Flippable Pairs and Subset Comparisons in Comparative Probability Orderings |
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Authors: | Robin Christian Marston Conder Arkadii Slinko |
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Institution: | (1) Department of Combinatorics and Optimization, University of Waterloo, Waterloo, Canada;(2) Department of Mathematics, University of Auckland, Auckland, New Zealand |
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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. (Math. Oper. Res. 27:227–243, 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.
Research partially supported by the N.Z. Centres of Research Excellence Fund (grant UOA 201). |
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Keywords: | Comparative probability Flip relation Elicitation Subset comparisons Additively representable linear orders |
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