Total unimodularity of the Xi matrix

Determine whether the matrix Xi, defined by stacking the binary indicator row vectors xi^{J} that mark undominated patches for each subfamily of budgets J in the random utility model framework, is totally unimodular in general across arbitrary budget configurations and patch constructions.

Background

The paper constructs a matrix Xi whose rows xi{J} indicate, for each subfamily of budgets J, which patches are undominated with respect to direct revealed preference. This matrix generates an H-representation for the polytope of RUM-consistent stochastic demand systems via the linear inequalities Xi·π ≥ 1. Establishing integrality of this polytope is central to the authors' approach.

A common route to proving integrality is to show that the defining matrix is totally unimodular (TUM). While the authors’ numerical examples yield TUM matrices, they note that the general status is unclear because their construction typically violates a standard sufficient condition for total unimodularity (e.g., at most two non-zero entries per column and certain row-sum properties). Clarifying whether Xi is TUM in general would strengthen the theoretical foundations and could simplify computational and statistical procedures.

References

While the matrices in our numerical examples are TUMs, it is not at all clear if it is generally the case. At least, a matrix Xi obtained in our procedure typically violates a well known sufficient condition for being a TUM, which prohibits a matrix from having more than two non-zero entries in each columns, in addition to another requirement concerning the property of row sums.

— A dual approach to nonparametric characterization for random utility models (2403.04328 - Koida et al., 7 Mar 2024) in Remark 2, Section 3.3 (Integrality of polytopes and the proof of Theorem 1)