It is true that we start in G but at the beginning of the ko fight I assumed both players are able to use big ko threats (I mean greater than the freezing ko threat).
In that case before reaching G with the freezing ko threat the sequence from H was:
white takes ko to reach G'
is a black ko threat and G' becomes G
now, from G, it is true that white can end the ko but in that case white has to ignore the last black ko threat. For that reason when you reach C2 the count is C2 = 2δ - t (2δ for the ko threat and the execution of the ko threat, and -t for cooled game) and not simply t (for the cooled game)
With a similar analysis the count for C1 at the freezing ko threat is C1 = K - 2δ

Are you comparing the following scenarios

H => G' => G + d => ... => G + d => C2 + d => C2 + 2d
G => H' => H - d => ... => H - d => C1 - d => C1 - 2d

I don't know if it is this or something else that you mean.

No, surely I was not clear enough. Sorry for that.

The starting position is position G, black to play, with both players having a lot of ko threats including big ones.
The two scenarios begin with the same moves : black takes first the ko and a ko fight takes place until we reach the freezing ko threat.
At the freezing ko threat I say simply that the result do not depend on which player decide to ignore the opponent ko threat:
C1 : black wins this ko while white gets compensation from her ko threat and the sequence is globally sente for black
C2 : white wins this ko while black gets compensation from her ko threat and the sequence is globally gote for black (=> -t appears for cooling)

If your t is still a temperature of something other than the ko, isn't it then a free variable like d?

How do you solve the equation
K = 3t + 4d
without changing variables?

t is for me the tenperature of the environment. As such t is logically used as the cooling value.

What now about d? Assume a ko fight takes place. As far as each player uses big ko threats then the ko fight will continue. At a certain time a player will use a ko threat not big enough and the opponent will ignore this ko threat. Here we have reached the freezing ko threat and the value d is defined for that freezing ko threat. The player playing this freezing ko threat will be able to play the ko threat and to continue by executing the threat to gain 2t + 2d points.

The main result of my analysis is the equation K = 3t + 4d allowing to calculate d. That way you know when you have to answer a ko threat and when you have to ignore it. The stake of the ko is really the 4d points of the equation. If a player decide to not fight a ko she will loose 4d points. By fighting the ko each player will be able to gain 2d points.

If now you assume one player have better ko threats available then the sharing of the 4d points might be 3d for the player with the better ko threats and only d for the other player. OC if one player have no ko threat at all the opponent will take the 4d points. I will be able to show you in detail how to calculate the freezing ko threat in case a player have better ko threats than the other but it is another issue.

If t = 7 then d = (K - 21) / 4 and if k = 18 then d = -0.75
if t = 6 then d = (K - 18) / 4 and if k = 18 then d = 0
...
if t = 1 then d = (K - 3) / 4 and if k = 18 then d = 3.75
if t = 0 then d = K / 4 and if k = 18 then d = 4.5

This doesn't seem right.

What looks wrong for you?
If t = 7 then d = -0.75 => the ko is not active you have better to play in the environment instead of the ko area.
If t = 6 then d = 0 => it does not matter if you play in the environment or you play in the ko area. No ko fight is needed.
IF t = 0 then d = 4.5 => the compensation 2d for the ko is 9 points (fighting the ko is crucial) => The 18 points of the ko are equally shared between the two players which is correct when you assume the two players have about the same ko threats.

This isn't true.


This isn't true.


This isn't true. Also if you wished to derive a result based on the ko being equally shared then that shouldn't require more than one sentence.


You ignored this and every value where the t matters in your equation. It just doesn't add up.

Obviously there is a misunderstaning somewhere but for the time being I do not know where.
Let's take this ko with a swing value K = 18.
According to you, at which temperature t of the environment you will start the ko?

One has little to do with the other as I explained earlier. You often don't play the hottest move and a good example of that is when there is an even number of moves of similar size. Another example are gote moves where there is a large gap in size between the available moves if you'd order them by size.

Just imagine this. Two people put a pile of ten euro bills on a table, each contributing the same cash, and then play the game of taking one bill at a time. Playing first in this game isn't worth anything. Both put down the same cash and the number of ten euro bills is even.

If the players instead use various euro bills then it starts to get interesting, but not very. The players should first take the largest bills that are left in odd numbers. The value of playing first will be determined by the denomination of the bills that are in odd numbers. If you order the odd numbered bills by the size of their denomination then the value of going first is the difference between the sum of the 1st, 3rd, 5th,... and the sum of the 2nd, 4th, 6th, ... But sometimes this wouldn't be good enough.

