A smooth bounded choice model: Formulation and application in three large-scale case studies

Abstract

One-stage (implicit) choice set formation models offer a computationally efficient way to model how individuals consider alternatives. Among these, the Bounded Choice Model (BCM) stands out for its consistent, utility-based cutoffs. However, the BCM is non-differentiable, which limits its usefulness: key outputs such as elasticities and standard errors cannot be computed analytically. To overcome this, we introduce the Smooth Bounded Choice Model (SBCM). This model assumes a new smooth truncated logistic distribution for the error terms and applies a smooth approximation to the maximum function used in defining the reference utility. As a result, the SBCM is infinitely differentiable, while preserving core features of the BCM, such as bounding, continuity, and the ability to collapse to the Multinomial Logit (MNL) model under specific conditions. Importantly, the SBCM is not just a smoother version of the BCM. Its more flexible distributional assumptions can better capture actual choice behaviour and allow for meaningful differences in predicted probabilities. We derive closed-form expressions for choice probabilities, gradients, Hessians, elasticities, and standard errors, and present a practical estimation method. The SBCM is tested in three case studies: one mode choice and two route choice settings (bicycle and public transport). In all cases, it outperforms both the BCM and MNL in terms of model fit and interpretability. While the BCM has so far been limited to car route choice, we show that the SBCM is widely applicable across various discrete choice contexts.