ | O O O X . O X X X | $$ | . . . . . . . . . | $$ | . . . . . . . . . | $$ | . . . . . . . . . | $$ | . . . . . . . . . | $$ | . . . . . . . . . | $$ -----------------[/go]

Click Here To Show Diagram Code: [go]$$Wc $$ ----------------- $$ | 1 X 2 X . O O O X | $$ | X O X X . O O X . | $$ | O O O X . O X X X | $$ | . . . . . . . . . | $$ | . . . . . . . . . | $$ | . . . . . . . . . | $$ | . . . . . . . . . | $$ | . . . . . . . . . | $$ -----------------[/go]

The result is -1 without a ko fight, as the position after

is 0 on the board by mirror go.

Click Here To Show Diagram Code: [go]$$Wc $$ ----------------- $$ | . X 2 X . O O O X | $$ | X O X X . O O X 1 | $$ | O O O X . O X X X | $$ | . . . . . . . . . | $$ | . . . . . . . . . | $$ | . . . . . . . . . | $$ | . . . . . . . . . | $$ | . . . . . . . . . | $$ -----------------[/go]

The same is true when White takes this ko.

Click Here To Show Diagram Code: [go]$$Bc $$ ----------------- $$ | 2 X 1 X . O O O X | $$ | X O X X . O O X . | $$ | O O O X . O X X X | $$ | . . . . . . . . . | $$ | . . . . . . . . . | $$ | . . . . . . . . . | $$ | . . . . . . . . . | $$ | . . . . . . . . . | $$ -----------------[/go]

The same is true with this sequence.

Click Here To Show Diagram Code: [go]$$Bc Possible ko fight $$ ----------------- $$ | 2 X . X . O O O X | $$ | X O X X . O O X 1 | $$ | O O O X . O X X X | $$ | . . . . . . . . . | $$ | . . . . . . . . . | $$ | . . . . . . . . . | $$ | . . . . . . . . . | $$ | . . . . . . . . . | $$ -----------------[/go]

But in this case there is a possible ko fight.

If we assume that Black has no ko threat, then Black fills the ko and then White does, too. With that assumption this is fine as a difference game, IMO. But really, there is no theory to back this up.

RobertJasiek · Post by **RobertJasiek** » Sun Feb 12, 2023 1:54 am

INTRODUCTION

What are the move value, gains and count?

Click Here To Show Diagram Code: [go]$$B Initial position $$ --------------- $$ | . X . X . X . $$ | X O X X X X X $$ | O O O O . . . $$ | . O . O . . .[/go]

Transformation to a follow-up:

Click Here To Show Diagram Code: [go]$$W Transformation $$ --------------- $$ | 1 X 2 X . X . $$ | X O X X X X X $$ | O O O O . . . $$ | . O . O . . .[/go]

Click Here To Show Diagram Code: [go]$$B Prisoners = -1, count = -2/3 $$ --------------- $$ | O X X X . X . $$ | . O X X X X X $$ | O O O O . . . $$ | . O . O . . .[/go]

The prisoner difference is -1 and the ko has the count 1/3 so the follow-up position has the count -1 + 1/3 = -2/3.

Now, Bill has used this count of the follow-up to derive the alleged count -2/3 of the initial position.

Click Here To Show Diagram Code: [go]$$B Miai? $$ --------------- $$ | W X B X . X . $$ | X O X X X X X $$ | O O O O . . . $$ | . O . O . . .[/go]

His argument has been that the circled stones could be exchanged in either order in a sente-like sequence so would be miai and therefore the initial position would inherit the count -2/3 from its sente-like follower.

However, "miai" is only an informal term. In the transformation above, Black need not play at 2 but he has the alternative option of playing elsewhere and fighting the stage ko. There is no a priori justification why the initial count would have to be determined by the basic ko fight due to the connection Black 2.

The initial position allows a basic ko fight or a stage ko fight. The ko(s) can be fought differently. Such kos can be 'active' or 'hyperactive', their move value, gains or count can depend on the environment and possibly its ko threats. Due to different fights (here: basic ko fight or stage ko fight) in different environments, the initial values can differ. There need not be only one move value, only one black gain, only one white gain and only one count.

BASIC KO FIGHT

If there is the basic ko fight, the initial count is -2/3 indeed, the move value is 1/3 and each gain is 1/3.

STAGE KO FIGHT

However, if there is the stage ko fight, we must not use the initial count -2/3 of the basic ko fight and must not derive the gain (-2/3 - (-3)) / 3 = 7/9 from this abused initial count and the count -3 of the settled white follower after three successive white plays.

Instead, for the stage ko fight, we must calculate its own values.

Click Here To Show Diagram Code: [go]$$B Stage ko fight, Black starts, count = -1/3 $$ --------------- $$ | . X 1 X . X . $$ | X O X X X X X $$ | O O O O . . . $$ | . O . O . . .[/go]

If Black starts in the stage ko fight, the local sequence comprises one play. It does not comprise two plays (with White answering by capturing the ko) because the gain 1/3 of White's ko capture would be smaller than the move value calculated then. Black's start results, after the only one play, in the count -1/3 for the ko with its black ko stone.

Click Here To Show Diagram Code: [go]$$W Stage ko fight, White starts, Black 2 and 4 elsewhere $$ --------------- $$ | 1 5 3 X . X . $$ | X O X X X X X $$ | O O O O . . . $$ | . O . O . . .[/go]

Click Here To Show Diagram Code: [go]$$W Prisoners = -2, count = -3 $$ --------------- $$ | O O O X . X . $$ | C O X X X X X $$ | O O O O . . . $$ | . O . O . . .[/go]

The stage ko fight and White's start result in the prisoner difference -2 and the count -3.

Now, for the stage ko fight, we know that Black's start leads to 1 black excess play and the count -1/3 while White's start leads to 3 white excess plays and the count -3. The tally of both numbers of excess plays is 1 + 3 = 4. We also use the difference of resulting counts to calculate the move value and count of the initial position in the case of a stage ko fight:

Move value = (-1/3 - (-3)) / 4 = 2/3.

We can derive the count from Black's or White's follower:

Count = -1/3 - 1 * 2/3 = -3 + 3 * 2/3 = -1.

Gain of Black's play: -1/3 - (-1) = 2/3.

The gains of White's plays need not all be the same. First, we would have to determine the counts after White 1 and White 3 in the stage ko fight.

Schachus · Post by **Schachus** » Mon Feb 13, 2023 12:30 am

I disagree. I think it is completely uneccessary to count the position where white plays 3 times. It will just never happen, unless it is used as a ko threat in another ko. The reasoning is similar to any other connect and die: white will just never capture there, unless its big for black to answer, because back can never prevent white from capturing.

As such, this position is much more like a connect and die then like a usual stage ko (in which black could protect the first stage without self-atari).

Th only point of ko remaining is that if white captures, black protects and then white tenuki it is actually worse for white than not capturing to begin with(not in terms of counts, but ko threats, white would allow black to take the ko first later. One more reason why white would never start playing there unless the global temperature is so low, that both players will continue playing locally

Your count on the other hand makes it seem like a move in the „stage ko“ would be worth twice as much as a move in the simple ko remaining after white and black play, which might just make white believe it was good to play there once the ambient temperature is 0,5. This would be a mistake, as black could later potentially take the ko first

Life In 19x19

Engame value of ko

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