I hope this isn't original but to humor Gerard who seems to
want to discuss models.
Preliminary about Computer EvaluationTo start with let G = { L | R } be a game where black, or left, can move to game L and white, or right, can move to game R.
Programs like KataGo can compute or rather estimate left's
negamax value and right's negamax value for a game. This estimate can be related to the mean and temperature of G as shown
lnv(G) ~= mean(G) + temperature(G) , left's negamax value
rnv(G) ~= mean(G) - temperature(G) , right's negamax value
and importantly
mean(G) = (mean(G) + temperature(G) + mean(G) - temperature(G)) / 2 ~= (lnv(G) + rnv(G)) / 2
I'll most of the time use equality signs instead of some other sign that may be more correct, just for convenience.
Later proof reading: I might have to check the previous part again later
Simple ModelNow let's look at the simple for total effect of a local position
e =
t +
s +
xThe meaning of the variables are as follows, but briefly:
e : for total effect
t or
k*
t : for a penalty or a tax for extra moves in the local position. Sometimes I'll write
k*
t instead of
t and let
k=-7 (or other value) and
t be the difference between black and white moves in the local position.
s : for the difference in territory
x : for outside influence and other effects
To give total effect for a local position P in game G a meaning I’ll refer to a null game N. The null game is a proxy for G – P, or game G without P, which we may be able to realize in some simple cases and be unable to in more intricate cases. One way to realize G – P would be to remove the position P from G or replace it with something neutral.
Let’s assume that we can find a suitable null game and define the total effect as follows
e = mean(G) - mean(N)
This definition suffices to solve for the outside effect
x in some situations as follows.
Let
e^L =
k * (
t + 1) +
s^L +
xe^R =
k * (
t - 1) +
s^R +
xbe models for the left and right subgames of the local position P. P is embedded in the game G but I'll sometimes treat it as if it were a game, without trying to get into when that is justified and when not.
Using little bit of algebra we can write
mean(G) - mean(N) =
e = (
e^L +
e^R) / 2 = ((
k * (
t + 1) +
s^L +
x) + (
k * (
t - 1) +
s^R +
x)) / 2 = (
s^L +
s^R) / 2 +
xNote: The last equality is only justified when there are equal number of moves by both players and therefore t = 0.
Which gives us a formula for the outside effect
xx = mean(G) - mean(N) - (
s^L +
s^R) / 2
In the next post I'll apply this model to a position suggested by Gerard.
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