A proof that a cardinal utility function is determined if changes between outcomes, as well as the outcomes themselves, can be ranked in order of desirability. It is assumed that a numerical ordinal utility function with convex range is already given, and that the ranking of changes is continuous and enjoys a certain "crossover" property that is characteristic of difference comparisons.
Shapley, Lloyd S., Cardinal Utility from Intensity Comparisons. Santa Monica, CA: RAND Corporation, 1975. https://www.rand.org/pubs/reports/R1683.html. Also available in print form.
Shapley, Lloyd S., Cardinal Utility from Intensity Comparisons, Santa Monica, Calif.: RAND Corporation, R-1683-PR, 1975. As of September 08, 2021: https://www.rand.org/pubs/reports/R1683.html