The Polytope of Probability Functions on a Finite Poset
Jan Snellman
TL;DR
This work introduces a polyhedral framework for probability functions on finite posets by constructing an auxiliary derived poset and defining the probability functions polytope \\mathcal{Q}_{P} as an affine section of an order polytope. The approach yields a concrete, computable description of all probability functions on a poset and reveals that the polytope need not be lattice or 0/1, with the smallest non-lattice example arising at \\mathcal{Q}_{C_{2}\\times C_{2}\\times C_{2}}$. By relating \\mathcal{Q}_{P} to the order polytope of the antichains poset \\mathcal{A}(P) and detailing vertices, faces, and specific poset constructions (chains, antichains, ordinal sums, disjoint unions), the paper provides both general theory and concrete small-case data. The results illuminate how probability constraints interact with poset structure and open avenues for deeper study of the linear-extension polytope’s relation to probability functions. The findings have potential implications for combinatorial probability, poset polyhedra, and related optimization problems where order constraints govern probabilistic relations.
Abstract
Kim, Kim, and Neggers (2019) defined probability functions on a poset, by listing some very natural conditions that a function \(π: P \times P \to [0,1]\) should satisfy in order to capture the intuition of "the likelihood that \(a\) precedes \(b\) in \(P\)". In particular, this generalizes the common notion of poset probability for finite posets, where \(π(a,b)\) is the proportion of linear extensions of \(P\) in which \(a\) precedes \(b\). They constructed a family of such functions for posets embedded in the ordered plane; that is two say, for posets of order dimension at most two. We study probability functions of a finite poset \(P\) by constructing an ancillary poset \(\tilde{P}\), that we call *probability functions posets*. The relations of this new poset encodes the restrictions imposed on probability functions of the original poset by the conditions of the definition. Then, we define the probability functions polytope, which parameterizes the probability functions on \(P\), and show that it can be realized as the order polytope of \(\tilde{P}\) intersected by a certain affine subspace. We give a partial description of the vertices of probability functions polytope and show that, in contrast to the order polytope, it is not always a lattice polytope.