Assortment optimization has received active explorations in the past few decades due to its practical importance. Despite the extensive literature dealing with optimization algorithms and latent score estimation, uncertainty quantification for the optimal assortment still needs to be explored and is of great practical significance. Instead of estimating and recovering the complete optimal offer set, decision makers may only be interested in testing whether a given property holds true for the optimal assortment, such as whether they should include several products of interest in the optimal set, or how many categories of products the optimal set should include. This paper proposes a novel inferential framework for testing such properties. We consider the widely adopted multinomial logit (MNL) model, where we assume that each customer will purchase an item within the offered products with a probability proportional to the underlying preference score associated with the product. We reduce inferring a general optimal assortment property to quantifying the uncertainty associated with the sign change point detection of the marginal revenue gaps. We show the asymptotic normality of the marginal revenue gap estimator, and construct a maximum statistic via the gap estimators to detect the sign change point. By approximating the distribution of the maximum statistic with multiplier bootstrap techniques, we propose a valid testing procedure. We also conduct numerical experiments to assess the performance of our method.

翻译：分类优化因其实际重要性在过去数十年中受到广泛关注。尽管已有大量文献涉及优化算法和潜在得分估计，但最优分类的不确定性量化仍需探索，且具有重要实践意义。决策者可能无需估计和恢复完整的优选集，而仅需检验最优分类是否满足特定属性，例如是否应将某些感兴趣的产品纳入优选集，或优选集应包含多少产品类别。本文提出一种新颖的推断框架，用于检验此类属性。我们采用广泛使用的多项式对数（MNL）模型，假设每位顾客会以与产品潜在偏好得分成比例的概率购买被提供的商品。我们将一般最优分类属性的推断问题简化为对边际收益缺口变号点检测的不确定性量化问题。我们证明了边际收益缺口估计量的渐近正态性，并基于缺口估计量构造了最大值统计量以检测变号点。通过使用乘子自助法近似最大值统计量的分布，我们提出一个有效的检验程序。此外，我们通过数值实验评估了所提方法的性能。