Logit unfolding choice models for binary data

Discrete choice models with non-monotonic response functions are important in many areas of application, especially political sciences and marketing. This paper describes a novel unfolding model for binary data that allows for heavy-tailed shocks to the underlying utilities. One of our key contributions is a Markov chain Monte Carlo algorithm that requires little or no parameter tuning, fully explores the support of the posterior distribution, and can be used to fit various extensions of our core model that involve (Bayesian) hypothesis testing on the latent construct. Our empirical evaluations of the model and the associated algorithm suggest that they provide better complexity-adjusted fit to voting data from the United States House of Representatives.

💡 Research Summary

The paper addresses a notable limitation of standard binary discrete‑choice models: their response functions are monotonic in the latent trait, which makes them ill‑suited for phenomena where extreme individuals sometimes vote together against the middle (“end‑against‑the‑middle” behavior). To overcome this, the authors propose a novel “logit unfolding” model that embeds three policy positions for each item—one representing a negative (status‑quo) option, one representing a positive (proposed) option, and a third representing an extreme alternative. Utility for legislator i on item j is defined as the negative squared distance between the legislator’s ideal point β_i and each policy position ψ_{j,k}, plus an independent Gumbel shock ε_{i,j,k}. A positive vote occurs only when the utility of the positive position exceeds both negative utilities.

Through algebraic manipulation, the choice probability simplifies to

p(y_{ij}=1 | β_i, α_j, δ_j) = 1 /