This 'n' that

[Gérard TAILLE]

As I said above, I would prefer not talking about statistical averages, because, AFAIK, nobody has done the relevant statistics, or is likely to do them anytime soon. So we are lelt with guesswork. Informed guesswork, perhaps, but guesswork, nonetheless.

Average territorial or area values for combinatorial games (go positions without ko fights in the game tree) do not rely upon statistics, and are fine in that context. Average gains of plays or sequences of play which are the differences between the average territorial or area values of the final positions and average values of the starting positions are also fine.

As for domination, it is possible, for instance, to say that a play by Black that moves to a final net score of Black 4 dominates a play that moves to a final net score of Black 3, but that occurs at temperature 0 or below. It is not in general possible to say that a Black play that moves to an average value of 4 dominates a Black play that moves to an average value of 3. Von Neumann game theory requires final scores to assert domination, unless one can prove otherwise. CGT domination can be proven with difference games, which, perforce occur at temperature 0. Now, there are cases where we may feel confident that one play dominates another, even if we haven't proven that, but in general I would prefer not to talk about domination except at temperature 0 or with difference games.

BTW, does it make sense for you if I use the wording : move "a" is better than move "b" at temperature t ?

My preference would be to restrict that to cases where the miai value of move a is greater than the miai value of move b. Otherwise, we are talking about statistics, which requires guesswork.

Let's take an example to understand your proposal

$$W White to play
$$ --------------------
$$ | O O . O O b O . .
$$ | X X X X X X O . .
$$ | X O O O O a O . .
$$ | X X X X X X O . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .

Click Here To Show Diagram Code

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

A move at "a" is worth 4 points and a move at "b" is worth 3,5 points.
In this case do you say a move at "a" is better than a move at "b", though "b" is better than "a" at temperature 0 ?
Isn'it better to specify at which temperature "a" is "better" than "b" and here to say "a" is better than a move at "b" if temperature > 1 ?

[Bill Spight]

As I said above, I would prefer not talking about statistical averages, because, AFAIK, nobody has done the relevant statistics, or is likely to do them anytime soon. So we are lelt with guesswork. Informed guesswork, perhaps, but guesswork, nonetheless.

Average territorial or area values for combinatorial games (go positions without ko fights in the game tree) do not rely upon statistics, and are fine in that context. Average gains of plays or sequences of play which are the differences between the average territorial or area values of the final positions and average values of the starting positions are also fine.

As for domination, it is possible, for instance, to say that a play by Black that moves to a final net score of Black 4 dominates a play that moves to a final net score of Black 3, but that occurs at temperature 0 or below. It is not in general possible to say that a Black play that moves to an average value of 4 dominates a Black play that moves to an average value of 3. Von Neumann game theory requires final scores to assert domination, unless one can prove otherwise. CGT domination can be proven with difference games, which, perforce occur at temperature 0. Now, there are cases where we may feel confident that one play dominates another, even if we haven't proven that, but in general I would prefer not to talk about domination except at temperature 0 or with difference games.

BTW, does it make sense for you if I use the wording : move "a" is better than move "b" at temperature t ?

My preference would be to restrict that to cases where the miai value of move a is greater than the miai value of move b. Otherwise, we are talking about statistics, which requires guesswork.

Let's take an example to understand your proposal

$$W White to play
$$ --------------------
$$ | O O . O O b O . .
$$ | X X X X X X O . .
$$ | X O O O O a O . .
$$ | X X X X X X O . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .

Click Here To Show Diagram Code

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

A move at "a" is worth 4 points and a move at "b" is worth 3,5 points.
In this case do you say a move at "a" is better than a move at "b", though "b" is better than "a" at temperature 0 ?
Isn'it better to specify at which temperature "a" is "better" than "b" and here to say "a" is better than a move at "b" if temperature > 1 ?

If you restrict the comparison to combinatorial games, proper, and environments of combinatorial games, proper, fine. But as long as there is the possibility of some godawful ko at the specified temperature, t, that could allow Black to gain more than 1 - t, better to leave well enough alone.

Edited for correctness.

[Gérard TAILLE]

Let's take an example to understand your proposal

$$W White to play
$$ --------------------
$$ | O O . O O b O . .
$$ | X X X X X X O . .
$$ | X O O O O a O . .
$$ | X X X X X X O . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .

