Ranking Committees,Income Streams or Multisets

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.A significant part of this work was written when both authors were visiting professors at Bilkent University, Ankara. Slinko thanks Semih Koray and Mefharet Kocatepe for making this possible and Serguei Stepanov for discussion about Farey fractions. Sertel thanks the Institut des Hautes Etudes Scientifiques (IHES), Bures-sur-Yvette, France, for a couple of invitations, in 1999 and 2000, during which he had a chance to elaborate on some of the questions addressed in this paper. The authors thank students of The University of Auckland Irene Peng and Mark Lui who participated at an early stage of this project and Marston Conder for checking the result of Theorem 6 with MAGMA and correcting it. Murat Sertel was deceased (1942 – 2003).An erratum to this article is available at .