E-commerce is shifting from search-based shopping to agentic purchasing. Rather than relying on keywords, AI shopping agents learn customer preferences through targeted multi-round conversations and then recommend a tailored set of products. We develop a solicit-then-suggest framework to study this setting. In a d-dimensional preference space, an agent conducts m rounds of solicitation to refine its belief about the customer's ideal product, then recommends k products from which the customer chooses. Our analysis identifies the key economic tradeoff. Under a Gaussian prior, we establish an uncertainty decomposition: solicitation depth and assortment breadth are substitutes, with total prior uncertainty split between what solicitation resolves and what assortment breadth hedges. The two instruments improve match quality at very different rates. Expected loss decreases on the order of 1/m with solicitation depth, but only on the order of k^(-2/d) with assortment breadth, reflecting a curse of dimensionality. Thus, a few well-designed questions can achieve what would otherwise require far more recommendations. We also characterize the optimal policy. The optimal assortment forms a Voronoi partition, assigning each product to the posterior region it best serves. With a single recommended product, the optimal solicitation follows a water-filling rule that equalizes posterior uncertainty across dimensions. With multiple products, the optimum may allocate less precision to dimensions that the assortment can hedge. This single-product water-filling rule also yields a general approximation guarantee for larger assortments, and the gap vanishes as dimension grows. Beyond the Gaussian case, the uncertainty decomposition and substitutability between solicitation depth and assortment breadth continue to hold for non-Gaussian priors.

翻译：电子商务正从基于关键词的搜索式购物转向代理式购物。AI购物代理不再依赖关键词，而是通过有针对性的多轮对话学习客户偏好，然后推荐一组量身定制的产品。我们开发了一个“先征集后建议”框架来研究这一场景。在一个d维偏好空间中，代理通过m轮征集来修正其对客户理想产品的信念，然后推荐k个产品供客户选择。我们的分析揭示了关键的经济权衡。在高斯先验下，我们建立了一种不确定性分解：征集深度和品类广度是相互替代的，总先验不确定性在征集所能解决的部分和品类广度所能对冲的部分之间分配。这两种工具以截然不同的速度改善匹配质量。预期损失随征集深度以1/m的阶数下降，但随品类广度仅以k^(-2/d)的阶数下降，这体现了维数灾难。因此，几个设计得当的问题就能达到原本需要远更多推荐才能实现的效果。我们还刻画了最优策略。最优品类形成一个Voronoi划分，将每个产品分配给它服务最好的后验区域。当推荐单个产品时，最优征集遵循一个“注水”规则，使各维度的后验不确定性相等。当推荐多个产品时，最优策略可能会在品类能够对冲的维度上分配较少的精度。这种单产品“注水”规则也为更大的品类集合提供了一般性的近似保证，且该差距随维度增加而消失。在高斯情形之外，征集深度与品类广度之间的不确定性分解和可替代性对于非高斯先验仍然成立。