On the Number of Facets of Polytopes Representing Comparative Probability Orders

TL;DR

This paper investigates the maximum number of facets of polytopes representing comparative probability orders, establishing Fibonacci-based lower bounds and providing near-sharp upper bounds through combinatorial methods.

Contribution

It introduces a Fibonacci-based lower bound for the facets of these polytopes and conjectures its sharpness, advancing understanding of their geometric complexity.

Findings

Lower bound on facets is at least Fibonacci number F_{n+1}.

Upper bound on facets is close to the lower bound.

Proof uses combinatorial concepts like flippable pairs.

Abstract

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.