My previous long posts; it’s kind of a series:

- Combinatorial games in general (Section 3.5 - 3.5.3)
- Go as a combinatorial game (Section 4.1 and Appendix A/B?)

## Values of Go positions and motivation for chilling (~Section 3.6)

I realized there was a little more to say about Go as a combinatorial game. Assuming Japanese rules, what types of games are possible? We have these values:

- Hot games (like my examples)
- Integers (finished games with a score that can be counted)
- Integers plus * (a dame)

In fact these are the only possible values of Go positions possible assuming nothing weird like ko or seki. That seems like kind of an important fact about Go, but strangely it’s hidden in the proof of Lemma 1.2 in Section 3.6.

The dame are not any more interesting than they are in normal Go, since they are worth less than the typical 1/2-point komi. And notably, you *do not* get 1/2-point positions on the board or any other non-integer numbers.

So the interesting positions are really just the hot ones. But normal combinatorial game theory doesn’t provide a great way to analyze those games. The fun values for combining games are infinitesimals. So we would like it if Go could be improved to have fewer hot values and more interesting infinitesimal values.

Luckily there’s a way to do this in general, called *cooling*, and the book uses something slightly different called *chilling*, defined (again strangely for such an important concept) in the proof of Theorem 1. For Go positions, since there are no non-integer numbers, I think chilling is exactly the same as cooling by 1 point.

You can think of chilled Go as a simple modification of Japanese rules: whoever moves loses 1 point. We also skip filling dame (since nobody would want to be the one to fill) and go straight to scoring at the end (that’s the point of "*G* is of the form *n* or *n** " in the book’s definition).

So here’s where I’m stuck on chilling: is the strategy in a chilled game necessarily the same as for Japanese rules? Does the book even claim that it is anywhere? It seems that there would often be two terminal positions resulting in scores of, say, -1/2 and 1/2, that both chill to -1/2 because they end on different players’ moves, and you would miss the winning play.

I suppose I should be able to think up an example, but I’ve been sitting on this post for too long. Anyone know what I’m missing?