Flippable Pairs and Subset Comparisons in Comparative Probability Orderings and Related Simple Games

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, Peke?c 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 regions in the corresponding hyperplane arrangement can have no more than 13 faces and that there are 20 regions with 13 faces. All the neighbours of the 20 comparative probability orders which correspond to those regions are representable. Finally we define a class of simple games with complete desirability relation for which its strong desirability relation is acyclic, and show that the flip relation carries all the information about these games. We show that for n = 6 these games are weighted majority games

On the Number of Facets of Polytopes Representing Comparative Probability Orders

Fine and Gill (1973) introduced the geometric representation for those comparative probability orders on n atoms that have an underlying probability measure. In this representation every such comparative probability order is represented by a region of a certain hyperplane arrangement. Maclagan (1999) asked how many facets a polytope, which is the closure of such a region, might have. We prove that the maximal number of facets is at least F_{n+1}, where F_n is the nth Fibonacci number. We conjecture that this lower bound is sharp. Our proof is combinatorial and makes use of the concept of flippable pairs introduced by Maclagan. We also obtain an upper bound which is not too far from the lower bound.Comment: 13 page

EXTREME POINTS OF THE CREDAL SETS GENERATED BY COMPARATIVE PROBABILITIES

ABSTRACT. When using convex probability sets (or, equivalently, lower previsions) as uncertainty models, identifying extreme points can help simplifying various computations or the use of some algorithms. In general, sets induced by specific models such as possibility distributions, linear vacuous mixtures or 2-monotone measures may have extreme points easier to compute than generic convex sets. In this paper, we study extreme points of another specific model: comparative probability orderings between the singletons of a finite space. We characterise these extreme points by mean of a graphical representation of the comparative model, and use them to study the properties of the lower probability induced by this set. By doing so, we show that 2-monotone capacities are not informative enough to handle this type of comparisons without a loss of information. In addition, we connect comparative probabilities with other uncertainty models, such as imprecise probability masses

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

Flippable Pairs and Subset Comparisons in Comparative Probability Orderings and Related Simple Games

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, Peke?c 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 regions in the corresponding hyperplane arrangement can have no more than 13 faces and that there are 20 regions with 13 faces. All the neighbours of the 20 comparative probability orders which correspond to those regions are representable. Finally we define a class of simple games with complete desirability relation for which its strong desirability relation is acyclic, and show that the flip relation carries all the information about these games. We show that for n = 6 these games are weighted majority games.Additively reesentable linear orders, comrative obability, elicitation, subset comrisons, sime game, weighted majority game, desirability relation.