2.1 Finite games and birthdays
Let \(\mathcal{G}\) denote the collection of all finite (or short) combinatorial games. An element \(x\in \mathcal{G}\) is a finite rooted tree written \(x=\{ X^L\mid X^R\} \), where the left- and right-option collections \(X^L\) and \(X^R\) are finite lists of games. The basic games are
The rank \(\operatorname {bd}(x)\in \mathbb N\) is defined recursively by
Thus our convention is \(\operatorname {bd}(0)=1\).
If \(u\in X^L\) or \(u\in X^R\), then \(\operatorname {bd}(u){\lt}\operatorname {bd}(x)\).
Write \(x=\{ X^L\mid X^R\} \), and let \(B\) be the finite collection of birthdays of all members of \(X^L\) and \(X^R\). If \(u\) belongs to either option collection, then \(\operatorname {bd}(u)\in B\); in particular, \(B\) is nonempty and
By Definition 2.2, \(\operatorname {bd}(x)=1+\max B\), where adjoining \(0\) makes the same formula valid when there are no options. Hence \(\operatorname {bd}(u){\lt}\operatorname {bd}(x)\). The left- and right-option assertions are the two specializations of this argument.