This 'n' that

[Bill Spight]

Combinatorial games as defined by CGT, including your game, G, have been proven to have average values, which in territorial go, mean average territorial values. The proof does not involve any environment for the games. Your statement says that that proof is false.

Let's try to proceed slowly because it seems there is some misunderstanding.

Consider only my first question:
Question 1) what is the best move for the global game H? Is it a move in the local game G (in that case which move?) or is it a tenuki (a move in the environment E)?

This question may be answered with von Neumann game theory, in which the final score after correct play is the value of the game.

The average value of the game is not part of von Neumann game theory. If the game is a combinatorial game, then its average value may be calculated without regard to any other game, such as an environment, by the method of multiples. Either the average value is produced by some finite number of multiples, or it is approached in the limit as the number of multiples approaches infinity. You seem to dispute that idea.

[Gérard TAILLE]

Combinatorial games as defined by CGT, including your game, G, have been proven to have average values, which in territorial go, mean average territorial values. The proof does not involve any environment for the games. Your statement says that that proof is false.

Let's try to proceed slowly because it seems there is some misunderstanding.

Consider only my first question:
Question 1) what is the best move for the global game H? Is it a move in the local game G (in that case which move?) or is it a tenuki (a move in the environment E)?

This question may be answered with von Neumann game theory, in which the final score after correct play is the value of the game.

The average value of the game is not part of von Neumann game theory. If the game is a combinatorial game, then its average value may be calculated without regard to any other game, such as an environment, by the method of multiples. Either the average value is produced by some finite number of multiples, or it is approached in the limit as the number of multiples approaches infinity. You seem to dispute that idea.

OK I proposed to handle question 1 but you prefer to handle question 2.

The problem is then to evaluate a position without knowing the environment. Because you mentionned on many occasions Berlekamp's komaster theory I keep in mind that ko are not excluded.

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Comparing these to diagrams any go player with white will without any hesitation prefer the first diagram because white has potentially a ko threat at "a".
But if I am correct the method of multiples gives the same value to the two positions.
Maybe nobody knows if the difference between the two diagrams is 0.001 or 0.01 or 0.1 or 0.2 or 0.5 or whatever you want but the differnece is > 0.
That does not mean that I do not like the value given by the method of multiples. On contrary I consider it is the best estimation in the context of our current knowledge of go theory. I am convinced we can invent a new count for positions, taking into account ko threats, but it is for future work ...
BTW if these two positions are part of the same game and it is white to play white will obviously choose to play at b rather than c.

[Gérard TAILLE]

for a human, a picture with a left wall and a right wall is far more easier and pleasant to read than equations

I disagree because

- I find equations easier to read than graphs of mappings,

- graphs of mappings rely on equations and we must use the equations anyway to justify correctness of the graphs of mappings,

- after every move, a new graph of mappings occurs,

- when applied while playing a game, calculating equations is simpler than imagining and mentally constructing graphs of mappings.

The thermography approach is quite interesting : we define an ideal environment depending of only one parameter called temperature

This is not thermography. Even my non-thermographic ideal environment has a granularity, such as 2, 1 or 1/2. Thermography also relies on such a second paramater: the arbitrarily small granularity so that a RICH environment is formed.

I do not understand how you present the results of your analyse when using two parameters (temperature + granularity)
Taking my example: G = {9||4|0} + {3|-3}
With only the temperature t the results are:
1) average territorial value of the local game G : 5½
2) temperature of the local game may be 3½
3) best black move if t ≤ 1 is in {3|-3}
4) the best black move if 1 ≤ t ≤ 3½ is in {9||4|0}
5) the best black move if 3½ ≤ t is tenuki.
I suspect your result are completly identical for the point 1) and 2) but what about the results for the best moves in your envrironments with two parameters?

[Bill Spight]

Combinatorial games as defined by CGT, including your game, G, have been proven to have average values, which in territorial go, mean average territorial values. The proof does not involve any environment for the games. Your statement says that that proof is false.

Let's try to proceed slowly because it seems there is some misunderstanding.

Consider only my first question:
Question 1) what is the best move for the global game H? Is it a move in the local game G (in that case which move?) or is it a tenuki (a move in the environment E)?

This question may be answered with von Neumann game theory, in which the final score after correct play is the value of the game.

