A triangulation for pointed order polytopes

Abstract

In this paper we propose a way to triangulate a pointed order polytope. Pointed order polytopes are a generalization of order polytopes that include some important groups of polytopes appearing when bipolar scales arise in Decision Making or Game Theory, as the set of bi-capacities or the set of normalized bi-games, even for cases with restricted cooperation. Triangulating polytopes is an important and difficult problem that allows an elegant way to generate uniform random points in the polytope. For order polytopes, there exists a nice result that allows a way to triangulate this family of polytopes based on generating linear extensions. In this paper we prove a similar result for pointed order polytopes. The results in this paper allow to derive a procedure to generate random points inside a pointed order polytope that depends only on the structure of the subjacent poset, a problem that usually is simpler to tackle. In particular, this could be applied to generate bi-capacities or bi-capacities belonging to some subfamilies (e.g. k-symmetric, k-interactive, ...) in a random way.