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}

Why I said I prefer thermography calculation ?

Click Here To Show Diagram Code

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

For the position above I showed you we can reach a good result by an average calculation on 256 environments. It is not easy of course but it is not the most important point.

Click Here To Show Diagram Code

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

For a position with a ko like the position above I am convinced the average score of the position is 2⅔ but I did not manage to reach this figure by an average calculation on various environments.

As a consequence, though I like an environment like E_t = {½|-½} + {1|1} + {1½|-1½} + {2|2} + {2½|2½} ... + {t|-t}, it is better to use it for practical (or confirmation) reasons but not for try and prove theoritical results.
Anyway we can claim that a result coming from an analysis in such environment is very near the best possible result isn't it?
I suspect I am in alignement with Robert on this point (but I am not quite sure).

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)?

Thank you. Here are my preferences, for your consideration.

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.



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.

Edit: If the statement is based upon unstated information about subsequent plays, IMO, that information should be stated. If not, then my preference is as I said.



Well, a move in the environment at temperature t, gains t. So that would mean that move b gains less that t.

Is it a good understanding to say that if a reference to ko exist then you do not want to use the term "better than" ?
Isn'it a pity in certain position like:

Click Here To Show Diagram Code

[go]$$B Black to play
$$ ---------------------
$$ | . . . . . O b . . |
$$ | . . c a O O X X X |
$$ | O O O O X X X . . |
$$ | . . . . . . . . . |
$$ | . . . . . . . . . |
$$ | . . . . . . . . . |
$$ | . . . . . . . . . |
$$ | . . . . . . . . . |
$$ | . . . . . . . . . |
$$ ---------------------[/go]

Though it is common sense for me to say black "a" is "better" than black "b", obviously I agree that the miai value of black "a" is not greater than black "b".
BTW black "a" is more sente than black "b" because after black a and a white tenuki, black may follow by black c.

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.



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

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".

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

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) ???

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?

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.

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

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?

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



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.