The average value of the game is not part of von Neumann game theory. If the game is a combinatorial game, then its average value may be calculated without regard to any other game, such as an environment, by the method of multiples. Either the average value is produced by some finite number of multiples, or it is approached in the limit as the number of multiples approaches infinity. You seem to dispute that idea.

OK I proposed to handle question 1 but you prefer to handle question 2.

The problem is then to evaluate a position without knowing the environment. Because you mentionned on many occasions Berlekamp's komaster theory I keep in mind that ko are not excluded.

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Comparing these to diagrams any go player with white will without any hesitation prefer the first diagram because white has potentially a ko threat at "a".

I edited the diagrams to keep the number of stones the same. Given that the player knows nothing about the environment, then we agree.

But if I am correct the method of multiples gives the same value to the two positions.

Right again.

Maybe nobody knows if the difference between the two diagrams is 0.001 or 0.01 or 0.1 or 0.2 or 0.5 or whatever you want but the differnece is > 0.

Make that ≥ 0. Once enough is known about the environment.

That does not mean that I do not like the value given by the method of multiples. On contrary I consider it is the best estimation in the context of our current knowledge of go theory. I am convinced we can invent a new count for positions, taking into account ko threats, but it is for future work ...

Bueno.

All of the go textbooks I know of will produce the same average territorial value for both diagrams, and, I expect, so would have Hayashi Genbi.

And for any ko ensemble including the first position, thermography allows us to compute a mast value, even if it is not, strictly speaking, an average value.

[Gérard TAILLE]

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I edited the diagrams to keep the number of stones the same. Given that the player knows nothing about the environment, then we agree.

But if I am correct the method of multiples gives the same value to the two positions.

Right again.

Maybe nobody knows if the difference between the two diagrams is 0.001 or 0.01 or 0.1 or 0.2 or 0.5 or whatever you want but the differnece is > 0.

Make that ≥ 0. Once enough is known about the environment.

That does not mean that I do not like the value given by the method of multiples. On contrary I consider it is the best estimation in the context of our current knowledge of go theory. I am convinced we can invent a new count for positions, taking into account ko threats, but it is for future work ...

Bueno.

All of the go textbooks I know of will produce the same average territorial value for both diagrams, and, I expect, so would have Hayashi Genbi.

And for any ko ensemble including the first position, thermography allows us to compute a mast value, even if it is not, strictly speaking, an average value.

Yes Bill all of the go textbooks we know produce the same average territorial value for both diagrams but how many books try to handle ko threats.

Let me add explicitly a very small ko:

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What is the average territorial value of this diagram?
We may ignore the ko threat and say that the territorial value is 6⅓.
But after white "b" we may also take into account the ko threat and decide for the remaining ko white black is komaster. In that case are we allowed to change the territorial value and how ?

[Bill Spight]

Yes Bill all of the go textbooks we know produce the same average territorial value for both diagrams but how many books try to handle ko threats.

OC, books on technique do so. But I know of no Chinese, Japanese, or Korean books that talk about mast values or inclined masts.

Let me add explicitly a very small ko:

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What is the average territorial value of this diagram?

I assume that this is the whole game, that the environment has temperature 0 with no ko threats for either player.

We cannot, strictly speaking, speak of an average territorial value, although traditional go books do so, omitting the word, "average". The mast value is 6⅓, as you indicate.

We may ignore the ko threat and say that the territorial value is 6⅓.
But after white "b" we may also take into account the ko threat and decide for the remaining ko white black is komaster. In that case are we allowed to change the territorial value and how ?

After White b, the ko still has a mast value of ⅓, but the mast is inclined along the line, s = t, up to t = ⅓. Since t = 0, s = 0.

White is what I dubbed a komonster, because White not only wins the ko, but gains from the drop in temperature. Kim Yonghoan also developed a different komonster theory. Berlekamp called our theories pseudothermography. Under our theories after White b the result is 0 with a vertical mast at s = 0.

See the first note in this topic.

[RobertJasiek]

I do not understand how you present the results of your analyse when using two parameters (temperature + granularity) [...] what about the results for the best moves in your envrironments with two parameters?

I will explain this elsewhere: https://www.lifein19x19.com/viewtopic.p ... 48#p265648

Until then, see the CGT literature for the topics of, and related to, rich environment, mean value = count, [local] temperature = move value, thermograph(y), sentetrat, T-orthodox, orthodox forecast, orthodox accounting.

