  Bolker gives axioms constraining preference, and shows that any preferences satisfying his axioms can be represented by a probability measure \(P\) and a utility measure \(U\). However, Bolker's axioms do not ensure that \(P\) is unique, or that \(U\) is unique up to positive linear transformation. Nor do they allow us to define comparative probability in terms of preference. Instead, where \(P\) and \(U\) jointly represent a preference ordering, Bolker shows that the pair \(\langle P, U \rangle\) is unique up to a fractional linear transformation. 