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.