Cardinality-Constrained Randomized Assortments with Balanced Market Share

Abstract

Assortment optimization asks a seller which of $n$ products to display to each customer. Under the multinomial-logit (MNL) model, a revenue-maximizing policy may concentrate most purchases on only a few products. Balanced market share (BMS) limits this disparity by requiring every product to receive either zero aggregate sales or at least an $α$-fraction of the largest product's sales. We study randomized BMS under a per-assortment cardinality constraint: every displayed assortment contains at most $K$ products, although the seller may rotate a larger catalog across customers. This couples per-realization feasibility with aggregate balance over an exponential policy space. We resolve the model on all three algorithmic fronts. First, reweighting assortments by their MNL denominators turns the cardinality restriction into a single uniform-matroid rank inequality, yielding an exact compact sales-space formulation in which every feasible sales vector can be implemented by a policy supported on $O(n)$ assortments. Second, deciding whether there exists a feasible policy attaining a given revenue target is NP-complete even for $α=1$ and $K=2$, whereas optimization is polynomial-time solvable once the set of products with positive sales is prescribed. Third, despite this hardness, BMS confines all positive sales to one multiplicative band, and a geometric scale search combined with a standard multiple-choice-knapsack dynamic program yields an FPTAS. For every $\varepsilon\in(0,1)$, it achieves at least a $(1-\varepsilon)$ fraction of the optimal expected revenue, preserves BMS and per-assortment cardinality exactly, and outputs a rational policy supported on $O(n)$ assortments.