Technical Note—An Expectation-Maximization Method to Estimate a Rank-Based Choice Model of Demand

Garrett J. Van Ryzin · Gustavo J. Vulcano

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Consumer Market Behavior and Pricing Supply Chain and Inventory Management Economic and Environmental Valuation

Abstract

We propose an expectation-maximization (EM) method to estimate customer preferences for a category of products using only sales transaction and product availability data. The demand model combines a general, rank-based discrete choice model of preferences with a Bernoulli process of customer arrivals over time. The discrete choice model is defined by a probability mass function (pmf) on a given set of preference rankings of alternatives, including the no-purchase alternative. Each customer is represented by a preference list, and when faced with a given choice set is assumed to either purchase the available option that ranks highest in her preference list, or not purchase at all if no available product ranks higher than the no-purchase alternative. We apply the EM method to jointly estimate the arrival rate of customers and the pmf of the rank-based choice model, and show that it leads to a remarkably simple and highly efficient estimation procedure. All limit points of the procedure are provably stationary points of the associated incomplete data log-likelihood function, and the output produced are maximum likelihood estimates (MLEs). Our numerical experiments confirm the practical potential of the proposal. The online appendix is available at https://doi.org/10.1287/opre.2016.1559 .

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References

2007Demand Estimation and Assortment Optimization Under Substitution: Methodology and Application 2001Stocking Retail Assortments Under Dynamic Consumer Substitution 2013A Nonparametric Approach to Modeling Choice with Limited Data 1998Estimation of Consumer Demand with Stock-Out Based Substitution: An Application to Vending Machine Products 2001Inventory Competition Under Dynamic Consumer Choice 2005Revenue Management for Parallel Flights with Customer-Choice Behavior 2010Assortment Planning and Inventory Decisions Under Stockout-Based Substitution 2013Demand Estimation Under Incomplete Product Availability

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