My ideal environment I use mainly as a simple model for an early endgame estimate of the value of starting in the environment. Otherwise, I assume an ordinary environment of simple gotes without follow-ups to derive more specific results than those of CGT.

However, usually I presume no kos (other than basic endgame kos). For advanced endgame evaluation of kos (other than basic endgame kos, see for example http://home.snafu.de/jasiek/kodame.pdf ), continue to listen to Bill!

[Gérard TAILLE]

Well Bill, we cannot agree on all points.

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Though I am not able to put a figure for the average territorial value of the first diagram I am convinced that this value is strictly smaller than 6 because the existence of white potentiel ko threat. For me the existence of a ko threat must add a value on the position.

Let me formulate the same thing in a completly different way.
Assume that a god play for normal go game leads to a win for black by 7 points. IOW assume the god komi is 7.
Assume now another game in which you assume white receives at the beginning 10 tokens white can use as he wants during the game, each token being an absolute ko threat.
What is the god komi of this game. Knowbody knows OC but what is your guess if we ask to all professionnels in order to have a statistic? Are you sure the result of such statistic will be still a komi still equal to 7 ?

[Bill Spight]

Well Bill, we cannot agree on all points.

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Though I am not able to put a figure for the average territorial value of the first diagram I am convinced that this value is strictly smaller than 6 because the existence of white potentiel ko threat. For me the existence of a ko threat must add a value on the position.

Suppose that White to play creates a ko threat, which is the only ko threat ever created. There is a simple ko which arises later with Black to take the ko. As in your diagram. However, the environment as such has a pair of simple gote of the same size (Edit: at least as hot as the ko) such that with correct play Black takes the ko, White plays the threat, Black answers the threat, White takes the ko back,Black takes one of the simple gote, White wins the ko, and then Black takes the other gote. Having the threat gains nothing, because of the structure of the environment.

Let me formulate the same thing in a completly different way.
Assume that a god play for normal go game leads to a win for black by 7 points. IOW assume the god komi is 7.
Assume now another game in which you assume white receives at the beginning 10 tokens white can use as he wants during the game, each token being an absolute ko threat.
What is the god komi of this game. Knowbody knows OC but what is your guess if we ask to all professionnels in order to have a statistic? Are you sure the result of such statistic will be still a komi still equal to 7 ?

Things are not clear, but it seems that an infinite number of absolute ko threats is worth a finite amount. That does not mean that the player with no threats cannot win a ko. For instance, three copies of a simple ko position combine to yield the same result, regardless of ko threats.

Edit: I am assuming a superko rule which does not allow the player with unlimited threats to hang the game. Or perhaps we could give one player 100 or 200 tokens, which would pretty much come to the same thing. Each token would be worth a very small amount. And the second 100 tokens would be worth much less than the first 100. The player with no token against a very large but finite number of tokens should be able to win the second largest simple ko by taking the largest simple ko at every opportunity, and going back and forth between the two.

[RobertJasiek]

Without a fully developed theory including ko threats, they need not alter local counts, gains and move values. Other accounting of ko threats can be imagined: a) as an extra local value, b) as a component of a global value.

[Gérard TAILLE]

Well Bill, we cannot agree on all points.

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Though I am not able to put a figure for the average territorial value of the first diagram I am convinced that this value is strictly smaller than 6 because the existence of white potentiel ko threat. For me the existence of a ko threat must add a value on the position.

Suppose that White to play creates a ko threat, which is the only ko threat ever created. There is a simple ko which arises later with Black to take the ko. As in your diagram. However, the environment as such has a pair of simple gote of the same size such that with correct play Black takes the ko, White plays the threat, Black answers the threat, White takes the ko back,Black takes one of the simple gote, White wins the ko, and then Black takes the other gote. Having the threat gains nothing, because of the structure of the environment.

Sure, depending of the environment, the gain due to the ko threat may be different.

Let me formulate the same thing in a completly different way.
Assume that a god play for normal go game leads to a win for black by 7 points. IOW assume the god komi is 7.
Assume now another game in which you assume white receives at the beginning 10 tokens white can use as he wants during the game, each token being an absolute ko threat.
What is the god komi of this game. Knowbody knows OC but what is your guess if we ask to all professionnels in order to have a statistic? Are you sure the result of such statistic will be still a komi still equal to 7 ?

