Numerical representations of imperfectly ordered preferences (a unified geometric exposition)

This paper uses "generalized numerical representations" to extend some of the result of utility theory regarding imperfectly ordered preferences in general and semiordered preferences in particular. It offers a unified geometric approach, which helps visualize how the increasingly stringent conditions of suborders, interval orders, semiorders, and weak orders give rise to increasingly intuitive representations. The differences between the proposed framework and the more traditional utility representations are especially significant in the context of uncountable sets. The main new results are axiomatizations for fixed threshold representations of the "just noticeable difference" (jnd) when the set of alternatives is infinite. It is also noted that all fixed-jnd representations give rise to an "almost cardinal" utility function, since permissible transformations must be linear "in the large.".