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



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.



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.

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

Visualize whirled peas.

Everything with love. Stay safe.

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!

Well Bill, we cannot agree on all points.

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]

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.

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 ?

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.



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.

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

Visualize whirled peas.

Everything with love. Stay safe.

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.

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



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:

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.

Even zero.



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.

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

Visualize whirled peas.

Everything with love. Stay safe.

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

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:

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?

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.



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



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.

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.

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 .

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

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.



As the ko caveat indicates.

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

Visualize whirled peas.

Everything with love. Stay safe.

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 ?

In a non-ko environment, the hanetsugi dominates the descent, even though they gain the same, on average.

In a ko environment, the descent may be better than the hanetsugi, but we cannot say that it dominates it. Neither dominates the other. As to which is statistically better, my bet goes to the hanetsugi. And I understand komonster analysis.

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

Visualize whirled peas.

Everything with love. Stay safe.

Oops your are not allowed to use the wording "dominate" in a ko environment are you?
Anyway and more generally, seeing you use this term "dominate" twice, it seems you do not accept to say that a move could be "on average" (I mean typically in an ideal environment at temperature t) better than another. In a certain sense "dominate" is the contrary of "average". When you use a difference game basically you look indirectly for very specific environments to prove that a move do not dominate another. You are right OC but why you want to ignore that a move may be "on average" the better than another.

Click Here To Show Diagram Code

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

It was exactly in this context I said taking the ko with was not a good move. Surely you can find an environment in which taking the ko is good (you find the subtle {u||||2u|0||-u|||-2u} area to prove that point) but, on average (typically in an ideal environment or in the majority of cases if you prefer) this move is not a good move.

OK Bill I can also understand that you do not want to use the wording "move better than another" if it is not a move that "dominates the other". You are right and for that reason I use the wording "on average better".
BTW it is the same idea for "the average territorial value". this "average" makes sense for me but I know also that for a specific environment the territorial value may be different.

You are in von Neumann game theory for specific games, but the term has a somewhat different meaning in CGT.



Of course it can. That's one point of the heuristic.



I am not ignoring that fact, as I have said over and over.



It looks like White's best local move.



If you are using on average in the statistical sense instead of the CGT sense, then I agree that if there is a play elsewhere that gains the same, thermographically, as taking the ko, then White should usually play elsewhere.



For kos, strictly speaking, we have to talk about mast values instead of average values. White means that we cannot talk about the average values of moves, either. in CGT.

We can talk about statistical averages. But, AFAIK, the statistics have not been done for this hanetsugi vs. descent comparison or any other such close comparison, nor is there anyone who wants to spend the time and energy to do so. {shrug}

----

There are two problems with language, here. One is using "dominate" in two senses, one in the sense of CGT, the other in the sense of von Neumann game theory. The other is using "on average" in two senses, one in the sense of CGT, the other in the sense of statistics.

----

Statistically speaking, here is my general belief about a choice between a ko play and a non-ko play. It is based not only upon my own thinking and experience, but also upon what I know about professional preferences. If komonster analysis applies, I lean towards what it says. Otherwise, if one play gains more thermographically, I lean towards it. Otherwise, if the plays gain the same, thermographically, I lean towards the non-ko play.

But as I say, this is guesswork, because nobody has actually done the statistics, AFAIK.

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

Visualize whirled peas.

Everything with love. Stay safe.

Let me go over this question again, using komonster analysis, stopping play when the area temperature drops to 0. Edited for correctness.



Let's assume that there are no dame, and that there are no ko threats, which is what I think you also mean.

Komonster analysis, stopping when the area temperature is 0.

Click Here To Show Diagram Code

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

gains 2 points.
gains 2 points.
gains 2 points.

White gains net 2 points.

Click Here To Show Diagram Code

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

fills ko

gains 2 points.
gains 2 points.
gains 4 points. (2 points on top plus 2 points for lifting the ko ban and changing the bottom from 2 points for Black to 0.)
gains 2 points.
gains 2 points.
gains 2 points.

White gains net 2 points.

Now, we can call a ko threat, but should not answer .

Click Here To Show Diagram Code

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

takes ko

gains 2 points.
gains 2 points.
gains 4 points.
gains 2 points.
gains 2 points.

White gains net 4 points. (White need not fill the ko at area temperature 0.)

Click Here To Show Diagram Code

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

- gains 2 points.
gains 2 points. (Again, Black need not fill the ko at area temperature 0.)

Black gains net 2 points.

It looks like, in von Neumann game theory under these conditions, and komonster analysis, the descent is better than the hanetsugi.

After the descent, descent dominates taking the ko.

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

Visualize whirled peas.

Everything with love. Stay safe.

I think you have found here the misunderstanding. How could we avoid such misunderstanding in the future?
Because I am far to be as expert as you on such issue I accept all wording you can propose, provided the proposal looks unambiguous.

Can you clarify, for the future, in which meaning you would like to use the words "dominate" or "average"?

BTW, does it make sense for you if I use the wording : move "a" is better than move "b" at temperature t ? In such sentence can move "a" be a tenuki (I mean a move in the environment at temperature t)?