Things are not clear, but it seems that an infinite number of absolute ko threats is worth a finite amount. That does not mean that the player with no threats cannot win a ko. For instance, three copies of a simple ko position combine to yield the same result, regardless of ko threats.

Edit: I am assuming a superko rule which does not allow the player with unlimited threats to hang the game. Or perhaps we could give one player 100 or 200 tokens, which would pretty much come to the same thing. Each token would be worth a very small amount. And the second 100 tokens would be worth much less than the first 100. The player with no token against a very large but finite number of tokens should be able to win the second largest simple ko by taking the largest simple ko at every opportunity, and going back and forth between the two.

I agree with you Bill, no doubt all tokens have different values with: first token > second token > third token > ...
That is one reason (among others) it is difficult to evaluate the ko threat in the diagram:

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If it does not exist any ko threat in the enviroment than the value of the ko threat "a" can be based on the value of first token but it it exist 100 ko threat in the environment then the value of the ko threat "a" should be based on the value of 100th token that is negligeable.
If, on average (what does mean average here?) we assume let say 5 ko threats in the environment then the value of the ko threat "a" can be based on the value of fifth token.

[Bill Spight]

Well Bill, we cannot agree on all points.

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

$$
$$ ---------------------------
$$ | . . X X O O O c O . . . -
$$ | X X . X X X X X O . . . -
$$ | O X X X O O O O O . . . -
$$ | O O O O . . . . . . . . -
$$ | . . . . O . . . . . . . -
$$ | . . . . . . . . . . . . -
$$

Click Here To Show Diagram Code

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

Though I am not able to put a figure for the average territorial value of the first diagram I am convinced that this value is strictly smaller than 6 because the existence of white potentiel ko threat. For me the existence of a ko threat must add a value on the position.

Suppose that White to play creates a ko threat, which is the only ko threat ever created. There is a simple ko which arises later with Black to take the ko. As in your diagram. However, the environment as such has a pair of simple gote of the same size such that with correct play Black takes the ko, White plays the threat, Black answers the threat, White takes the ko back,Black takes one of the simple gote, White wins the ko, and then Black takes the other gote. Having the threat gains nothing, because of the structure of the environment.

Sure, depending of the environment, the gain due to the ko threat may be different.

Even zero.

Let me formulate the same thing in a completly different way.
Assume that a god play for normal go game leads to a win for black by 7 points. IOW assume the god komi is 7.
Assume now another game in which you assume white receives at the beginning 10 tokens white can use as he wants during the game, each token being an absolute ko threat.
What is the god komi of this game. Knowbody knows OC but what is your guess if we ask to all professionnels in order to have a statistic? Are you sure the result of such statistic will be still a komi still equal to 7 ?

Things are not clear, but it seems that an infinite number of absolute ko threats is worth a finite amount. That does not mean that the player with no threats cannot win a ko. For instance, three copies of a simple ko position combine to yield the same result, regardless of ko threats.

Edit: I am assuming a superko rule which does not allow the player with unlimited threats to hang the game. Or perhaps we could give one player 100 or 200 tokens, which would pretty much come to the same thing. Each token would be worth a very small amount. And the second 100 tokens would be worth much less than the first 100. The player with no token against a very large but finite number of tokens should be able to win the second largest simple ko by taking the largest simple ko at every opportunity, and going back and forth between the two.

I agree with you Bill, no doubt all tokens have different values with: first token > second token > third token > ...
That is one reason (among others) it is difficult to evaluate the ko threat in the diagram:

$$
$$ ---------------------------
$$ | . a X X O O O b O . . . -
$$ | X X . X X X X X O . . . -
$$ | X O X X O O O O O . . . -
$$ | O O O O . . . . . . . . -
$$ | . . . . O . . . . . . . -
$$ | . . . . . . . . . . . . -
$$

Click Here To Show Diagram Code

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

If it does not exist any ko threat in the enviroment than the value of the ko threat "a" can be based on the value of first token but it it exist 100 ko threat in the environment then the value of the ko threat "a" should be based on the value of 100th token that is negligeable.
If, on average (what does mean average here?) we assume let say 5 ko threats in the environment then the value of the ko threat "a" can be based on the value of fifth token.

