Ranking committees, income streams or multisets

Abstract

Multisets are collections of objects which may include several copies of the same object. They may represent bundles of goods, committees formed of members of several political parties, or income streams. In this paper we investigate the ways in which a linear order on a finite set A can be consistently extended to an order on the set of all multisets on A of some given cardinality k and when such an extension arises from a utility function on A. The condition of consistency that we introduce is a close relative of the de Finetti’s condition that defines comparative probability orders. We prove that, when A has three elements, any consistent linear order on multisets on A of cardinality k arises from a utility function and all such orders can be characterised by means of Farey fractions. This is not true when A has cardinality four or greater. It is proved that, unlike linear orders that can be represented by a utility function, any non-representable order on the set of all multisets of cardinality k cannot be extended to a consistent linear order on multisets of cardinality K for sufficiently large K. We also discuss the concept of risk aversion arising in this context.