Compromises Between Cardinality and Ordinality, with an Application to the Convexity of Preferences

By taking sets of utility functions as a primitive description of agents, we define an ordering over the measurability classes of assumptions on utility functions. Cardinal and ordinal assumptions constitute two types of measurability classes, but several standard assumptions lie strictly between these extremes. We apply the ordering to arguments for the convexity of preferences and show that diminishing marginal utility, which implies convexity, is an example of a compromise between cardinality and ordinality. Moreover, Arrow's (1951) explanation of convexity, proposed as an ordinal theory, in fact relies on utility functions that lie in the cardinal measurement class. In addition, we show that transitivity and order-density (but not completeness) fully characterize the ordinal preferences that can be induced from sets of utility functions. Finally, we derive a more general cardinality theorem for additively separable preferences.