I would be shocked if the value of a very large number of tokens to take a ko back was as much as one extra play in the opening, i.e., around 14 points. Especially with the possibility of multiple kos arising, some of which the player with no tokens can win. Something like half that, or 7 points, might be closer to the mark. OC, this is all speculation.

[Gérard TAILLE]

Difference games (ii)

I was quite surprised when David Wolfe told me that the White descent in the next diagram was not as good as the hane-and-connect.

$$W Not so good
$$ -------------------
$$ . . . . . 2 1 . . .
$$ . X . X X X O . O .
$$ . . . X O O O . . .
$$ . . . X . . . . . .
$$ . . . . . . . . . .

Click Here To Show Diagram Code

[go]$$W Not so good
$$ -------------------
$$ . . . . . 2 1 . . .
$$ . X . X X X O . O .
$$ . . . X O O O . . .
$$ . . . X . . . . . .
$$ . . . . . . . . . .[/go]

The reason being, he explained, was that it allowed Black to play .

This surprised me for two reasons. One, in estimating territory here, the exchange of and , or vice versa, is standard. And correct. Two, Takagawa, in his Igo Reader series (囲碁読本), aimed at kyu players, had said that the descent was as good as the hane-and-connect, and I had often played it in positions like this, where I had follow-ups that I would not have had with the hane-and-connect.

The difference game makes David's point clear.

$$W Difference game
$$ -------------------
$$ . . . . . . W . . .
$$ . X . X X X O . O .
$$ . . . X O O O . . .
$$ . . . X . . . . . .
$$ . . . . . . . . . .
$$ . . . O . . . . . .
$$ . . . O X X X . . .
$$ . O . O O O X . X .
$$ . . . . W B B . . .
$$ -------------------

Click Here To Show Diagram Code

[go]$$W Difference game
$$ -------------------
$$ . . . . . . W . . .
$$ . X . X X X O . O .
$$ . . . X O O O . . .
$$ . . . X . . . . . .
$$ . . . . . . . . . .
$$ . . . O . . . . . .
$$ . . . O X X X . . .
$$ . O . O O O X . X .
$$ . . . . W B B . . .
$$ -------------------[/go]

For the difference game we set up the negative of the original position, then in the top we let White play the descent and in the bottom we let Black play the hane-and-connect.

$$B Black first
$$ -------------------
$$ . . . . C 1 W . . .
$$ . X . X X X O . O .
$$ . . . X O O O . . .
$$ . . . X . . . . . .
$$ . . . . . . . . . .
$$ . . . O . . . . . .
$$ . . . O X X X . . .
$$ . O . O O O X . X .
$$ . . . . W B B . . .
$$ -------------------

Click Here To Show Diagram Code

[go]$$B Black first
$$ -------------------
$$ . . . . C 1 W . . .
$$ . X . X X X O . O .
$$ . . . X O O O . . .
$$ . . . X . . . . . .
$$ . . . . . . . . . .
$$ . . . O . . . . . .
$$ . . . O X X X . . .
$$ . O . O O O X . X .
$$ . . . . W B B . . .
$$ -------------------[/go]

Black to play makes one point of territory (marked).

$$W White first
$$ -------------------
$$ . . . . 2 1 W . . .
$$ . X . X X X O . O .
$$ . . . X O O O . . .
$$ . . . X . . . . . .
$$ . . . . . . . . . .
$$ . . . O . . . . . .
$$ . . . O X X X . . .
$$ . O . O O O X . X .
$$ . . . . W B B . . .
$$ -------------------

Click Here To Show Diagram Code

[go]$$W White first
$$ -------------------
$$ . . . . 2 1 W . . .
$$ . X . X X X O . O .
$$ . . . X O O O . . .
$$ . . . X . . . . . .
$$ . . . . . . . . . .
$$ . . . O . . . . . .
$$ . . . O X X X . . .
$$ . O . O O O X . X .
$$ . . . . W B B . . .
$$ -------------------[/go]

White to play only gets jigo. The hane-and-connect is correct. Takagawa was wrong.

