A geometrical approach to multiset orderings

2007

Order

Abstract. We study additive representability of orders on multisets (of size k drawn from a set of size n) which satisfy the condition of Independence of Equal Submultisets (IES) introduced by Slinko (2002, 2007). Here we take a geometric view of those orders, and relate them to certain combinatorial objects which we call discrete cones. Following Fishburn (1996) and Conder and Slinko (2004), we define functions f (n, k) and g(n, k) which measure the maximal possible deviation of an arbitrary order satisfying the IES and an arbitrary almost representable order satisfying the IES, respectively, from a representable order. We prove that g(n, k) = n − 1 whenever n ≥ 3 and (n, k) = (5, 2). In the exceptional case, g(5, 2) = 3. We also prove that g(n, k) ≤ f (n, k) ≤ n and establish that for small n and k the functions g(n, k) and f (n, k) coincide.

Section: Remaining Casesmentioning

confidence: 98%

Orders on Multisets and Discrete Cones

Conder

Marshall

2007

Order

“…It has to be noted that orders on multisets and its subsets which extend a given linear order on the underlying set on which multisets are constructed have been instrumental in proofs of program termination, in equational reasoning algorithms based on term rewriting systems, in computer algebra, the theory of invariants, and the theory of partitions. We refer the reader to the two surveys by Martin [27] and Dershowitz [8] and to the book by Kreuzer and Robbiano [25] on these topics.…”

Section: Introductionmentioning

confidence: 99%

Ranking committees, income streams or multisets

Sertel

2006

Economic Theory

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.

“…Rankings of P(A) and its subsets which extend a given ranking of A have been instrumental in proofs of program termination, in equational reasoning algorithms based on term rewriting systems, in computer algebra, the theory of invariants, and the theory of partitions. We refer the reader to the two surveys by Martin [3] and Dershowitz [4] on these topics.…”

Section: Introductionmentioning

confidence: 99%

“…It guarantees that there are no infinitely decreasing chains of multisets, which is helpful in proving program termination [4]. The interest of this property has decreased significantly since Martin [3] showed that when P = P(A), i.e. P is the family of all multisets, all orders preserved under the operation of multiset union are additive (i.e.…”

Section: Introductionmentioning

confidence: 99%

Ranking Committees, Words or Multisets

Sertel

2002

SSRN Journal

Standard-Nutzungsbedingungen:Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden.Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen.Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. The opinions expressed in this paper do not necessarily reflect the position of Fondazione Eni Enrico Mattei Terms of use: Documents in EconStor may Ranking Committees, Words or Multisets SummaryWe investigate the ways in which a linear order on a finite set A can be consistently extended to a linear order on a set P k (A) of multisets on A of fixed cardinality k. We show that for card(A) = 3 all linear orders on P k (A) are additive and classify them by means of Farey fractions. For card(A) ≥ 4 we show that there are non-additive consistent linear orders of P k (A), we prove that they cannot be extended to a linear order of P k (A) for K sufficiently large. We give the lower bounds for the number of additive linear orders in P 2 (A) and the total number of consistent linear orders in P 2 (A) .