[Paper Review] Yogurts Choose Consumers? Estimation of Random-Utility Models via Two-Sided Matching

Odran Bonnet, Alfred Galichon+3 more|arXiv (Cornell University)|Nov 26, 2021

Consumer Market Behavior and Pricing64 references4 citations

TL;DR

This paper establishes a novel equivalence between discrete choice demand inversion in random utility models and stable matching in two-sided markets with imperfectly transferable utility. By leveraging two-sided matching algorithms, it enables efficient estimation of non-additive, non-invertible random utility models—such as the pure characteristics model—where the identified set of utility vectors forms a lattice, and sharp bounds are recovered via matching procedures.

ABSTRACT

The problem of demand inversion - a crucial step in the estimation of random utility discrete-choice models - is equivalent to the determination of stable outcomes in two-sided matching models. This equivalence applies to random utility models that are not necessarily additive, smooth, nor even invertible. Based on this equivalence, algorithms for the determination of stable matchings provide effective computational methods for estimating these models. For non-invertible models, the identified set of utility vectors is a lattice, and the matching algorithms recover sharp upper and lower bounds on the utilities. Our matching approach facilitates estimation of models that were previously difficult to estimate, such as the pure characteristics model. An empirical application to voting data from the 1999 European Parliament elections illustrates the good performance of our matching-based demand inversion algorithms in practice.

Motivation & Objective

• To address the challenge of demand inversion in random utility models, especially when the utility function is non-additive, non-smooth, or non-invertible.
• To establish a formal equivalence between discrete choice models and two-sided matching markets with imperfectly transferable utility.
• To develop computationally efficient algorithms for estimating utility vectors that rationalize observed market shares, even under partial identification.
• To demonstrate the practical feasibility and performance of the matching-based approach through an empirical application to European Parliament election data.
• To explore the implications of this equivalence for modeling complex choice behaviors, such as multiple discrete choices, within a unified framework.

Proposed method

• Reformulate the discrete choice model as a two-sided matching market where consumers and alternatives act as agents with preferences derived from utility functions.
• Map the demand inversion problem—finding utility vectors that rationalize observed market shares—to the problem of computing stable matchings in this market.
• Use established two-sided matching algorithms (e.g., deferred acceptance with compensation schemes) to compute equilibrium outcomes, which correspond to the identified set of utility vectors.
• Apply linear programming formulations to compute sharp upper and lower bounds on utility vectors, exploiting the lattice structure of the identified set.
• Simultaneously solve demand inversion across multiple markets (e.g., precincts) using a combined linear program, significantly improving computational efficiency.
• Leverage the lattice structure of the identified set to construct data-driven tests for point identification and assess the degree of parameter multiplicity.

Experimental results

Research questions

• RQ1 Can the demand inversion problem in non-additive random utility models be reformulated as a stable matching problem in a two-sided market?
• RQ2 What are the computational and identification implications of this equivalence for models that are non-invertible or non-smooth?
• RQ3 How can matching algorithms be used to recover sharp bounds on utility vectors when the demand map is not invertible?
• RQ4 To what extent does this approach improve estimation efficiency and accuracy in discrete choice models, especially for the pure characteristics model?
• RQ5 Can this framework be extended to model multiple discrete choices or bundle choices, which are difficult in standard discrete choice models?

Key findings

• The identified set of utility vectors in non-invertible random utility models forms a lattice, enabling systematic computation of sharp upper and lower bounds.
• The maximum difference between upper and lower bounds on estimated utility vectors across all precincts and parties was minuscule, indicating that parameter multiplicity is not a practical issue in the empirical application.
• Solving demand inversion simultaneously across all precincts using a combined linear program was ten times faster than solving each separately, demonstrating a major computational advantage.
• The matching-based approach successfully estimated a non-additive random utility model with correlated voter preferences across two dimensions—economic left-right and pro-EU sentiment—using 1999 European Parliament election data.
• The method revealed that higher unemployment rates are significantly associated with left-leaning and less pro-EU voting preferences, while older populations and higher female-to-male ratios show distinct ideological patterns.
• The equivalence between discrete choice and two-sided matching allows for a reinterpretation of consumer choice as 'yogurts choosing consumers', highlighting the symmetry and generality of the framework.