Axioms for unique subjective probability on finite sets

Abstract

This paper discusses axioms for a comparative probability relation on a finite set which imply the existence of a unique probability measure that agrees with the comparative relation. We focus on three axiomatizations that extend a sufficient condition for finite uniqueness developed by Duncan Luce to cases of uniqueness not covered by his condition. Two of the axiomatizations, due to Van Lier and the present authors, use only simple and intuitively straightforward conditions for uniqueness. Their conditions are sufficient but not necessary. Our third axiomatization presents conditions that are necessary and sufficient for uniqueness but which are much more complex than the merely sufficient conditions. We also investigate the structure of the family of unique representations for each axiomatization.