Due to numerous applications in retail and (online) advertising the problem of assortment selection has been widely studied under many combinations of discrete choice models and feasibility constraints. In many situations, however, an assortment of products has to be constructed gradually and without accurate knowledge of all possible alternatives; in such cases, existing offline approaches become inapplicable. We consider a stochastic variant of the assortment selection problem, where the parameters that determine the revenue and (relative) demand of each item are jointly drawn from some known item-specific distribution. The items are observed sequentially in an arbitrary and unknown order; upon observing the realized parameters of each item, the decision-maker decides irrevocably whether to include it in the constructed assortment, or forfeit it forever. The objective is to maximize the expected total revenue of the constructed assortment, relative to that of an offline algorithm which foresees all the parameter realizations and computes the optimal assortment. We provide simple threshold-based online policies for the unconstrained and cardinality-constrained versions of the problem under a natural class of substitutable choice models; as we show, our policies are (worst-case) optimal under the celebrated Multinomial Logit choice model. We extend our results to the case of knapsack constraints and discuss interesting connections to the Prophet Inequality problem, which is already subsumed by our setting.

翻译：由于在零售和（在线）广告中的广泛应用，分类选择问题已在多种离散选择模型与可行性约束的联合场景下得到广泛研究。然而，在许多实际情境中，分类选择必须逐步构建，且无法准确获知所有可能的备选项；此时，现有的离线方法不再适用。我们考虑分类选择问题的一个随机变体，其中决定每个商品收益与（相对）需求的参数共同从已知的特定商品分布中抽样。商品以任意未知顺序依次出现；决策者在观察到每个商品的参数实现后，需不可撤销地决定是否将其纳入构建的分类中，或永久放弃。目标是最大化所构建分类的期望总收益，相较于那些能预知所有参数实现并计算最优分类的离线算法。针对无约束与基数约束两种版本的问题，我们在可替代选择模型的自然类别下提出了基于阈值的简单在线策略；如我们所示，这些策略在著名的多项式逻辑选择模型下具有（最坏情况）最优性。我们将结果扩展到背包约束场景，并讨论了与先知不等式问题的有趣联系——该问题实际上已被我们的设定所涵盖。