This is good example showing how ko threat may impact the analysis.

$$W
$$ -------------------
$$ . . . . b a 1 . . .
$$ . X . X X X O . O .
$$ . . . . O O O . . .
$$ . . . . . . . . . .
$$ . . . . . . . . . .
$$ . . . . . . . . . .
$$ . . . . . . . . . .
$$ . . . . . . . . . .
$$ -------------------

Click Here To Show Diagram Code

[go]$$W
$$ -------------------
$$ . . . . b a 1 . . .
$$ . X . X X X O . O .
$$ . . . . O O O . . .
$$ . . . . . . . . . .
$$ . . . . . . . . . .
$$ . . . . . . . . . .
$$ . . . . . . . . . .
$$ . . . . . . . . . .
$$ -------------------[/go]

After the descent we may expect that a reverse sente black move at "a" is equivalent to the exchange white "a" black "b". Then we can conclude that, on average (!) the descent is equivalent to the hane tsugi.
Now is the point : if a small ko appear in the environment, the possibility for white to exchange white "a" black "b" may act as a ko threat and in this case the descent may appear better than the hane tsugi.
Are you really sure Takagawa was wrong?
Certainly you can build an environment in which the assumption a reverse black move at "a" is equivalent to the exchange white "a" black "b" is wrong and in which the reverse sente black "a" is better but it is also possible to build an environment in which the ko threat white "a" black "b" makes the descent better:

$$W
$$ -------------------
$$ . . . . . . . . . .
$$ . X . X X X O . O .
$$ . . . . O O O . . .
$$ . . . . . . . . . .
$$ . . . . . . . . . .
$$ . . . . . . . . . .
$$ . . . . . . . . . .
$$ . . X X O O . . . .
$$ . . X O . O . . . .
$$ -------------------

Click Here To Show Diagram Code

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

Assume we are in an area counting context.
In this very simple position with only two small yose points remaining, the descent is best isn't it?

OC do not conclude that I do not like difference games but we must avoid to take the result too quickly as granted, due to hidden ko threat aspects.

edit : in one sense the descent can also be seen as an application of the one-two-three rule can't it?

[Bill Spight]

Difference games (ii)

I was quite surprised when David Wolfe told me that the White descent in the next diagram was not as good as the hane-and-connect.

$$W Not so good
$$ -------------------
$$ . . . . . 2 1 . . .
$$ . X . X X X O . O .
$$ . . . X O O O . . .
$$ . . . X . . . . . .
$$ . . . . . . . . . .

Click Here To Show Diagram Code

[go]$$W Not so good
$$ -------------------
$$ . . . . . 2 1 . . .
$$ . X . X X X O . O .
$$ . . . X O O O . . .
$$ . . . X . . . . . .
$$ . . . . . . . . . .[/go]

The reason being, he explained, was that it allowed Black to play .

This surprised me for two reasons. One, in estimating territory here, the exchange of and , or vice versa, is standard. And correct. Two, Takagawa, in his Igo Reader series (囲碁読本), aimed at kyu players, had said that the descent was as good as the hane-and-connect, and I had often played it in positions like this, where I had follow-ups that I would not have had with the hane-and-connect.

The difference game makes David's point clear.

$$W Difference game
$$ -------------------
$$ . . . . . . W . . .
$$ . X . X X X O . O .
$$ . . . X O O O . . .
$$ . . . X . . . . . .
$$ . . . . . . . . . .
$$ . . . O . . . . . .
$$ . . . O X X X . . .
$$ . O . O O O X . X .
$$ . . . . W B B . . .
$$ -------------------

Click Here To Show Diagram Code

[go]$$W Difference game
$$ -------------------
$$ . . . . . . W . . .
$$ . X . X X X O . O .
$$ . . . X O O O . . .
$$ . . . X . . . . . .
$$ . . . . . . . . . .
$$ . . . O . . . . . .
$$ . . . O X X X . . .
$$ . O . O O O X . X .
$$ . . . . W B B . . .
$$ -------------------[/go]

For the difference game we set up the negative of the original position, then in the top we let White play the descent and in the bottom we let Black play the hane-and-connect.

$$B Black first
$$ -------------------
$$ . . . . C 1 W . . .
$$ . X . X X X O . O .
$$ . . . X O O O . . .
$$ . . . X . . . . . .
$$ . . . . . . . . . .
$$ . . . O . . . . . .
$$ . . . O X X X . . .
$$ . O . O O O X . X .
$$ . . . . W B B . . .
$$ -------------------

