By taking sets of utility functions as a primitive description of agents, we deﬁne an ordering over assumptions on utility functions that gauges their implicit measurement requirements. Cardinal and ordinal assumptions constitute two types of measurement requirements, but several standard assumptions in economics lie between these extremes. We ﬁrst apply the ordering to diﬀerent theories for why consumer preferences should be convex and show that diminishing marginal utility, which for complete preferences implies convexity, is an example of a compromise between cardinality and ordinality. In contrast, the Arrow-Koopmans theory of convexity, although proposed as an ordinal theory, relies on utility functions that lie in the cardinal measurement class. In a second application, we show that diminishing marginal utility, rather than the standard stronger assumption of cardinality, also justiﬁes utilitarian recommendations on redistribution and axiomatizes the Pigou-Dalton principle. We also show that transitivity and order-density (but not completeness) characterize the ordinal preferences that can be induced from sets of utility functions, present a general cardinality theorem for additively separable preferences, and provide suﬀicient conditions for orderings of assumptions on utility functions to be acyclic and transitive.
Mandler, Michael, "Compromises between Cardinality and Ordinality in Preference Theory and Social Choice" (2001). Cowles Foundation Discussion Papers. 1584.