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. The derived model is referred to as the piecewise MNL model and provides greater modeling flexibility by allowing consumers to exhibit asymmetric price sensitivities around unknown inflection points. Methodology/results: We study the model identification for the piecewise MNL and further propose a maximum likelihood estimator (MLE) for its calibration by real data. Due to the presence of inflection points, the log-likelihood function is non-differentiable, which poses major challenges to both numerical and statistical analyses. We propose a novel profile-based numerical optimization procedure which locates the MLE efficiently and further establish statistical guarantees for the MLE based on the empirical process theory. We fully solve price optimization 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 price may not even be unique. For price competition, we show that multiple equilibria may exist depending on the magnitude of the inflection point and then fully characterize the conditions for the existence of each equilibrium. Managerial implications: Our extensive numerical experiments on synthetic and real data suggest that the piecewise MNL can improve model fitting and prediction accuracy compared with popular choice models in the literature, and ignoring varying price sensitivities could lead to sub-optimal solutions or even substantial losses for firms