Click Here To Show Diagram Code

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

A move at "a" is worth 4 points and a move at "b" is worth 3,5 points.
In this case do you say a move at "a" is better than a move at "b", though "b" is better than "a" at temperature 0 ?
Isn'it better to specify at which temperature "a" is "better" than "b" and here to say "a" is better than a move at "b" if temperature > 1 ?

If you restrict the comparison to combinatorial games, proper, and environments of combinatorial games, proper, fine. But as long as there is the possibility of some godawful ko at the specified temperature, t, that could allow Black to gain more than 1 - t, better to leave well enough alone.

Concerning kos, I understand perfectly; we must be very careful before claiming a result. Agreed indeed.

In my example above there are no kos. The point is only to know if you mention a temperature when saying a move "a" is better than a move "b".

[Bill Spight]

If you restrict the comparison to combinatorial games, proper, and environments of combinatorial games, proper, fine. But as long as there is the possibility of some godawful ko at the specified temperature, t, that could allow Black to gain more than 1 - t, better to leave well enough alone.

Concerning kos, I understand perfectly; we must be very careful before claiming a result. Agreed indeed.

In my example above there are no kos. The point is only to know if you mention a temperature when saying a move "a" is better than a move "b".

The problem being that the phrase, "at temperature t", leaves a lot of possibilities open, when t > 0.

[Bill Spight]

How I redefine thermography in terms of play in ideal environments

Let there be a game, G₀, with possible subgames, G₁, G₂, .... Let the temperature of G_i be t(G_i). Let there be an environment consisting of an odd number of simple gote of the form {t_j|-t_j}, t_j > 0, for each t_j, such that t_j - t_j+1 = ∆, and such that the net score by alternating play for Black playing first for each t(G_i) = t(G_i)/2, and the net score by alternating play for White playing first for each t(G_i) = -t(G_i)/2; and furthermore that there are sufficient gote in the environment at each t(G_i) for all ko and superko fights at that temperature to be resolved at that temperature.

I hope that is clear and correct.

Edit: I see that I need to say that there needs to be an ideal environment for each game or subgame, G_i, with t(G_i) > 0, and to specify that for each t_j, all games in the environment of the form, {t_j|-t_j}, are played before continuing to t_j+1.

The problem with the net final score of G₀ plus an ideal environment, E, is that it does not tell us the mast value of G₀. But the Lord giveth and the Lord taketh away. To get the walls of the thermograph of G₀ for each player we take the net results of G₀ + E and subtract the net results of E.

Let me illustrate with G₀ = {a||0|-2b}, a,b > 0.

Black first plays to a. The net score with alternating play is s = a - t(G₀)/2. To find the Black scaffold we subtract t(G₀)/2 from that to get

s = a - t(G₀).

Next, White first plays to {0|-2b}.

1) Black replies to 0. The net score with alternating play is s = -t(G₀)/2. To find the White scaffold we subtract -t(G₀)/2 from that to get

s = 0.

The two scaffolds intersect at (s,t) = (0,a).

Because {0|-2b} is a simple gote, we know that t({0|-2b}) = b. If a < b then {a||0|-2b} is a White sente, and we are done. The mast rises vertically from (0,a). We may color it red up to t = b.

But what if a > b? Then

2) Black replies in the environment. Since {0|-2b} is a simple gote, we may add it to the simple gote of the form, {t_j|-t_j}, with t_j = b, without affecting the final score in E with alternating play. {0|-2b} + {b|-b} = -b. So Black plays to -b, and the final score with alternating play is s = -b + t(G₀)/2. To find the White scaffold we subtract -t(G₀)/2 from that to get

s = -b + t(G₀).

The two scaffolds intersect at (s,t) = ((a-b)/2,(a+b)/2).

G₀ is a gote, and the mast rises vertically from ((a-b)/2,(a+b)/2).

[Gérard TAILLE]

If you restrict the comparison to combinatorial games, proper, and environments of combinatorial games, proper, fine. But as long as there is the possibility of some godawful ko at the specified temperature, t, that could allow Black to gain more than 1 - t, better to leave well enough alone.

Concerning kos, I understand perfectly; we must be very careful before claiming a result. Agreed indeed.

In my example above there are no kos. The point is only to know if you mention a temperature when saying a move "a" is better than a move "b".

The problem being that the phrase, "at temperature t", leaves a lot of possibilities open, when t > 0.

