So move trees and values are still a bit confusing, but I suppose we can try get a handle on this “chilling” idea, or at least get some ideas down.

They say in 2.2 that they effectively “tax away” some integer number of points in the position so that the move values are close to zero. The motivation was that in a lot of the corridor positions the values of moves are close to 1 point, so they focus on the differences to the nearest point I guess.

It says for the moment, just accept the methodology even though it’s not very well motivated as it will be useful later.

So it then talks about playing the “chilled game”.

If you jump ahead to chapter 3, the motivation seems to be that Go is a game about gaining as many points as you can, while Combinatorial Game theory is about playing the last move. Then chilling is supposed to turn Go endgame into a “tepid” game where combinatorial game theory applies and has been used successfully.

It’s mentioned on page 18, that there’s no real rule on how to do the markings but just some way to normalise the position near zero.

So…

I think you take a position, tax away a certain number of points and mark this in the position. Then every move by either player has to add a marking, or remove an opponents marking from the area they play in.

It looks like say when you have a position like:

you chill and mark it in some way (I would think since

black is making between 2-3 points, you add two markings to normalise near zero.)

So in the above it seems like it matches pg18’s example of a similar looking position.

I think the markings are kind of keeping track of the tax, so playing a move costs a point. Meaning of black plays it goes to

and if white plays it goes to

The actual scores in these positions in the chilled game should be something like the points you get minus the tax of the markings. So when black gets three points but has three markings they get a net of zero.

I think that’s how some of the examples work also - so maybe we can look at a couple of those?

Without chilling the example above shows how to write the unsettled position in CGT notation. You write the options for BLack (L for left) in the left side of some set brackets, and White’s (right) options in the right side. In this case Black can make two points, and White can turn it to zero so the unsettled position is {2|0}. This is probably a fuzzy number where the first player can win, so while it has mean 1 and temperature 1, it isn’t the number 1.

## The number 1

Actually the number 1 can be written in a number of ways like {0|2}. Typically numbers have all the left options lower than all the right options. When it represents a number you can kind of think of the left options as lower bounds for a number, and the right as upper bounds, and there’s a simplicity theorem that tells you, when it’s a number it’s the simplest number that fits in that interval (we can come back to that if it’s relevant).

In the chilled game you tax away one point, and if Black plays they need to add an additional black marking, while if white plays, they remove a black marking:

So then, I think, when evaluating the position, you kind of see the markings as a tax or a penalty on points. When black captures 1 stone for two points, they have two markings, so it’s a net of 0. When white saves the one stone, they remove a marking, and so again the local score is zero.

The position is written {0|0}=*, which is the value * we mentioned before for dame. I think the idea is that in this chilled game, it being tepid, is that players aren’t really incentivised to make moves like this until there’s really not too many other moves left. You’re just trying to avoid running out of moves, and a move like * here or a dame in Go, you really only want to play when there’s nothing else left to do (except special circumstances with ko in Chinese rules).

There’s a comment about a chilled dame being worth 0, “because the tax is so great”, but I think one would have to read the rules of the chilled game properly. If it was playable and it gave you a -1 due to the marking, you have to know what that means. Maybe it’s simply an illegal move in the chilled game. 0 is kind of like no move available in some situations, in others it’s just a second player win.

The next example is where the players can both make a point, but also leave a dame behind:

In the normal game, Black can play and make one point, or white can play and make one point, and there’s always a dame left. So the position is {1*|-1*}, where 1 means a point for black and -1 is a point for white, * is the dame value in area rules, and 1* is a shorthand for 1+*.

When the game is chilled, no markings are added, but when the player captures a point they also add a marking which cancels out the point, and chilled dame are worth zero. So in the chilled game again this is {0|0}=*.

Then it introduces a positive infinitesmal called up ↑. It’s negative is down ↓. ↑ is defined to be {0|*}, where black can move to a 0 position (where there’s no moves left or it’s a second player win) and White can only turn the position into a * (like one move left or a first player win). So the advantage is to Black, which is why it is a positive number. It shows up in another corridor like position in the chilled game

## An aside on ↑

I actually came across ↑ in another combinatorial game I like to play by designer Mark Steere. The game is called Hadron.

The rules are fairly simple, if you can’t play you lose. You can only place your colour on an empty space when there is an equal number of neighbours of both colours. So 0 neighbours, 1 red and 1 blue or 2 red and 2 blue. In the top left corner, if blue plays there’s no more legal moves there so that’s a 0. If red plays, now the corner square is a legal move because it will have 1 red and 1 blue neighbour. That single square is like a dame in go so it’s also worth *. That means the game in the top left is {0|*}=↑. It turns out corridors like these in hadron are kind of multiples of ↑, so it almost feels like a fundamental unit in some sense You can also prove that because both players always have the same set of moves available to them at all times, that the only values in this game are all infinitesmal

This corridor is {0|↑}= ⇑* or ↑+↑+* (double up star)

The longer corridors then I believe alternate having and not having a *, and also they’re integer multiples of ↑.

Then maybe we can look at the game in 2.7 in a separate post, assuming the above knowledge in some sense.