'Marginal Estimation+Price Optimization' with Robust Choice Models

Abstract

We develop an estimation and optimization framework for the multi-product pricing problem by exploiting the properties of the marginal distributions of the random utilities in discrete choice models. We provide a novel interpretation of the marginal distribution choice model from a robust optimization viewpoint where the modeler evaluates the choice probabilities assuming a worst-case realization of the utilities of the products in a budgeted uncertainty set. Moreover, we show that the multi-product pricing problem under this robust choice model is a convex optimization problem when the marginal distributions are log-concave. Multinomial Logit and the Nested Logit models are special cases of this model.We use this approach to address partially the problem of model identification in choice modeling. Instead of presupposing the structural forms of the consumer’s utility functions, we use experimentations to guide us to the appropriate marginal distribution models for the pricing problems. More specifically, we develop a data-driven linear program to estimate the marginal distributions from a set of price experiments and to find the optimal prices. Mixed Integer Linear Programming models can be used to find the prices when side constraints are present. Tests using both simulated data and real data from the automobile industry illustrates the benefits of the “marginal estimation price optimization” approach.