pnprog wrote:
It certainly makes more sense (to me at least) than the following:
Quote:
Apparently O wants to compensate uncertainty of non-ideal environments by a defensive error margin to predict whether a player has a guaranteed win.
Since O uses an error margin, the questions are: why does he use any; is the used error margin meaningful; what properties has it?
During the early endgame, we can distinguish the "ensemble" of the largest local endgames / hot local endgame regions from its "environment" of smaller, peaceful local endgames. A typical simplifying assumption about, and model of, the environment is that it only consists of simple local gote endgames.
If furthermore their move values drop constantly, we have an "ideal environment". An environment has a largest move value, which is called the "temperature T" (not in O's book though). E.g., the move values of the ideal environment might drop in steps of 1. Then the move values of the ideal environment would be T, T - 1,..., 3, 2, 1. E.g., if the temperature is T = 4, we have the ideal environment 4, 3, 2, 1.
O assumes without proof and I have proven that playing first in an ideal environment is worth half the temperature, that is, T/2. I think that O does not even speak of "ideal environment", but then he assumes this concept implicitly. E.g., in the ideal environment 4, 3, 2, 1, the first playing player gains +4, his opponent lets the player lose -3, the player gains +2 during the sequence, his opponent causes the player to lose -1. In total, we have + 4 - 3 + 2 -1 = 2 as the net profit of the sequence. Since the temperature of this ideal environment is T = 4, half the temperature, or the value of playing first in the ideal environment, is T/2 = 4/2 = 2. This is the player's previously calculated net profit during the sequence of playing out the ideal environment.
If each environment were an ideal environment, in which playing first were worth exactly T/2 (half the temperature, that is, half the value of the largest move value of the environment), we would not need any error margin at all!
However, environments can be non-ideal environments with different drops of move values. If still we assume that an environment only consists of simple gotes, the exact value of playing first in such an environment is at least 0 and at most T. During the early endgame, we do not know what the exact value is even if there should only be simple gotes. Therefore, O in his book and modern endgame theory estimate the value of playing first in the environment. On average, this value is T/2.
Since this is a model value, or estimate, we can just be aware that it can be imprecise or one can also try to estimate an error margin for this value. In his book, O prefers to consider an explicit error margin, which he assumes to be half the value of playing first in the environment. Since T/2 is the value of playing first in the environment, half of it is T/4. In our example of the temperature (largest move value in the environment) 4, we have T/4 = 4/4 = 1.
Error margins can be introduced in different manners. O has the, somewhat arbitrary, preference of taking the player's perspective and a defensive attitude. He calculates a defensive error margin - defensive from the player's perspective. Therefore, O adds the error margin to the opponent's points.
After calculating points, adding T/2 for the value of playing first in the environment and adding the error margin T/4 to the opponent's points, if the player is still ahead, O seems to speak of his "certainty" of winning the game, assuming reasonable play by both players.
Non-ideal environments (maybe also with local sente endgames and other excitements) have an uncertainty in them whether accounting T/2 for the value of playing first in the environment predicts the winner well enough. O's error margin T/4 for half the value of playing first in the environment shall remove most of the uncertainty of whether the player has a guaranteed win. In the translation, O throws in 99% certainty achieved thereby, but this symbolic figure cannot be derived from the error margin he defines.
There are many ways of defining an error margin. E.g., one could also define it as the maximum possible error, T/2, for a non-ideal enviroment of simple gotes. Even then, we would not have 100% certainty because there are also local sentes, kos and other excitements.
A simpler model would not use any error margin at all. We might simply consider T/2 for the value of playing first in the environment. John likes to claim that my theory would be an overkill, but we can ask whether O's use of his error margin at all is an overkill. I am not convinced that we need such an error margin. Not using it is simpler. Then we also need not worry how good or bad the specific error margin is.
To start with, why should it be more meaningful to add the error margin to the opponent's points instead of using it as a plus-minus tolerance of the next moving player's points?