Mallows-Smoothed Distribution Over Rankings Approach for Modeling Choice

65 Pages Posted: 21 May 2018 Last revised: 28 Sep 2019

See all articles by Antoine Désir

Antoine Désir

INSEAD

Vineet Goyal

Columbia University - Department of Industrial Engineering and Operations Research (IEOR)

Srikanth Jagabathula

New York University (NYU) - Department of Information, Operations, and Management Sciences

Danny Segev

University of Haifa - Department of Statistics

Date Written: May 3, 2018

Abstract

Assortment optimization is an important problem that arises in many applications including retailing and online advertising. The goal in such problems is to determine a revenue/profit maximizing subset of products to offer from a large universe of products when customers exhibit stochastic substitution behavior. We consider a mixture of Mallows model for demand, which can be viewed as a “smoothed” generalization of the class of sparse rank-based choice models, designed to overcome some of its key limitations. In spite of these advantages, the Mallows distribution has an exponential support size and does not admit a closed-form expression for choice probabilities.

We first conduct a case study using a publicly available data set involving real-world preferences on sushi types to show the benefits of Mallows-based smoothing. We show that smoothing significantly improves both the prediction and the decision accuracy on this data set. We then present an efficient procedure to compute the choice probabilities for any assortment under the mixture of Mallows model. Surprisingly, this finding allows to formulate a compact mixed integer program (MIP) that leads to a practical approach for solving the constrained assortment optimization problem under a general mixture of Mallows model. To complement this MIP formulation, we also exploit additional structural properties of the underlying distribution to propose several polynomial-time approximation schemes. These are the first efficient algorithms with provably near-optimal performance guarantees for the assortment optimization problem under the Mallows or the mixture of Mallows model in such generality.

Suggested Citation

Désir, Antoine and Goyal, Vineet and Jagabathula, Srikanth and Segev, Danny, Mallows-Smoothed Distribution Over Rankings Approach for Modeling Choice (May 3, 2018). Available at SSRN: https://ssrn.com/abstract=3172997 or http://dx.doi.org/10.2139/ssrn.3172997

Antoine Désir

INSEAD ( email )

Boulevard de Constance
77305 Fontainebleau Cedex
France

Vineet Goyal

Columbia University - Department of Industrial Engineering and Operations Research (IEOR) ( email )

331 S.W. Mudd Building
500 West 120th Street
New York, NY 10027
United States

Srikanth Jagabathula

New York University (NYU) - Department of Information, Operations, and Management Sciences ( email )

44 West Fourth Street
New York, NY 10012
United States

Danny Segev (Contact Author)

University of Haifa - Department of Statistics ( email )

Haifa 31905
Israel

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