birthday

    Keywords: Theory

Birthday is a concept in Combinatorial Game Theory. Informally it is the number of steps required to construct a game beginning from the zero game; or, equivalently, the length of the longest sequence of moves possible (without requiring alternating play).

More formally, for a game G={L1 L2 ... | R1 R2 ...}, the birthday b(G) is defined inductively as b(G)={ b(L1) b(L2) ... b(R1) b(R2)... | } This looks like a circular definition, but any apparent circularity ends when the game is reduced down to the zero game as in the worked example below.

A birthday is a "number" as defined at surreal numbers, and for finite games the birthday will always be a non-negative integer.

For an example, let's calculate the birthday of the game ^.

Recall that ^ is shorthand for {0|*}, where * is shorthand for {0|0} and 0 is shorthand for {|} [1].

Therefore we have b(^)={b(0) b(*) | }. Now we need to calculate b(0) and b(*)

Since 0={|}, we have b(0)={|}=0.

Since *={0|0}, we have b(*)={b(0) b(0) | 0} = {0 0|}

Therefore b(^)={ 0 {0 0 |} | }.

We can simplify this by deleting dominated options to reduce it to canonical form: {0 0|} is equal to {0|}, which is called 1, and then {0 1|} is equal to {1|}, which is called 2.

Therefore we say that b(^)=2.


Working from the other direction, we can build up a list of games according to birthdays.

0={|} is the only game of birthday 0.

We make games of birthday 1 by using 0:

The are three games of birthday 1. They are {0|} (called 1), {|0} (called -1) and {0|0} (called * or STAR).

We make games of birthday 2 by using games of birthday 1 or zero.

There are many games of birthday 2. We can start with:

{|1}, {|-1}, {|*}, {0|1}, {0|-1}, {0|*}, {1|}, {1|0}, {1|1}, {1|-1}, {1|*}, {-1|}, {-1|0}, {-1|1}, {-1|-1}, {-1|*}, {*|}, {*|0}, {*|1}, {*|-1}, and {*|*}.

(Are these 21 games all different? or are any two of them equal? -- xela)

Then there are things like {1 0 | 0}, which is equal to {1|0} (we can delete a dominated option); and {0 * | 1}, which is equal to {0|1} because * is a reversible option (is this correct? -- xela), and so on.

xela: I'm not sure if this last part is either correct or useful. I'm really just thinking out loud; feel free to correct or delete sections as appropriate.


[1] Is there a need for a separate page to define 0 and/or explain the abstract concept of a zero game? I looked at Zero in CGT terms but it didn't seem right for this particular context.


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