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Optimal Selection with Balanced Market Share: Static and Dynamic Assortment Optimization

Omar El Housni, Qing Feng, Huseyin Topaloglu

Abstract

Assortment optimization is a critical tool for online retailers aiming to maximize revenue. However, optimizing purely for revenue can lead to unbalanced sales across products, potentially causing a long tail of low-selling products and products with excessively large market shares, both of which could be harmful to the seller. To address these issues, we introduce a market share balancing constraint that limits the disparity in expected sales between any two offered products to a factor of a given parameter $α$. We study both static and dynamic assortment optimization under the multinomial logit (MNL) model with this fairness constraint. In the static setting, the seller selects a distribution over assortments that satisfies the market share balancing constraint while maximizing expected revenue. We show that this problem can be solved in polynomial time, and we characterize the structure of the optimal solution: a product is included if and only if its revenue and preference weight exceed certain thresholds. We further extend our analysis to settings with additional feasibility constraints on the assortment and demonstrate that, given a $β$-approximation oracle for the constrained problem, we can construct a $β$-approximation algorithm under the fairness constraint. In the dynamic setting, each product has a finite initial inventory, and the seller implements a dynamic policy to maximize total expected revenue while respecting both inventory limits and the market share balancing constraint in expectation. We design a policy that is asymptotically optimal, with its approximation ratio converging to one as inventories grow large.

Optimal Selection with Balanced Market Share: Static and Dynamic Assortment Optimization

Abstract

Assortment optimization is a critical tool for online retailers aiming to maximize revenue. However, optimizing purely for revenue can lead to unbalanced sales across products, potentially causing a long tail of low-selling products and products with excessively large market shares, both of which could be harmful to the seller. To address these issues, we introduce a market share balancing constraint that limits the disparity in expected sales between any two offered products to a factor of a given parameter . We study both static and dynamic assortment optimization under the multinomial logit (MNL) model with this fairness constraint. In the static setting, the seller selects a distribution over assortments that satisfies the market share balancing constraint while maximizing expected revenue. We show that this problem can be solved in polynomial time, and we characterize the structure of the optimal solution: a product is included if and only if its revenue and preference weight exceed certain thresholds. We further extend our analysis to settings with additional feasibility constraints on the assortment and demonstrate that, given a -approximation oracle for the constrained problem, we can construct a -approximation algorithm under the fairness constraint. In the dynamic setting, each product has a finite initial inventory, and the seller implements a dynamic policy to maximize total expected revenue while respecting both inventory limits and the market share balancing constraint in expectation. We design a policy that is asymptotically optimal, with its approximation ratio converging to one as inventories grow large.

Paper Structure

This paper contains 38 sections, 18 theorems, 188 equations, 2 figures, 5 tables, 7 algorithms.

Table of Contents

  1. Introduction
  2. Main Contributions
  3. Related Literature
  4. Static Assortment Optimization Problem
  5. Problem Formulation
  6. Model Discussion
  7. Value of Randomization
  8. Algorithms for the Static Problem
  9. Algorithm for \ref{['prob:randomized_0']}
  10. Static Problem with Additional Constraints
  11. Dynamic Assortment Optimization
  12. Problem Formulation
  13. Upper Bound Problem
  14. Algorithm for the Dynamic Problem
  15. Numerical Experiments on Real-World Data
  16. ...and 23 more sections

Key Result

Proposition 2.1

The optimal value $R^*$ of prob:randomized_0 and the optimal value $R^*_{det}$ of BMS-Deterministic satisfy Furthermore, for any $0<\alpha\leq 1$ and any $n\in\mathbb{Z}^+$, there exists an instance with $n$ products such that

Figures (2)

  • Figure 1: Optimal expected revenue with market share balancing constraint and absolute market share balancing constraint under different $\alpha$
  • Figure 2: Revenue-difference plot under our setting and the absolute market share balancing constraint for different product types

Theorems & Definitions (27)

  • Proposition 2.1
  • Theorem 3.1
  • Lemma 3.2
  • proof : Proof of Theorem \ref{['thm:randomized']}
  • Corollary 3.3
  • Example 3.4
  • Theorem 3.5
  • Lemma 3.6
  • Lemma 3.7
  • Lemma 3.8
  • ...and 17 more