Flippable Pairs and Subset Comparisons in Comparative Probability Orderings

Abstract

Abstract. We show that every additively representable comparative probability order on n atoms is determined by at least n − 1 binary subset comparisons. 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. We generalise an important theorem of Fishburn, Pekeč and Reeds, by showing that in any minimal set of comparisons that determine a comparative probability order, all comparisons are flippable.By calculating the characteristics of the flip relation for n = 6 we discover that the polytopes associated with the regions in the corresponding hyperplane arrangement can have no more than 13 facets and that there are 20 regions whose associated polytopes have 13 facets. All the neighbours of the 20 comparative probability orders which correspond to those regions are representable.

Additive Representability of Finite Measurement Structures

“…The converse as we will see later is not true. However, it appeared that IES alone implies additive representability on P k ( [3]) for all k. The following theorem was proved first in [39] and later appeared in [40]. We remind the reader of the definition of one of the main number-theoretic functions φ, which is Euler's totient function.…”

“…This approach was rediscovered by Fishburn [14] who pioneered their combinatorial study. Further combinatorial properties of discrete cones were studied in [10,15,16,28,6,30,3]. In this section we concentrate on combinatorics of rationality assessment.…”

“…= u n , then the order fails to be almost representable. Papers [39,40] present such an order belonging to P 3 [4].…”

“…Orders on the infinite set P[n] of all multisets on [n] satisfying the analogue of the de Finetti axiom (3), where the union is understood as the multiset union and the condition C ∩ (A ∪ B) = ∅ is not assumed, were considered by Danilov [7] and Martin [32]. Both independently prove that all orders on P[n] satisfying this axiom are additively representable.…”

“…However some interesting things have been observed. For example, it can be easily checked that the linear order on P ≤2 [3] 1 2 ≻ 12 ≻ 2 2 ≻ 13 ≻ 1 ≻ 23 ≻ 3 2 ≻ 2 ≻ 3 ≻ ∅ is not representable. Hence the analogue of Theorem 12 is not true.…”