Subset Comparisons for Additive Linear Orders

Abstract

This paper investigates algebraic and combinatorial properties of the set of linear orders on the algebra of subsets of a finite set that are representable by positive measures. It is motivated by topics in decision theory and the theory of measurement, where an understanding of such properties can facilitate the design of strategies to elicit comparisons between subsets that, for example, determine an individual's preference order over subsets of objects or an individual's qualitative probability order over subsets of states of the world. We introduce a notion of critical pairs of binary comparisons for such orders and prove that (i) each order is uniquely characterized by its set of critical pairs and (ii) the smallest set of binary comparisons that determines an order is a subset of its set of critical pairs. The paper then focuses on the minimum number of on-line binary-comparison queries between subsets that suffice to determine any representable order for a set of given cardinality n. It is observed that, for small n, the minimum is attained by first determining the ordering of singleton subsets. We also consider query procedures with fixed numbers of stages, in each stage of which a number of queries for the next stage are formulated.1. Introduction. This paper investigates algebraic and combinatorial properties of additive linear orders on the Boolean algebra n of subsets of an n-item set n = 1 2 n . We refer to a binary relation ≺ on n as an additive linear order if there exists a measure on n such that i = i > 0 for all i ∈ n , and with A = i∈A i for all A ∈ n ,

Flippable Pairs and Subset Comparisons in Comparative Probability Orderings

“…We show that there are many orders of this kind, not just the lexicographic order. These results provide answers to two questions of Fishburn et al (2002). We also study the flip relation on the class of all comparative probability orders introduced by Maclagan.…”

“…In terms of cones, this tool is given in Lemma 1, which says that in representable cones we can deduce new comparisons by forming linear combinations of the characteristic vectors of known comparisons with positive coefficients. This is also a reformulation of Axiom 3 from [9] in terms of discrete cones associated with .…”

“…This can be formulated in terms of multilists of comparisons (see, for example, [9,proof of Theorem 3.7]). In terms of cones, this tool is given in Lemma 1, which says that in representable cones we can deduce new comparisons by forming linear combinations of the characteristic vectors of known comparisons with positive coefficients.…”

“…A significant part of this research direction was initiated by Fishburn et al [9] and is related to problems in decision theory and the theory of measurement, especially to the problem of preference elicitation. We answer two questions posed in that paper.…”

“…Only one such example appears to have been given previously in the literature [10]. Fishburn et al [9] paid significant attention to elicitation of comparative probability orders representable by a probability measure. They proved that any minimal set of comparisons that determines an order on 2 X consists of critical pairs.…”