Discrete Choice Models with Piecewise Linear Utility: Modeling, Estimation, and Pricing

Problem definition: This paper incorporates a piecewise linear structure into the utility-price relationship of the classic multinomial logit (MNL) model and studies the associated operations problems such as estimation and pricing. Methodology/results: The derived model is referred to as the piecewise MNL model. We study its model identification and further propose a maximum likelihood estimator (MLE) for its calibration by real data. Because of the presence of inflection points, the log-likelihood function is nondifferentiable, which poses major challenges to both numerical and statistical analyses. We then propose a novel profile-based numerical optimization procedure that locates the MLE efficiently and further establish statistical guarantees for the MLE based on the empirical process theory. We fully characterize the price optimization problem under the piecewise MNL and show that the optimal pricing policy can be quite different from the standard MNL. In particular, the equal-(adjusted-)markup policy is no longer optimal, and the optimal solution may not even be unique. Furthermore, we propose an efficient approximation algorithm for the pricing problem that accounts for consumer heterogeneity within the piecewise MNL framework. Managerial implications: Our extensive numerical experiments on synthetic and real data suggest that the piecewise MNL can provide greater modeling flexibility, improve model fitting and prediction accuracy compared with popular choice models in the literature, and ignoring varying price sensitivities could lead to suboptimal solutions or even substantial losses for firms.

Funding: C. Ke acknowledges financial support from the National Natural Science Foundation of China [Grant 72101113].

Supplemental Material: The online appendix is available at https://doi.org/10.1287/msom.2023.0169.