Numbers/ Discussion

Migeru: Still trying to translate surreal numbers into go positions. In ZeroInCGTTerms I studied { | } = 0. Now I can construct {0| }=1, { |0}=-1 and {0|0} (not a number).

{0| } = 1  

Black can play to { | } = 0, but white has no legal moves. The fact that Black has one more legal move than white translates into a score of 1. This is very interesting, because it indicates that the original game of go was exactly a combinatorial game of the kind "the last player to legally move wins"^[1] (see tedomari).

{ |0} = -1 is the color-reversed version of {0| } = 1.

xela: I'm having trouble understanding fractions as explained in Mathematical Go. As a go position, "half a point" is {1|0}. This seems clear and intuitive: if Black plays first, there is a whole point; if White plays first, no points--so on average it's half a point, and this can be checked formally by playing a difference game: two lots of {1|0} plus -1 comes out to jigo whoever starts.

The problem though is that in CGT terms, {1|0} is not a number, because 1>0. On page 48 of Mathematical Go, the number 1/2 is defined to be {0|1}. This is certainly not equal to {1|0}: when viewed as an abstract game, {0|1} is always a win for Left, whereas {1|0} is a win for the first player (if Right goes first then the move is to the zero game, which Left loses). On page 49, Mathematical Go goes on to explain that the definition of 1/2={1|0}, while intuitively appealing, is incompatible with the Number Avoidance Theorem? (described in section 5.2 of http://arxiv.org/pdf/math/0410026). But in the rest of the book, it looks to me like 1/2 is treated as though it really is the same as {1|0}, without further comment.

Similary for 1/4 and other fractions.

Have I misunderstood something here?

Bill: Let me summarize briefly. {1 | 0} is not a number, {0 | 1} is, because 0 < 1. {1 | 0} is hotter than {0 | 1}; each player would rather play in the former rather than the latter, in accordance with the number avoidance theorem.

{0 | 1} + {0 | 1} = 1

which we can easily verify by playing it out. So {0 | 1} = 1/2.

{1 | 0} + {1 | 0} = 1

which we can easily verify by playing it out. So the mean value of {1 | 0} = 1/2.

So far so good. Now, if {1 | 0} is not a number, why does Mathematical Go treat it as such? Because Mathematical Go is mainly concerned with chilled go, and {1 | 0} chills to {0 | 1}.

Half point (all stones alive)  

Now we count the marked area as 1/2 point of territory for Black, but that is its mean value, not the final score. Its game tree is

 {1 | *}

because Black can play to a position worth 1 point while White can play to a position with a dame (which is a star, *, since each player has a move). This also chills to 1/2.

The relation of chilled go to territory scoring is like that of territory scoring to area scoring. In area scoring each played stone is worth one point. If you tax each move by one point, it cancels out the value of the played stone and you get a form of territory scoring (although there are other differences between specific rule sets). If ko is not a consideration, winning this form of territory scoring means that you win by area scoring as well. If you tax each move under territory scoring by one point you get chilled go. Similarly, if ko is not a consideration, winning chilled go means that you win by territory scoring as well. In chilled go we could simply stop playing and score the above position as 1/2 point for Black. That's why Mathematical Go treats it as a number. It is a score in chilled go.

xela: Thanks! "{1 | 0} chills to {0 | 1}" -- that is the missing link for me -- it makes a lot more sense in that light.

[1] Bill: No pass go is not the same as the original game surmised by Zhang Hu, which is a kind of territory scoring with a group tax.