[Paper Review] A Solicit-Then-Suggest Model of Agentic Purchasing

Shengyu Cao, Ming Hu|arXiv (Cornell University)|Mar 21, 2026

Auction Theory and Applications0 citations

TL;DR

The paper formalizes a solicit-then-suggest framework where an AI shopping agent learns a customer’s preferences via multi-round questioning and then offers a small, tailored assortment; solicitation depth and assortment breadth act as substitutes with different efficiency.

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.

Motivation & Objective

• Motivate and define agentic purchasing as a departure from traditional search by enabling multi-round preference elicitation through conversation.
• Develop a tractable probabilistic model (solicit-then-suggest) that jointly optimizes information gathering and downstream assortment design.
• Characterize how solicitation depth and assortment breadth interact, including their substitution relationship and effect on match quality.
• Provide optimal policies for single- and multi-product assortments under Gaussian priors, and extend results to non-Gaussian priors.
• Offer design guidance for practical systems, including when a simple single-question policy suffices for larger assortments.

Proposed method

• Model the customer with a latent ideal point θ in a d-dimensional space and Gaussian prior N(μ0, Σ0).
• Use m rounds of directional queries with unit-norm y_t and noisy responses z_t = θᵀy_t + ε_t to update beliefs via Kalman filter equations (κ_t, μ_t, Σ_t).
• After m rounds, select k products to form a Voronoi-partitioned optimal assortment, placing each product at the posterior centroid of its region.
• Prove that for k=1 the optimal recommendation is the posterior mean and the expected loss equals half the posterior variance.
• Derive a water-filling solicitation policy for the single-product case and show an efficiency gap bounds when extending to larger assortments.
• Extend the analysis beyond Gaussian priors, establishing the uncertainty decomposition identity and substitutability between solicitation and assortment breadth.

Experimental results

Research questions

• RQ1 How do solicitation depth and assortment breadth interact in agentic purchasing?
• RQ2 What are the optimal policies for choosing query directions and product assortments under Gaussian priors?
• RQ3 What are the rates at which expected loss decreases with more questions vs. more products, and why do these differ?
• RQ4 Do the core results extend to non-Gaussian priors, and how does the Gaussian benchmark perform asymptotically?
• RQ5 What practical design guidance follows from the interaction between information gathering and product hedging?

Key findings

• Uncertainty decomposability: total prior uncertainty is split between what solicitation eliminates and what tailored assortment hedges; solicitation and assortment are substitutes.
• Expected loss decays at O(1/m) with solicitation depth, whereas for assortments with k products the loss decays at O(k^{-2/d}), revealing a curse of dimensionality for breadth but not for querying.
• Optimal single-product assortments place the product at the posterior mean, with loss equal to half the posterior variance; multi-product assortments form Voronoi partitions with products at posterior centroids.
• Water-filling solicitation policy equalizes posterior uncertainty across actively learned dimensions, yielding near-optimality for small m and larger d; selective focus can arise in joint optimization.
• Gaussian results provide conservative benchmarks for non-Gaussian priors, with asymptotic exactness as conversation length grows.