Now that was with typical banknotes, those will have denominations that are far apart. It gets more interesting when we have notes with various numbers that could be spaced in any way. In that case it will also matter what the gaps between the notes are. You should now select the odd notes that have a large size difference to the next odd note. It will be this difference that you gain on your opponent, not the value printed on the note.

Oh I see. All my calculations and evaluations assume an "ideal" or a "rich enough" environment for which a temperature is well defined and, with this assumption, I think my equations are correct. In addition it is quite easy for me, always with my assumptions on the environment, to define what is the temperature of a local position or the value of a move in a local position.

Now OC I can try to use your approach and understand another calculation. Can you firstly clarify how you define the value of a move or the temperature or whatever else of a local position ? That way I will understand the result you found (especially in your post https://lifein19x19.com/viewtopic.php?p=280342#p280342) and we can have a common understanding.

You didn't say anything about assuming more about the environment than that there is a temperature. If you have two type of environments in mind, an ideal environment and a rich enough environment, then shouldn't you provide two complete definitions, one fore each environment?

This doesn't appear to have much to do with the temperature of the ko position. You appear to conflate the temperature of what you call the environment and that of the ko. In the following you subtract t not because there are moves in the environment but because there are moves in the ko. I did point this out to you previously.



I don't have any issue with it if t = 6 or t = 0. You defined K > 3t an that should mean that t = 6 is out of the question but maybe you meant K >= 3t. The problem is when you implicitly claim that by playing one more or less move in the ko you can gain something in the environment. I have demonstrated multiple times why that isn't so simple.

The problem in the course of the derivation (besides being too terse) that I see is that you change the meaning of t back and forth. At one point it is something in the environment then it is in the ko and at the end it is a mixture. Often times one knows the answers already but has difficulty with deriving or communicating them. So maybe K = 3t + 4d is a marvelous answer to the right question but to me it looks like the environment and the ko have just been mixed together.

Maybe I was not clear enough but if you look at my posts see https://lifein19x19.com/viewtopic.php?p=280333#p280333, https://lifein19x19.com/viewtopic.php?p=280355#p280355 and https://lifein19x19.com/viewtopic.php?p=280382#p280382 you will find explicitly that in my own analysis t is always the temperature of the (ideal or rich) environment.

The only posts where I use t differnetly if when I commented your own analysis where t was either the cooling value or the value of a move in the ko (when you claim t = (a-b)/6 or t = (a-b)/5 or t = 1/3 v or t = 2/3 v ...)

You noted yourself that I defined the variables differently for my own analyse. That's true and for my own analysis, keep in mind that t is always the temperature of the environment and keep in mind that the environment is always an "ideal" or "rich enough" environment.


That's true. My results show in particular two points:
1) the value of the move taking the ko depends on the temperature t of the environment : (K-t)/2 (edit)
2) the value of the freezing ko threat depends also on the temperature t of the environment : (K+t)/2 (taking both the ko threat and the execution of this ko threat)

I understand why you cannot agree with my results : you simply do not assume an "ideal" or "rich enough" environment. I agree with you, without this assumption on the environment, my results cannot be true.

In order to progress and discuss efficiently could you give us your own results for this direct ko : when start fighting the ko? When ignoring an opponent ko threat? That way I am sure we can reach a common understanding.

When you said your t was equivalent to cooling, I chose to believe it until it became clear to me that this was not the case. I was trying to follow the argument.

I don't know why but you appear unwilling to provide definitions for the two kinds of environments.

When you say the environment is rich enough and has temperature t, I understand this to be an oxymoron. Playing first in the environment would have less benefit if it was any richer and it should be "sufficiently impoverished environment" to have that specific benefit. If you choose richer and richer environments you will ends up with something like that game of Euro banknotes in piles, in the sense that it doesn't matter who goes first in the environment. This is what I understand is the whole point with rich environments, the impetus of playing first in the environment is being eliminated from the analysis but you are trying to add it back by just saying that it is there.

Anyway this is my understanding. Maybe it helps clarify my problem with your argument.

For me t is always the temperature of the environment. When cooling with a value t I always assume t being the temperature of the environment but obviously your approach is different. So forget this comparaison with your cooling value.



