Homogeneous families of semiorders and the theory of probabilistic consistency

Abstract

Beginning with the data p( a, b), interpreted as the frequency with which a is preferred to b, we define various conditions of probabilistic consistency on this data. Most of these are expressed as conditions on a family of binary relations defined from the p( a, b)′s, where a is in the λth relation to b if p( a, b) > λ. The conditions essentially state that this family of relations is a homogenous or consistent family of semiorders. The conditions are then restated in a more general setting, starting with a family of binary relations rather than with the data p( a, b) and the particular family of relations defined from p( a, b). The main result is that almost all the probabilistic consistency conditions studied are equivalent. The logical relations among all these conditions are described. Finally, some recent work of Tversky-Russo on conditions of probabilistic consistency on the data p( a, b) is shown to hold in the more general setting of this paper.