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

TL;DR

This paper introduces a market share balancing constraint in assortment optimization to ensure fair sales distribution, providing polynomial-time solutions and asymptotically optimal dynamic policies under the MNL model.

Contribution

It formulates and solves a novel fairness-constrained assortment optimization problem, extending static and dynamic models with polynomial-time algorithms and asymptotic optimality.

Findings

Polynomial-time solution for static fairness-constrained assortment optimization.

Characterization of optimal static assortments based on revenue and preference thresholds.

Asymptotically optimal dynamic policy as inventories grow large.

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 $\alpha$. 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 $\beta$-approximation oracle for the constrained problem, we can construct a $\beta$-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.