Oops I certainly do not want to write strictly move "a" is better than vove "b" at temperature t but more specifically something like for example move "a" is better than move "b" if temperature is 3 < t < 5

$$W White to play
$$ --------------------
$$ | O O . O O b O . .
$$ | X X X X X X O . .
$$ | X O O O O a O . .
$$ | X X X X X X O . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .

Click Here To Show Diagram Code

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

It is just a matter of wording Bill.

In this example theory tells us that the miai value of move "a" is greater than the miai value of move "b" (4 > 3½) but, more precisely, theory tells us that, in an ideal environment at temperature t > 1 then move "a" is better than move "b".

What is your wording to express these results?

1) simply move "a" is better than move "b" meaning only that miai value of move "a" is greater than the miai value of move "b"?
2) in an ideal environment at temperature t > 1 then move "a" is better than move "b"
3) at temperature t > 1 then move "a" is better than move "b" meanning implicitly you use an ideal environment ?
4) you never accept to say move "a" is better than move "b" because you know it is false for some non ideal environment
5) ???

[Bill Spight]

If you restrict the comparison to combinatorial games, proper, and environments of combinatorial games, proper, fine. But as long as there is the possibility of some godawful ko at the specified temperature, t, that could allow Black to gain more than 1 - t, better to leave well enough alone.

Concerning kos, I understand perfectly; we must be very careful before claiming a result. Agreed indeed.

In my example above there are no kos. The point is only to know if you mention a temperature when saying a move "a" is better than a move "b".

The problem being that the phrase, "at temperature t", leaves a lot of possibilities open, when t > 0.

Oops I certainly do not want to write strictly move "a" is better than vove "b" at temperature t but more specifically something like for example move "a" is better than move "b" if temperature is 3 < t < 5

$$W White to play
$$ --------------------
$$ | O O . O O b O . .
$$ | X X X X X X O . .
$$ | X O O O O a O . .
$$ | X X X X X X O . .
$$ | . . . . . . . . .
$$ | . . . . . . . . .

Click Here To Show Diagram Code

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

It is just a matter of wording Bill.

Well, yes. But that's what we are talking about.

In this example theory tells us that the miai value of move "a" is greater than the miai value of move "b" (4 > 3½) but, more precisely, theory tells us that, in an ideal environment at temperature t > 1 then move "a" is better than move "b".

What is your wording to express these results?

1) simply move "a" is better than move "b" meaning only that miai value of move "a" is greater than the miai value of move "b"?
2) in an ideal environment at temperature t > 1 then move "a" is better than move "b"
3) at temperature t > 1 then move "a" is better than move "b" meanning implicitly you use an ideal environment ?
4) you never accept to say move "a" is better than move "b" because you know it is false for some non ideal environment
5) ???

It isn't just that I know it is false for some non ideal environment, it is that that fact is where misunderstandings have occurred in the past. So some additional verbal modification would be a good idea. Such as, "better thermographically" at some temperature or temperatures, or "better with no ko fight" at some temperature or temperatures. Personally, my choice would be to say, "gains more thermographically" in some temperature range, since that is where you see the difference in the wall of the thermograph.

[Gérard TAILLE]

How I redefine thermography in terms of play in ideal environments

Let there be a game, G₀, with possible subgames, G₁, G₂, .... Let the temperature of G_i be t(G_i). Let there be an environment consisting of an odd number of simple gote of the form {t_j|-t_j}, t_j > 0, for each t_j, such that t_j - t_j+1 = ∆, and such that the net score by alternating play for Black playing first for each t(G_i) = t(G_i)/2, and the net score by alternating play for White playing first for each t(G_i) = -t(G_i)/2; and furthermore that there are sufficient gote in the environment at each t(G_i) for all ko and superko fights at that temperature to be resolved at that temperature.

I hope that is clear and correct.

I understand what you mean Bill but for a theorical point of view it is quite unclear.
Basically you take a game G₀, with possible subgames, G₁, G₂ you put this game in an ideal environment and then you find various results concerning the game G₀, including in particular the value (s,t) for the bottom of the mast. That's fine but in your definition of your ideal environment you use the temperature t(G_i) of G_i which is not known at the beginning of the process.
Can the definition of the ideal environment depends on still unkonwn characteristics of the game G₀ and its subgames?
Why not define the ideal environment without any reference to the game studied?

