Gérard TAILLE wrote:
Bill Spight wrote:
Well, for convenience let's look at temperatures that are integers and an environment where the maximum temperature is 4. Also for convenience, let's write the gote {t | -t} as ±t.
Here are some of those environments.
1) ±4. Playing first in this environment gains 4. Max error =2.
2) ±4, ±4. Playing first gains 0. Max error = 2.
3) ±4, ±4, ±4. Playing first gains 4. Max error = 2.
Etc.
Regardless of the parity of the number of gote in the environment, the maximum error will always be the same.
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I am not trying to prove you wrong. Your approach is valid.
But an ideal environment is an analytical device. Given a non-ko position to analyze, you can always find an ideal environment such that, when the time comes for the players to play in the environment, the gain from playing in the environment will be ½ the ambient temperature. That simplifies the task.
(Analyzing kos may require environments with negative temperatures, and the rules are different.)
Though you think my approach may be valid it seems to me there remains some misunderstanding.
I looked at your reference:
https://senseis.xmp.net/?MethodOfMultiples and I feel it is really an very interesting approach.
It is not mine, it is the way that mathematicians proved what is called the mean value theorem for games such as independent non-ko go positions. I call it the method of multiples. With it you can derive the average value of positions and moves without appealing to an environment. If you evaluate them by means of an environment but get different answers, you are talking about something else. To get the same answers by your approach you have to take the difference between successive gote and the value of the last gote to 0 in the limit.
Quote:
In an other hand when in you post above you take an environmemnt this only ±4 gote points I think you miss my point.
I assure you I have not missed your point. Your model of the environment is quite similar to the one I started off with many years ago.
But in that particular note I thought that you were allowing a more general model of the environment than your own, for the purpose of discussion.
Quote:
When I wrote
env = g_{0}/2 + g_{n-1}/4
you have to understand (of course you may not agree on that !) that you cannot forget that, at the very end of the yose, a player will play the last yose point which is the smallest one.
Consider this environment:
±4, ±3, ±3 ±2, ±2, ±1, ±1
Here the smallest gote gains 1 pt. for the player who takes it. However, the gain from playing first in this environment is 4 pts. the same as it is from playing first in this environment.
±4
Consider this environment:
±4, ±4, ±3, ±3 ±2, ±2, ±1, ±1
The gain from playing first in it is 0.
If we allow both environments with a maximum gain from the largest play of 4 pts., then we can estimate the gain from playing first in that environment as 2 pts. with a maximum error of 2 pts. Any other estimate will have a larger maximum error. Now, such a large error may be undesirable for certain purposes, but the estimate yields the same values as the method of multiples.
Quote:
Let's take an area counting which is far easier to understand:
Putting aside ko complications g_{n-1} is typically equal to 1 which the value of a move on a dame point. Thus
env = g_{0}/2 + 0.25
Putting aside ko considerations, the method of multiples indicates that the average value of a dame is 0 and taking a dame gains 1 pt. at area scoring. Playing first in an environment of an unknown number of dame points gains ½ ±½. But your equation gives a different estimate.
Quote:
The point is that the number of gote moves in the environment is finite (say equal to n) and, as a consequence one of the player will take the last dame point (it looks the CGT infinitesemals analysis of getting the last move in the game Robert Jasiek is refering to).
On SL and here I have written by far more than anyone else on go infinitesimals.
It seems to me that Robert's post addresses your concerns.
All models of actual environments on the go board are idealizations. When applied to evaluation of positions and moves, different models may give different answers. That does not mean that one model is right and the other one is wrong. No model pretends to do more than approximate actual boards. The method of multiples yields the same values for non-ko positions, and for many ko positions, as the traditional values. There are good reasons that these values are useful. Your model yields the same values in the limit. That's fine. The method of multiples yields the value of a sente position only in the limit, as well.
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