For Figure 2.7 there’s a game where it’s White to move and win, and it asks to calculate all the values of moves

I think if we were just going to calculate without infinitesmals, I would think something like

kind of using the corridor methods, and the sente methods for game trees we were half working out (treat E and A as a local sente, and threating to save the f1 stone as local sente).

So the black estimates add to say 11.5 and the white to say around 11.5 also.

Now I think to play the chilled game we need to mark the positions, and I think the estimates help do that. I think that means we should and can mark 11 points for each in the respective areas to tax away those integer points like so:

which seems to be fairly close to what is shown in figure 2.9 for the markings. Then I think we can try calculate chilled values.

a) is ↓ from the previous post (if white plays it’s 0 in the chilled game, 3 points minus 3 points tax), and if black plays it turns into *, so {*|0}= ↓.

b) It seems like it’s just 1/2? The only way I could reason it in the chilled game would be if black moves, it’s worth zero (territory is cancelled by the marking), but if white tries to move then they pick up a marking which is like -1? So maybe that would be 1 point to black. If so it would make the position {0|1}=1/2 which actually is the value 1/2 believe it or not. The “simplest” number between 0 and 1 is 1/2.

c) If black plays it’s 0, if white also zero, so this is just *.

d) I think if black plays it moves to -1/2, because they remove one white mark and then it’s the unmarked corridor from (b) except now it’s white so a minus sign. If white plays they move it to 0. So that would be {-1/2|0}=-1/4 and that genuinely is the number -1/4 in CGT also.

e) If white plays it’s 0, if Black plays it moves to a three space area with one mark. I might need a tree for this one:

It looks like it’s also worth down, like (a), which agrees with the book.

f) If white plays 0, if black plays it’s the 1/2 corridor for white so it’s -1/4 again.

g) looks more complicated. If Black plays 0, if white plays… tree time again I guess.

So it turns out it’s worth double up *, or ⇑* which is also ↑+↑+*, and I mentioned it in a tiny section in the last post.

h) is the same as (c) so it’s also *.

So that’s the whole position.

You can add it all up, to get

1/2 - 1/4 - 1/4 + ↓ + ↓ + ⇑* + * + * = *

because ⇑*= ↑+↑+*, ↑=-↓ and * + *=0.

So the whole position has an infinitesmal of * which is a first player or next player win. Meaning that if White can play first they can win, which is what the problem says.

So given that apparently you should play on something worth * to win so c or h. That leaves the total at 0 for the rest of the board which is supposed to be a second player win in the chilled game. Then I guess that rounds to one point somehow in the actual go game.

It say the best move is actually to play G on ⇑* which is kind of blacks advantage, since ⇑>* meaning ⇑*>0. In the game tree when white plays it turns that into just an ↑, which is still and advantage to black but smaller. overall it removes a ↑* by playing (the difference in the positions from the tree), and so I think it turns the whole board positions value to a ↓ which favours white no matter who moves first.

There’s some interesting comments about how playing on any of the numbers is terrible.

It’d be interesting again to understand how the rounding of infinitesimals works. There was some earlier comments of:

If the game totals to 5* or 5+1/4*? Apparently 5+1/4* rounds the same way as 5+1/4 - whatever that is. A game of 5* though if black moves first they win by 6, but if white moves first only 5. If the game was just 5, with no infinitesmals, then it wouldn’t matter who went first, it would just end with B+5.

If a game has value 3↑* then Black can win by either 4 or 2 points. If the game had value 3↑ or 3⇑* then Black can win by 4 or 3 points, as ↑ and ⇑* both exceed zero.

Hopefully there’s more examples of that kind of thing later. Probably this post is a lot to take in, if you haven’t looked at these infinitesmal numbers before, the CGT notation etc. I can try explain it more in another post also.

Edit - maybe a sample playout of the endgame with katago, W+1 in area scoring.