A Solicit-Then-Suggest Model of Agentic Purchasing

Abstract: 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.