基数约束下的市场份额均衡随机商品组合

中文摘要 AI 辅助

品类优化问题询问卖家应向每位顾客展示 n 种产品中的哪些。在多项式逻辑（MNL）模型下，收入最大化的策略可能将大部分购买集中在少数几种产品上。市场份额均衡（BMS）通过要求每种产品要么获得零总销售额，要么至少获得最大产品销售额的 α 比例，来限制这种差异。我们在每个品类组合的基数约束下研究随机 BMS：每个展示的品类组合最多包含 K 种产品，尽管卖家可以在顾客之间轮换更大的目录。这将每次实现的可行性问题与指数级策略空间上的总体均衡问题耦合在一起。我们在所有三个算法方面解决了该模型。首先，通过 MNL 分母对品类组合进行重新加权，将基数限制转化为单个均匀拟阵秩不等式，从而得到一个精确紧凑的销售空间公式，其中每个可行的销售向量都可以由支持在 O(n) 个品类组合上的策略实现。其次，即使在 α=1 和 K=2 的情况下，判断是否存在达到给定收入目标的可行策略也是 NP 完全的，而一旦指定了具有正销售的产品集合，优化问题则可在多项式时间内求解。第三，尽管存在这种困难，BMS 将所有正销售限制在一个乘法带内，并且几何尺度搜索结合标准的多选背包动态规划可产生 FPTAS。对于每个 ε∈(0,1)，该算法至少达到最优期望收入的 (1-ε) 比例，精确保持 BMS 和每个品类组合的基数约束，并输出支持在 O(n) 个品类组合上的有理策略。

英文摘要

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.

发表机构

• City University of Hong Kong（香港城市大学）
• Peking University（北京大学）
• Chinese Academy of Sciences（中国科学院）

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