[Bill Spight]

How I redefine thermography in terms of play in ideal environments

Let there be a game, G₀, with possible subgames, G₁, G₂, .... Let the temperature of G_i be t(G_i). Let there be an environment consisting of an odd number of simple gote of the form {t_j|-t_j}, t_j > 0, for each t_j, such that t_j - t_j+1 = ∆, and such that the net score by alternating play for Black playing first for each t(G_i) = t(G_i)/2, and the net score by alternating play for White playing first for each t(G_i) = -t(G_i)/2; and furthermore that there are sufficient gote in the environment at each t(G_i) for all ko and superko fights at that temperature to be resolved at that temperature.

I hope that is clear and correct.

I understand what you mean Bill but for a theorical point of view it is quite unclear.
Basically you take a game G₀, with possible subgames, G₁, G₂ you put this game in an ideal environment and then you find various results concerning the game G₀, including in particular the value (s,t) for the bottom of the mast. That's fine but in your definition of your ideal environment you use the temperature t(G_i) of G_i which is not known at the beginning of the process.
Can the definition of the ideal environment depends on still unkonwn characteristics of the game G₀ and its subgames?
Why not define the ideal environment without any reference to the game studied?

Oh, I did. It is just that the environments need to be ideal at the temperatures of the game and its subgames. The fact that we do not necessarily know those temperatures is the point. We are trying to find them.

[Gérard TAILLE]

It isn't just that I know it is false for some non ideal environment, it is that that fact is where misunderstandings have occurred in the past. So some additional verbal modification would be a good idea. Such as, "better thermographically" at some temperature or temperatures, or "better with no ko fight" at some temperature or temperatures. Personally, my choice would be to say, "gains more thermographically" in some temperature range, since that is where you see the difference in the wall of the thermograph.

When you use the word "thermographically" obviously I assume you use implicitly an "ideal environment" but, with this only word "thermographically" do you assume also implicitly there no ko fight?

BTW, considering a position with a potential ko fight, do you consider there is an interest of playing the game in an ideal environment?

$$B
$$ --------------
$$ | . . . . X . .
$$ | . . O O X . .
$$ | . . O X X . .
$$ | . O O X . . .
$$ | . O X X . . .
$$ | . X . . . . .
$$ | . X . . . . .
$$ | X X . . . . .
$$ | . . . . . . .
$$ | . . . , . . .

Click Here To Show Diagram Code

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

IOW, in the position above, does it make sense to try to play the position by assuming an ideal environment, or do you consider there are no interest?

[Gérard TAILLE]

How I redefine thermography in terms of play in ideal environments

Let there be a game, G₀, with possible subgames, G₁, G₂, .... Let the temperature of G_i be t(G_i). Let there be an environment consisting of an odd number of simple gote of the form {t_j|-t_j}, t_j > 0, for each t_j, such that t_j - t_j+1 = ∆, and such that the net score by alternating play for Black playing first for each t(G_i) = t(G_i)/2, and the net score by alternating play for White playing first for each t(G_i) = -t(G_i)/2; and furthermore that there are sufficient gote in the environment at each t(G_i) for all ko and superko fights at that temperature to be resolved at that temperature.

I hope that is clear and correct.

I understand what you mean Bill but for a theorical point of view it is quite unclear.
Basically you take a game G₀, with possible subgames, G₁, G₂ you put this game in an ideal environment and then you find various results concerning the game G₀, including in particular the value (s,t) for the bottom of the mast. That's fine but in your definition of your ideal environment you use the temperature t(G_i) of G_i which is not known at the beginning of the process.
Can the definition of the ideal environment depends on still unkonwn characteristics of the game G₀ and its subgames?
Why not define the ideal environment without any reference to the game studied?

Oh, I did. It is just that the environments need to be ideal at the temperatures of the game and its subgames. The fact that we do not necessarily know those temperatures is the point. We are trying to find them.

Just for fun, with the game {a||0|-2b} with a > b, the simplest "ideal environment" built for this game seems to be {g0|-g0} + {g1|-g1} with
g0 = (a+b)/2 and g1 = b/2

[Bill Spight]

Just for fun, with the game {a||0|-2b} with a > b, the simplest "ideal environment" built for this game seems to be {g0|-g0} + {g1|-g1} with
g0 = (a+b)/2 and g1 = b/2

It's a bit more complicated than that.

Edited below for correctness.

Let's take the game, {9||0|-4}, with t = (9+2)/2 = 5½. and t({0|-4}) = 2.