After some research I succeed to find where I defined what I call a rich environment : see https://lifein19x19.com/viewtopic.php?p=277735#p277735. In the same post you will find what I call the move value of a local position.

In addition you could also look at the post https://lifein19x19.com/viewtopic.php?p=277877#p277877 to see a theoritical application to direct ko but do not take this last post has a possible reasonning in practice. No the goal of this post was only a way to justify some results with a pure theoritical approach. In practice the reasonning is far more simple!

I am not an expert of the "ideal" environment. I guess you could find some material in Bill Spight works. For me I consider an "ideal" environment being a "rich" environment infinitly riched, but it is not really a definition is it?

Definition of rich environment from my [22]:
"±T := {T | -T}. For D > 0 and integer T/D, the rich environment of temperature T is the game E_T_D := ±T ± (T − D) ±... ± D."

The purpose is to have arbitrarily small values T/D so the environment is arbitrarily dense, called "rich". It is so rich that essentially shape exceptions become immaterial if a local shape is considered within a rich environment.

In properly formatted E_T_D, T_D is annotated as an index of E. In T_D, D is annotated as an exponent of T.

Bill Spight and I have defined different environments and called them "ideal environment". He has explained his but I have not found it useful for my study purposes so I cannot recall the definition by heart. It is somewhere in between rich environment and my ideal environment.

Definition of my ideal environment from my [22]:
"An ideal environment consists of simple gotes without follow-ups, with move values T ≥ T_1 ≥ T_2 ≥... ≥ T_N-1 > 0 and move values dropping constantly by D > 0 at N > 0 drops, with the smallest move value T_N-1 = D."

Examples of move values representing ideal environments:

0, 1, 2..

0, 1/2, 1,..

0, 2, 4,..

Fortunetly your definition of a rich environment looks exactly identical to mine.


I do not understand your definition of "ideal" environment. The examples given look like examples of rich environements aren't they?
As I said I do not know the definition of an ideal environment as used by Bill Spight but I understood that in an ideal environment and assuming each player play alternatively in this ideal environment then the gain for the first player is exactly T/2.
For a rich environment it is almost true but strictly speaking: if T/D is even then the first player will effectively gain T/2 points but if T/D is odd the first player will be able to play also the last gote to gain T/2 + D/2.
For me an ideal environment is a pure mathematical object you cannot build in practice. For that reason I prefer to use a rich environmnent knowing I may choose an environment as rich as I want.

Originated from Elwyn Berlekamp, of course. I think in the Economist's View. But then also found entry in Combinational Game Theory (Siegel).



Absolutely not. Mine are sparse. I think we have to read rich as "for every D in DyadicNumbers" or something like that.



By my theorem, it is approximately T/2. Beware odd cardinality!



See. Now average for parities and take the limit.



You can put an ideal environment on a board but not a rich environment. Therefore, the property of building is vice versa:)

Either environment is a mathematical tool for simplification of modelling the positional environment by numbers that are easy to calculate together with a local endgame. Berlekamp chose his to prove theorems for CGT. I chose mine to prove theorems for the ordinarily sized endgame.

You gave the following definition of a rich environment:
I do not see in this definition something saying that you to take some average for parities and I do not see something saying that you have to take a limit.

BTW in your defintion of an ideal environment
I still do not see an average and a limit.

In this context what do you mean by:


I do not understand.
An environment made of the gote points 7, 6, 5 , 4, 3, 2, 1 is a rich environment isn't it? You can build a board with such environment can't we?

Can you give an example of ideal environment which is not a rich environment?

In the proof, we must consider both parities so the average of both parity cases, or consider each case separately. I have used limit informally here; N >> 0 is a sufficient assumption. Hint: Case odd N » 0 => the net profit of all move values of the ideal environment is T/2 + T/(2N) ≈ T/2.



No. It is an ideal environment (without the 0).



Yours.

Please do understand that the definition of rich environment relies on contextual assumptions for considering every D. It is denser than you can and want to imagine. Arbitrarily, infinitely dense. When playing a rich ensemble in order of decreasing values, the remaining stack of smaller move values in the rest of the rich environment remains arbitrarily dense until the very end. You have taken the last local endgame move and the rest of the rich environment proceeds, e.g., by taking 1/128, then 1/128 - eps, where 1 >> eps > 0, then 1/128 - 2*eps etc. to, after arbitrarily many moves, eventually eps, then 0.