Multiutility representations for incomplete difference preorders

A difference preorder is a (possibly incomplete) preorder on a space of state changes (rather than the states themselves); it encodes information about preference intensity, in addition to ordinal preferences. We find necessary and sufficient conditions for a difference preorder to be representable by a family of cardinal utility functions which take values in linearly ordered abelian groups. We also discuss the sense in which this cardinal utility representation is unique up to affine transformations, and under what conditions it is real-valued. This has applications to interpersonal comparisons, social welfare, and decisions under uncertainty.