A lexicographic semiorder polytope and probabilistic representations of choice

I consider the convex polytope defined by the convex hull of the set of all simple lexicographic semiorders (Davis-Stober, 2010) compatible with a fixed linear ordering over a set of choice alternatives. Simple lexicographic semiorders have been previously used by decision theorists to model intransitive choice (e.g., Tversky, 1969). This convex polytope is a full dimensional 0/1 polytope in Rn(n−1)Rn(n−1). I present seven families of inequalities that constitute a complete linear description of this polytope for any finite nn. I prove that these inequalities are a minimal such list, i.e., they are facet-defining for any finite nn. I go on to demonstrate that this polytope is equivalent to a ‘mixture model’ of probabilistic choice.

► I present a new random utility model derived from lexicographic semiorders.
► This model takes the form of a 0/1 convex polytope in n(n−1)n(n−1) dimensional space.
► I provide a complete, facet-defining description of this polytope for any nn.
► I illustrate how ternary choice data can be used to test this mode.