 These games are important in the theory of definitions. Suppose we have a collection \(A\) of objects and a family \(S\) of properties; each property cuts \(A\) into the set of those objects that have the property and the set of those that don’t. Let \(\exists\) cut, starting with the whole set \(A\) and using a property in \(S\) as a knife; let \(\forall\) choose one of the pieces (which are subsets of \(A)\) and give it back to \(\exists\) to cut again, once more using a property in \(S\); and so on. Let \(\exists\) lose as soon as \(\forall\) chooses an empty piece. We say that \((A, S)\) has rank at most \(m\) if \(\forall\) has a strategy which ensures that \(\exists\) will lose before her \(m\)-th move. The rank of \((A, S)\) gives valuable information about the family of subsets of \(A\) definable by properties in \(S\).