Flippable Pairs and Subset Comparisons in Comparative Probability Orderings

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. AMS classification: 60A05, 91B08 (primary), 06A07, 91A12, 91E99 (secondary)