Mallows-Smoothed Distribution Over Rankings Approach for Modeling Choice

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.