A slight detour on move calculations.

Gote moves:

The above position has two possible outcomes depending on whether black plays first and captures three white stones, or white plays and saves the three white stones. The local score is written down as 6 when black captures (territory count), three stones and three prisoners, or 0 when white saves as nobody gains any points.

The expected territory of the unsettled positon is the average of these two outcomes (6+0)/2=3. Now when either player makes a move they change the expected territory by 3 points, either black plays the capturing move and gains three more points than expected or white saves and saves 3 more points than expected. So the value of the move is said to be 3 points.

In other contexts, other endgame books, one might just call it a 6 point move, which is the swing or difference between the two outcomes (6-0=6). I think this is supposed to work to compare similar kinds of moves to find out which is bigger, but I think, if I understand, the drawback is that you have to come up with additional adhoc rules like a sente or reverse sente move is like double the value of a gote move.

You can do something similar with two stones.

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If we know the move values, I guess you can just compare them and pick the bigger number, but if you didn’t, maybe one could do a similar comparison with the method in the book and double the position with the colours swapped.

The game should be a draw, if all else is somehow equal and these are the only point making moves left.

If white starts and captures the three black stones on the top (a three point move), Black’s only winning move is to copy and similarly capture the three stones

So in this case A is definitely better than B since A is a draw and playing B loses by 2 points at least.

However, White might start by capturing two stones.

Black can mirror white by playing B, and get a draw, but if Black plays A, it’s also just a draw.

So in this case B isn’t better, and from the previous case A is sometimes better.

I guess in these examples you have symmetric positions and one player has played a move A say, and another plays B, and when you add them together (in the same game) you can try comparing the result to 0 with optimal play to see which move was better.

I think that’s roughly the kind of argument being presented in the book

with these kinds of diagrams. I think an added caveat though is that not all games are directly comparable. The values of games and moves can be a bit “fuzzy”, so it’s not always the case that you get A>B or A=B or A<B. There’s another option A||B which is that A is confused with B. So there’s some range of positions where you can’t exactly say which is better, and some that you can.

I think the main example is that *||0, * is confused with 0. It’s an infinitesimal that’s smaller than all positive numbers, and bigger than all negative numbers, and so is arbitrarily close to zero.

What > etc means, we can dig into, but essentially games where Black (by convention) say always wins whoever moves first are positive in value, and if white always wins no matter who moves first they are negative.

If the second player always wins then the game has value 0

(it makes more sense when the rules are that you want to play last - but I guess in Go this could be like a draw in some ways. I think one also needs to make some rule adjustments to turn go into a proper combinatorial game about moving last)

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An additional note:

I think in games that just have normal integer or rational values, every move you make just loses points. So for example, kind of like in Japanese rules, playing in your territory loses a points. Or if you were to make Go more like a combinatorial game, where playing last wins, you might make it so that you can’t pass, and you can either choose to play a move or give a prisoner back to the opponent. That would also disincentivise playing in the opponents territory like in no pass go.

In games like amazons, at the very late stage of the game (kind of like counting in Go, if you filled in the stones)

you have boxed off areas that have an integer number of moves left, and every move in this area reduces your score by one or more. People tend to call this the cold phase of the game, where every move lowers your score. I think when you look at how ordinary numbers are represented in games, this typically happens also.

Whereas in Go, and earlier phases of Amazons etc, there are typically moves that increase your score. I think these are referred to as hot games or positions. Probably they don’t often have usual integral or rational values. Most likely they’re a bit fuzzy. A position could be worth somewhere between 3 and 5 points say, but definitely no more than 5, in the sense if the opponent had 6 points you’d lose no matter what, and you might always beat the opponent if they only have 2 points. So that’s some kind of fuzziness inherent in hot games.

Maybe something like in Amazons

White could make at most 7 points in the local area, while it could drop down to something like four points

So maybe it has some fuzzy value between 4 and 7 points. It’s hard to know without actually being better at calculating and knowing how to calculate properly The idea would be similar, you know you’ll lose if you give the opponent 8 or more points (if those are the last areas left), and you know you’ll win if you give them 3 or less. But if you give them between 4 and 7, it might depend a bit on the details of the game.

(The depending on the details, I think, comes up a tiny bit when Berlekamp/Wolfe talk about rounding infinitesimals in the first sets of problems they solve - so I kind of expect it to be something similar )