Click Here To Show Diagram Code

[go]$$B Black first
$$ -------------------
$$ . . . . C 1 W . . .
$$ . X . X X X O . O .
$$ . . . X O O O . . .
$$ . . . X . . . . . .
$$ . . . . . . . . . .
$$ . . . O . . . . . .
$$ . . . O X X X . . .
$$ . O . O O O X . X .
$$ . . . . W B B . . .
$$ -------------------[/go]

Black to play makes one point of territory (marked).

$$W White first
$$ -------------------
$$ . . . . 2 1 W . . .
$$ . X . X X X O . O .
$$ . . . X O O O . . .
$$ . . . X . . . . . .
$$ . . . . . . . . . .
$$ . . . O . . . . . .
$$ . . . O X X X . . .
$$ . O . O O O X . X .
$$ . . . . W B B . . .
$$ -------------------

Click Here To Show Diagram Code

[go]$$W White first
$$ -------------------
$$ . . . . 2 1 W . . .
$$ . X . X X X O . O .
$$ . . . X O O O . . .
$$ . . . X . . . . . .
$$ . . . . . . . . . .
$$ . . . O . . . . . .
$$ . . . O X X X . . .
$$ . O . O O O X . X .
$$ . . . . W B B . . .
$$ -------------------[/go]

White to play only gets jigo. The hane-and-connect is correct. Takagawa was wrong.

This is good example showing how ko threat may impact the analysis.

$$W
$$ -------------------
$$ . . . . b a 1 . . .
$$ . X . X X X O . O .
$$ . . c . O O O . . .
$$ . . . . . . . . . .
$$ . . . . . . . . . .
$$ . . . . . . . . . .
$$ . . . . . . . . . .
$$ . . . . . . . . . .
$$ -------------------

Click Here To Show Diagram Code

[go]$$W
$$ -------------------
$$ . . . . b a 1 . . .
$$ . X . X X X O . O .
$$ . . c . O O O . . .
$$ . . . . . . . . . .
$$ . . . . . . . . . .
$$ . . . . . . . . . .
$$ . . . . . . . . . .
$$ . . . . . . . . . .
$$ -------------------[/go]

After the descent we may expect that a reverse sente black move at "a" is equivalent to the exchange white "a" black "b". Then we can conclude that, on average (!) the descent is equivalent to the hane tsugi.

OC, we need to put reverse sente in quotes, because White a is not really sente. That aside, the fact that the descent gains the same, on average, as the hanetsugi was Takagawa's point.

You have also changed the diagram. In your diagram the territory is less settled, and there appear to be larger plays than the descent or hanetsugi. A White play at c, for example. So it is not clear why we are even talking about the descent vs. the hanetsugi. My diagram has a similar flaw. I should have added a White stone on the fourth line.

Now is the point : if a small ko appear in the environment, the possibility for white to exchange white "a" black "b" may act as a ko threat and in this case the descent may appear better than the hane tsugi.
Are you really sure Takagawa was wrong?

Yes. Takagawa made no mention of ko threats. And difference games come with the ko caveat.

Certainly you can build an environment in which the assumption a reverse black move at "a" is equivalent to the exchange white "a" black "b" is wrong and in which the reverse sente black "a" is better but it is also possible to build an environment in which the ko threat white "a" black "b" makes the descent better:

$$W
$$ -------------------
$$ . . . . . . . . . .
$$ . X . X X X O . O .
$$ . . . . O O O . . .
$$ . . . . . . . . . .
$$ . . . . . . . . . .
$$ . . . . . . . . . .
$$ . . . . . . . . . .
$$ . . X X O O . . . .
$$ . . X O . O . . . .
$$ -------------------

Click Here To Show Diagram Code

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

Assume we are in an area counting context.
In this very simple position with only two small yose points remaining, the descent is best isn't it?

