RobertJasiek wrote:
Bill, maybe now I sometimes understand how to calculate a count of a komaster ko. But what does it mean?! Why does the count of the local ko position include points elsewhere or of the temperature? How do we use such a count for analysing the whole board? What is the move value and what does it mean? Next, dreaming of monsters.
The basic idea of evaluating positions by play is that, if a player makes a play that gains a certain number of points and the opponent replies with a play that gains the same amount, the value is the same as that of the original position. The first player plays in the position to be evaluated, and the opponent may reply in the environment of other plays. The play in the environment is assumed to be the largest gote play available and how much it gains is called the temperature of the environment.
Now suppose that we wish to evaluate the play, {7|5}. Let the temperature be
t. Then the result of play at temperature t when Black plays first is 7 - t and the result when White plays first is 5 + t. When 7 - t = 5 + t, t = 1, and the result, no matter who plays first, is 6. So when the temperature is 1, Black can guarantee a result of at least 6 by playing locally to a local score of 7 or by replying in the environment when White plays locally, and White can guarantee a result of at most 6 by playing locally to a local score of 5 or by replying in the environment when Black plays locally. That means that the value of the game under these conditions is 6.
The mean value of the position is also 6.
OC, these conditions may not exist. Suppose, for instance, that the temperature is 2. Then if Black plays locally White can reply to a value of 5, and if White plays locally Black can reply to a value of 7. Neither player will wish to play locally. However, suppose that we wish to estimate the value of the whole board. We may take 6 as the mean value (i.e., count) of the local position. The mean value is 6 at all temperatures 1 or higher.
Now suppose that the temperature is ½. Then if Black plays locally White can reply to a value of 6½, and if White plays locally Black can reply to a value of 5½. Each player will prefer to play locally. Black to play can guarantee a result of at least the mean value of the position, and White to play can guarantee a result of at most the mean value of the position, .
The temperature is an abstraction by which we may calculate mean values of positions and how much a local play gains, on average.
OK. Now let's consider the ko, K = {6||K|0}. As I have said, to evaluate it we have to consider the results of exiting the ko. First, let White be komaster of this ko. Berlekamp's rule is that when the komaster plays in the ko (normally by taking it) she must continue locally on the next turn if the koloser makes a play in the environment. (I have amended Berlekamp's rule to conform with my redefinition of thermography in 1998.)
If Black plays first at temperature t the result will be 6 - t and if White plays first the result will be 0 + 2t = 2t. (White takes the ko, Black replies in the environment with a gain of t, then White wins the ko and Black replies in the environment again. When 6 - t = 2t then t = 2 and the value of the game is 4, as is the mean value of the ko. Similar to the case of the simple gote, when t > 2 neither player will wish to play the ko, and we may estimate the value of the ko position as 4. And when t < 2 each player will prefer to play in the ko; the result when Black plays first is 6 - t, which is greater than 4, and the result when White plays first is 2t, which is less than 4.
Next, let Black be the komaster of this ko. Just because she is komaster does not mean that she is required to take the ko back. At temperatures above 2 Black will indeed prefer not to take the ko back, but to let White win it. At temperature 2 Black is indifferent between taking the ko back or not. Where being komaster pays off is when the temperature is less than 2. Then when White takes the ko Black, instead of replying in the environment, plays a threat large enough that White must answer (called a primary threat in my earlier theory) and then takes the ko back. White must play in the environment and then Black wins the ko. The result is 6 - t, the same as when Black plays first. That means that the value of the game is 6 - t. Black can guarantee a result of at least 6 - t and White can guarantee a result of at most 6 - t. When Black is komaster the value of the game depends upon the temperature. The mean value of the ko is not relevant when t < 2, and we may regard the value of the ko as the value of the game. OC, the temperature is an explicit parameter.
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RobertJasiek
