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A Solicit-Then-Suggest Model of Agentic Purchasing

Shengyu Cao, Ming Hu

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.

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.
Paper Structure (11 sections, 12 theorems, 24 equations, 7 figures)

This paper contains 11 sections, 12 theorems, 24 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Literature Review
  3. Model
  4. Analysis
  5. Optimal Assortment
  6. Optimal Solicitation Policy
  7. Solicitation and Assortment Interplay
  8. Beyond Gaussian Priors
  9. The Two-Phase Partition Tree
  10. Asymptotic Gaussianity and Universality
  11. Conclusion

Key Result

Lemma 1

Given Gaussian prior $\boldsymbol{\theta} | \mathcal{H}_{t-1} \sim \mathcal{N}(\boldsymbol{\mu}_{t-1}, \boldsymbol{\Sigma}_{t-1})$ and observation $z_t = \boldsymbol{\theta}^\top \boldsymbol{y}_t + \epsilon_t$ with $\epsilon_t \sim \mathcal{N}(0, \sigma^2)$, the posterior distribution is Gaussian: where the mean and covariance are updated via the Kalman filter equations: Here, $\boldsymbol{\kapp

Figures (7)

  • Figure 1: Traditional product search versus agentic purchasing.
  • Figure 2: The solicit-then-suggest model.
  • Figure 3: Solicitation interface from OpenAI's shopping agent in ChatGPT (screenshots captured by the authors, January 2026).
  • Figure 4: Optimal two-product assortment under a bivariate Gaussian posterior ($d = 2$, $\lambda_1 = 4$, $\lambda_2 = 1$).
  • Figure 5: Comparison of solicitation depth versus assortment breadth.
  • ...and 2 more figures

Theorems & Definitions (14)

  • Lemma 1: Gaussian Posterior Update
  • Proposition 1: Optimal Single-Product Assortment
  • Proposition 2: Optimal Two-Product Hedging
  • Example 1: Two-Product Laptop Assortment
  • Proposition 3: Optimal Multi-Product Assortment
  • Proposition 4: Value of Assortment Breadth
  • Lemma 2: Solicitation Gains for $k=1$
  • Proposition 5: Optimal Solicitation Policy for $k=1$: Rank-Capped Water-Filling
  • Example 2: Two-Product Solicitation Policy
  • Lemma 3: Two-Product Solicitation under Isotropic Prior
  • ...and 4 more