  Joyce (1999) shows that with additional resources, Bolker's theorem can be modified to pin down a unique \(P\), and a \(U\) that is unique up to positive linear transformation.  We need only supplement the preference ordering with a primitive “more likely than” relation, governed by its own set of axioms, and linked to belief by several additional axioms. Joyce modifies Bolker's result to show that given these additional axioms, the “more likely than” relation is represented by a unique \(P\), and the preference ordering is represented by \(P\) together with a utility function that is unique up to positive linear transformation.