The simplest ideal environment that meets all the requirements is this:

{5½|-5½}, {5¼|-5¼}, {5|-5} . . . , {2|-2}, {1¾|-1¾}, {1½|-1½} . . . , {¼|-¼}.

This environment will also work.

{5½|-5½}, {5¼|-5¼}, {4¾|-4¾}, {4¼|-4¼} . . . , {2¼|-2¼}, {2|-2}, {1¾|-1¾}, {1¼|-1¼}, {¾|-¾}, {¼|-¼}.

[Gérard TAILLE]

Just for fun, with the game {a||0|-2b} with a > b, the simplest "ideal environment" built for this game seems to be {g0|-g0} + {g1|-g1} with
g0 = (a+b)/2 and g1 = b/2

It's a bit more complicated than that.

Let's take the game, {9||0|-4}, with t = (9+4)/2 = 6½. and t({0|-4}) = 2.

The simplest ideal environment that meets all the requirements is this:

{6½|-6½}, {6¼|-6¼}, {6|-6} . . . , {2|-2}, {1¾|-1¾}, {1½|-1½} . . . , {¼|-¼}.

This environment will also work.

{6½|-6½}, {6¼|-6¼}, {5¾|-5¾}, {5¼|-5¼} . . . , {2¼|-2¼}, {2|-2}, {1¾|-1¾}, {1¼|-1¼}, {¾|-¾}, {¼|-¼}.

Oops, I made a mistake in my equations and I have to correct my formulas. then:
g0 = (a+3b)/4 and g1 = b/2

Now let's take your example G0 = {9||0|-4} = {9|G1}
Firstly I think the temperature of G0 is 5½ and (not 6½).
Secondly, by applying my formula, why the environment E = {3¾|-3¾} + {1|-1} is not ideal for the game G0?
When t = 5½ (G0 temperature) then the value of a play in the environment is 3¾ - 1 = 2¾ = 5½ / 2
and when the temperature of the environment drops to 2 (G1 temperature) then the value of a play in the environment is 1 = 2/2

[Bill Spight]

Just for fun, with the game {a||0|-2b} with a > b, the simplest "ideal environment" built for this game seems to be {g0|-g0} + {g1|-g1} with
g0 = (a+b)/2 and g1 = b/2

It's a bit more complicated than that.

Let's take the game, {9||0|-4}, with t = (9+4)/2 = 6½. and t({0|-4}) = 2.

The simplest ideal environment that meets all the requirements is this:

{6½|-6½}, {6¼|-6¼}, {6|-6} . . . , {2|-2}, {1¾|-1¾}, {1½|-1½} . . . , {¼|-¼}.

This environment will also work.

{6½|-6½}, {6¼|-6¼}, {5¾|-5¾}, {5¼|-5¼} . . . , {2¼|-2¼}, {2|-2}, {1¾|-1¾}, {1¼|-1¼}, {¾|-¾}, {¼|-¼}.

Oops, I made a mistake in my equations and I have to correct my formulas. then:
g0 = (a+3b)/4 and g1 = b/2

Now let's take your example G0 = {9||0|-4} = {9|G1}
Firstly I think the temperature of G0 is 5½ and (not 6½).

Yes. How silly of me. I have corrected the above note.

Secondly, by applying my formula, why the environment E = {3¾|-3¾} + {1|1} is not ideal for the game G0?

First, it is not hot enough. 3¾ < 5½.
Second, it does not include {2|-2}.
Third, it is not ideal. 3¾ - 1 = 2¾, not 3¾/2 = 1⅞.

[Gérard TAILLE]

Yes. How silly of me. I have corrected the above note.

Secondly, by applying my formula, why the environment E = {3¾|-3¾} + {1|-1} is not ideal for the game G0?

First, it is not hot enough. 3¾ < 5½.
Second, it does not include {2|-2}.
Third, it is not ideal. 3¾ - 1 = 2¾, not 3¾/2 = 1⅞.

OK Bill, it was not clear in your definition that the environment must contain at least a gote point at the temperature of G0 and all its subgames (your first two points).
However I do not understand your third point. Why to calculate the score of the environment at temperature 3¾? I do not see in the defintion where you calculate the score of the environment for a temperature that does not correspond to the temperature of G0 or a subgame of G0.

Because in E = {3¾|-3¾} + {1|-1} the gote 5½ and 2 are missing why not to take the simple environment:

E = {5½|-5½} + {3¾|-3¾} + {2|-2} + {1|-1}