Yes Robert, RICH environment are very interesting indeed.
I tried to answer the following question: what is the best granularity for modelling the real environment a go player may encounter?
I am not able to answer really this question but at least I have an intuition.
OC we need to have some statistical information about real environment but ... these statistics AFAIK are not available. Let me guess some figures in order to try and start to answer the question.
Assume that 100 moves before the end of the game, the value of a move is 10 points. We may conclude that a rich environment like 0.1, 0.2, 0.3, ... 9.9, 10.0 may be good approximation of a real environment. But we have to take into account that in the real life a lot of yose areas are miai (same value) and thus we have to consider the real enviroment is not made of 100 different values but far less.

I imagine easily that we have maybe only 30 or 50 different values for the last 100 moves and a granularity 0.2 or 0.3 looks quite good. Anyway, as far as I am concerned I like the granularity 0.5 simply because all corresponding gote points are very easy to build!

Surely a granularity 0.01 is far too small and a granularity 2.0 is far to big.
What about the famous "ideal environment" ? This environment looks like the limit of a rich enviroment when the granularity decreases until 0. For a mathematical point of view it is a very good environment because the granularity disappear and the results are thus simplier.

For a go player who is looking for the best model I do not know what is the best choice.
Finally I like to use both the "ideal environment" for its simple results and the rich environment with granularity 0.5 because it is easy to build in practice.

More later

I have been struggling with Safari, and my draft got clobbered.

More later, but let me say this for now.

I would prefer not to base anything on statistics, because nobody has done the relevant statistics, nor does anybody seem inclined to do so. So we are left guessing.

For temperatures greater than 0, I would prefer to avoid talking about domination. Difference games without ko fights occur at temperature 0, and in that limited context, for difference games that compare two plays, it is OK to talk about domination.

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

Visualize whirled peas.

Everything with love. Stay safe.

I am pragmatic and use whichever model environment enables me to prove something.



Right.



A player can a) read the theorems and proofs we mathematicians write, or b) simply believe the results we proclaim as having been established as truths, not use any model environment but simply apply the results. For the early endgame, this means that the results are good (if not the best available) approximations for the ordinary environments of real games. For the late endgame, application might even be perfect play if the environments fulfil the made assumptions.

Though my preference goes to thermography calculation, my mathematical curiosity tells me to look at a theory which will be based on the environment
E_t = {½|-½} + {1|1} + {1½|-1½} + {2|2} + {2½|2½} ... + {t|-t}
Let's try to define the value of a position P in this environment, knowing the only thing at our disposal being the score resulting from a game. My proposal is the following : in order to calculate the value of position P, you choose an environment E_t with t big enough to be higher than the expecting value of a move in P and then you calculate:
Delta = ScoreGame(P + E_t) - ScoreGame(E_t)
one time with black to play and one time with white to play. The average of these two Deltas is, per defintion, the value of position P.

Example 1

Click Here To Show Diagram Code

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

Let's take the environment E_t with t = 4

Black to play
ScoreGame(E_t) = +2
ScoreGame(P + E_t) = +6
Delta1 = ScoreGame(P + E_t) - ScoreGame(E_t) = 6 - 2 = +4

White to play
ScoreGame(E_t) = -2
ScoreGame(P + E_t) = +1
Delta2 = ScoreGame(P + E_t) - ScoreGame(E_t) = 1 - (-2) = +3

The average (Delta1 + Delta2) / 2 = 3½ is the expected value of P isn't it ?

more later

Let's now take a more interesting example

Example 2

Click Here To Show Diagram Code

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

Let's take the environment E_t with t = 4

Black to play
ScoreGame(E_t) = +2
ScoreGame(P + E_t) = +7
Delta1 = ScoreGame(P + E_t) - ScoreGame(E_t) = 7 - 2 = +5

White to play
ScoreGame(E_t) = -2
ScoreGame(P + E_t) = +3
Delta2 = ScoreGame(P + E_t) - ScoreGame(E_t) = 3 - (-2) = +5

The average (Delta1 + Delta2) / 2 = 5
The expected score of this position was (8 + (3 - 0)/2)/2 = 4¾

How can we explain this difference?

The answer is quite simple : any environment, including the environment
E₄ = {½|-½} + {1|1} + {1½|-1½} + {2|2} + {2½|2½} + {3|3} + {3½|-3½} + {4|4}
has its specificity. If for example you suppress the gote point {½|-½} from E₄ then you have:

Black to play
ScoreGame(E_t - {½|-½}) = +2½
ScoreGame(P + E_t) = +6½
Delta1 = ScoreGame(P + E_t - {½|-½}) - ScoreGame(E_t - {½|-½}) = 6½ - 2½ = +4

White to play
ScoreGame(E_t - {½|-½}) = -2½
ScoreGame(P + E_t - {½|-½}) = +2½
Delta2 = ScoreGame(P + E_t - {½|-½}) - ScoreGame(E_t - {½|-½}) = 2½ - (-2½) = +5

The average (Delta1 + Delta2) / 2 = 4½

How can we eliminate the specificity of the environment E_t?

In theory it is not quite difficult.
Instead of taking only the environment E_t above, you consider all the environment made of the gote points {½|-½}, {1|1}, {1½|-1½}, {2|2}, {2½|2½} ... and you finally make on average on all these environments.
If for example you want to consider the environments at temperature 4. You consider the 8 gote points:
{½|-½}, {1|1}, {1½|-1½}, {2|2}, {2½|2½}, {3|3}, {3½|-3½}, {4|4}
and you form all possible environments made of these gote points. When you eliminate the miai points then it remains 256 possible environments by simply choosing to take or not to take each gote points.

more later