One thing I would like to do is make the connection between averages that Go players do when estimating sizes of moves with follow up and CGT means of hot games.

So for example in the following position (easier example with no follow-up)

one might write this as G = {4 | -2 }. Now as a game tree one might called the expected outcome here (4+ -2)/2 = 1 the expected amount of points, or 1 point for black. If you played a game for example with two copies of this position, Black would play and get 4 points, while White would play and get 2. Black gets a net of +2 points with two copies of the position, and so it’s like each one is worth +1 to black.

Actually it turns out that in some sense that is exactly what happens when one calculates the mean of a position in CGT.

It’s defined as

m(G) = lim

_{n → ∞ }L(nG)/n = lim_{n → ∞ }R(nG)/n

with L() and R() being the left and right stops.

So the mean is kind of the limiting average of having n copies of the same game.

It also turns out that genuinely in the above case G+G=2.

Another way the mean is calculated is with cooling.

The temperature t(G) is the least such t so that G_{t} is infinitesimally close to a number. It turns out there’s always one for short games and that G_{t} =m(G) for t>t(G).

So in the position with G={4 | -2}, a go player might assign a temperature or value of the move as the difference of the outcomes from the mean that was calculated (4 -1) = 3. Or |-2 -1| =3.

It’s also the case that G_{3} = { 1 | 1} = 1 + *, which is arbitrarily close to the number 1.

So then the mean m(G) =1 and temperature t(G) is 3 which kind of agrees with maybe a go players calculation.

So at least maybe in some gote looking situations probably what the Go player might call an expected territory and a move value, might correspond to the combinatorial mean value and temperature.

Edit: A thermograph of the above game