The Polytope of Probability Functions on a Finite Poset

Kim, Kim, and Neggers (2019) defined probability functions on a poset, by listing some very natural conditions that a function $\pi :P\times P\to [0,1]$ 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 $\pi (a,b)$ 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.