 Variations of this game, allowing a piece to be cut into infinitely many smaller pieces, are fundamental in the branch of model theory called stability theory. Broadly speaking, a theory is ‘good’ in the sense of stability theory if, whenever we take a model \(A\) of the theory and \(S\) the set of first-order formulas in one free variable with parameters from \(A\), the structure \((A, S)\) has ‘small’ rank. A different variation is to require that at each step, \(\exists\) divides into two each of the pieces that have survived from earlier steps, and again she loses as soon as one of the cut fragments is empty. (In this version \(\forall\) is redundant.) With this variation, the rank of \((A\),S) is called its Vapnik-Chervonenkis dimension; this notion is used in computational learning theory.