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disjunctive sums
Path: CGTPath   · Prev: NegativeOfAGame   · Next: OrderingOfGames
    Keywords: EndGame

This is a fundamental concept in combinatorial game theory, where one defines the sum of two games, which is an easy idea: players may play in one or the other game. This is 'exclusive or', whence the term disjunctive.

The real depth of the concept can be seen in go when you start to wonder whether the endgame marks the point at which the game breaks up into a sum of several games (independent endgame positions). That's certainly not quite right:

  • as long as you can have ko fights, there may be plays in one part of the board that influence other regions;
  • there are some tricky endgame coupling effects that are standard if not that well-known, as neighbouring positions do interact beyond what one might guess.

Also, from a pro point of view, the endgame starts earlier than we think, before weak groups are settled but as soon as there isn't a reason for them to die or cause major swings in territory.

Is this because pros have a better thermometer? - Migeru

But this is still a useful start. See for example the stacks of coins model.


Migeru: Having sorted out my confusion (see below) here's the general (recursive)definition of addition. If G" = { A, B, C, ..., Z | a, b, c, ..., z } and H" = { A', B', ..., Z' | a', b', ..., z' } are games, then
G" + H" = { A+H", B+H", ..., Z+H", G"+A', G"+B', ..., G"+Z' |

| a+H", b+H", ..., z+H", G"+a', G"+b', ..., G"+z' }




Migeru: I have a problem with associativity. Consider

(1|-1) + (2|-2) + (4|-4) = (3|-3)

We have (if n>m)

(n|-n) + (m|-m) = (n-m|m-n)

so

(1|-1) + (2|-2) = (1|-1).

Then ((1|-1)+(2|-2))+(4|-4)=(1|-1)+(4|-4)=(3|-3)

But also (1|-1)+((2|-2)+(4|-4))=(1|-1)+(2|-2)=(1|-1)

and ((1|-1)+(4|-4))+(2|-2)=(3|-3)+(2|-2)=(1|-1).

What's going on?


Bill: Dear Migeru,

{1 | -1} + {2 | -2} + {4 | -4} || {3 | -3}

They are confused.

{n | -n} + {m | -m} = {{n+m | n-m} | {m-n | -m-n}}

when m and n are numbers such that n >= m.


Migeru: Thanks. I definitely need to brush up on my surreal numbers, because I didn't remember the correct definition of addition.



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This is a copy of the living page "disjunctive sums" at Sensei's Library.
(OC) 2003 the Authors, published under the OpenContent License V1.0.