So since you guys keep posting all kinds of Go positions, I’d like to talk about:

## Go as a combinatorial game (Section 4.1 and Appendix A/B?)

It seems to me that the book doesn’t do a great job of explaining the basics of how we can represent Go as a combinatorial game. Maybe some of it is supposed to be obvious, and maybe some of my discomfort comes from a preference for Chinese rules, while the book mostly uses Japanese.

So I’ll make some attempt here. Does this kind of thing work to draw a game tree?

```
⚫⚪⚪⚪
⚫➕⚫⚪
⚫⚪⚪⚪
Black ↙ ↘ White
⚫⚪⚪⚪ ⚫⚪⚪⚪
⚫⚫⚫⚪ ⚫⚪⬛⚪
⚫⚪⚪⚪ ⚫⚪⚪⚪
0 -2
```

## It should look like this

To represent Go as a combinatorial game, we consider all the reasonable moves (ignoring for simplicity unnecessarily filling in your own eyes or playing dead stones in the opponent’s territory). We also assume (for now at least) that there is no ko, since otherwise the games would not combine properly, and there might be cycles. Then we get a tree like the one above.

However, instead of just ending like a normal combinatorial game, Go has a *score* at the end. In this case it’s 0 if Black moves and -2 (2 points for White) if White moves. To take this into account, we put the 0 and -2 combinatorial games at the leaves of the tree.

Specifically, this means if Black moves the game is over, but if White moves, White gets 2 more moves afterward. You could imagine actually playing these moves with some scheme involving prisoner return and eye-filling, but for analysis of the game it’s not necessary.

At least that’s my understanding, but did the book ever really say it like this?

The example above can be written as { 0 | -2 }, which is notably a *hot* game, meaning that it’s to both players’ advantage to play first. Most Go positions are hot; that’s why we keep playing instead of just passing. Hot games are notoriously hard to analyze since they don’t simplify to numbers or infinitesimals.

## This game is particularly NOT equal to its average value of -1.

To see that *G* ≹ -1, consider *G* + 1, meaning the combination of this game with a single extra point for Black. Clearly, whoever plays first will win, so *G* + 1 is not zero.

What if we use Chinese rules? Since stones count, we have to decide which ones to consider, and I think it makes the most sense to measure the score relative to the starting position, assuming all stones were alive. Then Black just gains one additional point by playing first, while White converts a black stone to white territory and gains one stone, getting to -3. So the game becomes { 1 | -3 }, apparently an even hotter game.

Interestingly you can get back to the Japanese game if you impose a “tax” of one point on whoever plays. I’m not sure if this relationship was mentioned in the book, but Sensei’s says it like this:

In general, the difference between Territory and Area scoring is that stones on the board are counted in area scoring but not in territory scoring. Suppose that you play by area scoring but each board play costs one point. You would actually be playing by territory scoring, since each board play results in a stone on the board or captured.

Since we know Japanese and Chinese rules generally have the same optimal play, that suggests taxing away yet another point from each move, which is called *chilling*. But is there more we should talk about first just with normal unchilled Go games? Can we work out the actual mathematical scores of the reversal example above or something like that?