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An Alternative Approach for Nonparametric Analysis of Random Utility Models

Christopher Turansick

TL;DR

This paper tackles the computational bottlenecks in nonparametric testing of random utility models by shifting from a vertex-based conic (V-representation) to a half-space (H-representation) framework, enabling more scalable inference. The authors develop a three-step methodology that starts with the known H-representation for full-domain data, reparameterizes non-negativity constraints via a single-variable change of variables, and introduces slack variables to extend partial-domain data to the full domain; this is complemented by a monotone-utility encoding and a new axiomatics based on local-to-global feasibility. Key contributions include a monotone-utility encoding using Möbius-inverse constraints, a preference-free axiom for random utility with unobserved menus, and substantial computational improvements driven by a smaller N-matrix relative to KS’s M-matrix. The approach broadens the practical applicability of nonparametric random utility testing to larger and more complex datasets and offers a generalizable framework for testing other convex choice models beyond the standard random utility setting.

Abstract

We readdress the problem of nonparametric statistical testing of random utility models proposed in Kitamura and Stoye (2018). Although their test is elegant, it is subject to computational constraints which leaves execution of the test infeasible in many applications. We note that much of the computational burden in Kitamura and Stoye's test is due to their test defining a polyhedral cone through its vertices rather than its faces. We propose an alternative but equivalent hypothesis test for random utility models. This test relies on a series of equality and inequality constraints which defines the faces of the corresponding polyhedral cone. Building on our testing procedure, we develop a novel axiomatization of the random utility model.

An Alternative Approach for Nonparametric Analysis of Random Utility Models

TL;DR

This paper tackles the computational bottlenecks in nonparametric testing of random utility models by shifting from a vertex-based conic (V-representation) to a half-space (H-representation) framework, enabling more scalable inference. The authors develop a three-step methodology that starts with the known H-representation for full-domain data, reparameterizes non-negativity constraints via a single-variable change of variables, and introduces slack variables to extend partial-domain data to the full domain; this is complemented by a monotone-utility encoding and a new axiomatics based on local-to-global feasibility. Key contributions include a monotone-utility encoding using Möbius-inverse constraints, a preference-free axiom for random utility with unobserved menus, and substantial computational improvements driven by a smaller N-matrix relative to KS’s M-matrix. The approach broadens the practical applicability of nonparametric random utility testing to larger and more complex datasets and offers a generalizable framework for testing other convex choice models beyond the standard random utility setting.

Abstract

We readdress the problem of nonparametric statistical testing of random utility models proposed in Kitamura and Stoye (2018). Although their test is elegant, it is subject to computational constraints which leaves execution of the test infeasible in many applications. We note that much of the computational burden in Kitamura and Stoye's test is due to their test defining a polyhedral cone through its vertices rather than its faces. We propose an alternative but equivalent hypothesis test for random utility models. This test relies on a series of equality and inequality constraints which defines the faces of the corresponding polyhedral cone. Building on our testing procedure, we develop a novel axiomatization of the random utility model.
Paper Structure (19 sections, 11 theorems, 26 equations, 1 figure, 1 table)

This paper contains 19 sections, 11 theorems, 26 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Random Utility and Testing
  3. Model
  4. Current Methodology
  5. A Conic Approach
  6. Column (or Row) Generation
  7. New Methodology
  8. Encoding Monotone Choice
  9. Axiomatics
  10. Discussion
  11. Related Literature
  12. Preliminary Results
  13. Proof of Lemma \ref{['qtop']}
  14. Proof of Lemma \ref{['GeneralRUM']}
  15. Omitted Proofs
  16. ...and 4 more sections

Key Result

Theorem 2.1

A subset $\mathcal{P}$ of $\mathbb{R}^H$ is a finitely generated cone if and only if it is a finite intersection of closed half-spaces

Figures (1)

  • Figure 1: Here we capture two linear budgets over the goods $a_1$ and $a_2$. These budgets are labeled as $B_1$ and $B_2$ and have a single point of intersection. By assuming that no choice occurs at the point of intersection, we are left with $\{w,x,y,z\}$ as the set of patches.

Theorems & Definitions (22)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Theorem 2.1: Weyl-Minkowski Theorem
  • Definition 3.1
  • Theorem 3.1: falmagne1978representation
  • Theorem 3.2: falmagne1978representation
  • Lemma 3.1
  • Theorem 3.3
  • Proposition 3.1
  • ...and 12 more