Let's assume that there are no dame, which is what I think you mean, and that White plays last, AGA style, so we can simply count the territory. And that there are no ko threats, which is what I think you also mean.

$$W Descent, var. 1
$$ -------------------
$$ . . . C C 2 1 C . .
$$ . X . X X X O C O .
$$ . . . . O O O . . .
$$ . . . . . . . . . .
$$ . . . . . . . . . .
$$ . . . . . . . . . .
$$ . . . . . . . . . .
$$ . . X X O O . . . .
$$ . . X O 3 O . . . .
$$ -------------------

Click Here To Show Diagram Code

[go]$$W Descent, var. 1
$$ -------------------
$$ . . . C C 2 1 C . .
$$ . X . X X X O C O .
$$ . . . . O O O . . .
$$ . . . . . . . . . .
$$ . . . . . . . . . .
$$ . . . . . . . . . .
$$ . . . . . . . . . .
$$ . . X X O O . . . .
$$ . . X O 3 O . . . .
$$ -------------------[/go]

Result: Zero.

$$W Descent, var. 2
$$ -------------------
$$ . . . 6 5 3 1 7 . .
$$ . X . X X X O C O .
$$ . . . . O O O . . .
$$ . . . . . . . . . .
$$ . . . . . . . . . .
$$ . . . . . . . . . .
$$ . . . . . . . . . .
$$ . . X X O O . . . .
$$ . . X O 2 O . . . .
$$ -------------------

Click Here To Show Diagram Code

[go]$$W Descent, var. 2
$$ -------------------
$$ . . . 6 5 3 1 7 . .
$$ . X . X X X O C O .
$$ . . . . O O O . . .
$$ . . . . . . . . . .
$$ . . . . . . . . . .
$$ . . . . . . . . . .
$$ . . . . . . . . . .
$$ . . X X O O . . . .
$$ . . X O 2 O . . . .
$$ -------------------[/go]

fills ko

Result: Zero.

Now, we can call a ko threat, but it is theoretically larger than or . should not answer .

$$W Descent, var. 3
$$ -------------------
$$ . . . 6 4 3 1 C . .
$$ . X . X X X O C O .
$$ . . . . O O O . . .
$$ . . . . . . . . . .
$$ . . . . . . . . . .
$$ . . . . . . . . . .
$$ . . . . . . . . . .
$$ . . X X O O . . . .
$$ . . X O 2 O . . . .
$$ -------------------

Click Here To Show Diagram Code

[go]$$W Descent, var. 3
$$ -------------------
$$ . . . 6 4 3 1 C . .
$$ . X . X X X O C O .
$$ . . . . O O O . . .
$$ . . . . . . . . . .
$$ . . . . . . . . . .
$$ . . . . . . . . . .
$$ . . . . . . . . . .
$$ . . X X O O . . . .
$$ . . X O 2 O . . . .
$$ -------------------[/go]

takes ko, fills ko

Result: White +2

$$W Hanetsugi
$$ -------------------
$$ . . . C 2 1 3 5 . .
$$ . X . X X X O 7 O .
$$ . . . . O O O . . .
$$ . . . . . . . . . .
$$ . . . . . . . . . .
$$ . . . . . . . . . .
$$ . . . . . . . . . .
$$ . . X X O O . . . .
$$ . . X O 4 O . . . .
$$ -------------------

Click Here To Show Diagram Code

[go]$$W Hanetsugi
$$ -------------------
$$ . . . C 2 1 3 5 . .
$$ . X . X X X O 7 O .
$$ . . . . O O O . . .
$$ . . . . . . . . . .
$$ . . . . . . . . . .
$$ . . . . . . . . . .
$$ . . . . . . . . . .
$$ . . X X O O . . . .
$$ . . X O 4 O . . . .
$$ -------------------[/go]

fills ko

Result: Black +2

So indeed, under these conditions the descent is better than the hanetsugi.

OC do not conclude that I do not like difference games but we must avoid to take the result too quickly as granted, due to hidden ko threat aspects.

As the ko caveat indicates.

[Gérard TAILLE]

$$W
$$ -------------------
$$ . . . . . . 1 . . .
$$ . X . X X X O . O .
$$ . . . X O O O . . .
$$ . . . X . . . . . .
$$ . . . . . . . . . .

Click Here To Show Diagram Code

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

What is my conclusion.
In a non-ko environment, the descent is on average as good as the hanetsugi (OC I agree that in certain environment the hanetsugi is better and this is for example obvious if the temperature of the environment is = 0).
In a ko environment however, the descent seems a little better.
For a go player who do not want to analyse in detail the real environment, can we conclude (even if Takagawa did not mention ko threat) that the descent is, on average, a little better than the hanetsugi ?