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Multi-Item Screening with a Maximin-Ratio Objective

Shixin Wang

TL;DR

We address robust multi-item screening with only valuation support information under a maximin ratio objective. The paper shows that the robustly optimal mechanism is separable when only marginal supports are known, while jointly leveraging the joint support yields a practical randomized pricing scheme; moreover, when marginal bundle information is available, a bundle-wise separable mechanism is optimal, and under rho-scaled invariant ambiguity sets, randomized grand bundling is optimal. The core technique is a saddle-point analysis that characterizes worst-case (comonotonic) distributions and reduces multi-item design to tractable scalar optimization, yielding a unique gamma_star that governs the mechanism design. The results provide implementable guidelines for robust pricing across ambiguity sets and extend to bundle-based information, offering practical mechanisms with provable performance guarantees.

Abstract

In multi-item screening, optimal selling mechanisms are challenging to characterize and implement, even with full knowledge of valuation distributions. In this paper, we aim to develop tractable, interpretable, and implementable mechanisms with strong performance guarantees in the absence of precise distributional knowledge. In particular, we study robust screening with a maximin ratio objective. We show that given the marginal support of valuations, the optimal mechanism is separable: each item's allocation probability and payment depend only on its own valuation and not on other items' valuations. However, we design the allocation and payment rules by leveraging the available joint support information. This enhanced separable mechanism can be efficiently implemented through randomized pricing for individual products, which is easy to interpret and implement. Moreover, our framework extends naturally to scenarios where the seller possesses marginal support information on aggregate valuations for any product bundle partition, for which we characterize a bundle-wise separable mechanism and its guarantee. Beyond rectangular-support ambiguity sets, we further establish the optimality of randomized grand bundling mechanisms within a broad class of ambiguity sets, which we term ``$\boldsymbolρ-$scaled invariant ambiguity set".

Multi-Item Screening with a Maximin-Ratio Objective

TL;DR

We address robust multi-item screening with only valuation support information under a maximin ratio objective. The paper shows that the robustly optimal mechanism is separable when only marginal supports are known, while jointly leveraging the joint support yields a practical randomized pricing scheme; moreover, when marginal bundle information is available, a bundle-wise separable mechanism is optimal, and under rho-scaled invariant ambiguity sets, randomized grand bundling is optimal. The core technique is a saddle-point analysis that characterizes worst-case (comonotonic) distributions and reduces multi-item design to tractable scalar optimization, yielding a unique gamma_star that governs the mechanism design. The results provide implementable guidelines for robust pricing across ambiguity sets and extend to bundle-based information, offering practical mechanisms with provable performance guarantees.

Abstract

In multi-item screening, optimal selling mechanisms are challenging to characterize and implement, even with full knowledge of valuation distributions. In this paper, we aim to develop tractable, interpretable, and implementable mechanisms with strong performance guarantees in the absence of precise distributional knowledge. In particular, we study robust screening with a maximin ratio objective. We show that given the marginal support of valuations, the optimal mechanism is separable: each item's allocation probability and payment depend only on its own valuation and not on other items' valuations. However, we design the allocation and payment rules by leveraging the available joint support information. This enhanced separable mechanism can be efficiently implemented through randomized pricing for individual products, which is easy to interpret and implement. Moreover, our framework extends naturally to scenarios where the seller possesses marginal support information on aggregate valuations for any product bundle partition, for which we characterize a bundle-wise separable mechanism and its guarantee. Beyond rectangular-support ambiguity sets, we further establish the optimality of randomized grand bundling mechanisms within a broad class of ambiguity sets, which we term ``scaled invariant ambiguity set".
Paper Structure (10 sections, 19 theorems, 55 equations, 8 figures)

This paper contains 10 sections, 19 theorems, 55 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Literature Review
  3. Model Formulation
  4. Discussion of the Other Separable Mechanisms
  5. Optimality of the Separable Mechanism $M_{\gamma^*}$
  6. Warm Up: Two-Item Problem
  7. General Multi-Item Problem
  8. Optimality Condition for Bundling
  9. Summary
  10. Omitted Proofs

Key Result

Lemma 1

For any seller's mechanism $M$, nature's optimal strategy $(\mathbb{F},M')$ is a single-point strategy, i.e., $\mathbb{F}$ is a one-point distribution at a valuation $\boldsymbol{v} \in \mathcal{V}$ and $M'$ sells the full bundle at price $\boldsymbol{1}^\top \boldsymbol{v}$, so Problem eq:original

Figures (8)

  • Figure 1: Illustration of the Separable Mechanism in \ref{['cor:2item']} for $\underline{v}_1=0.01$, $\underline{v}_2=0.5$, $\overline{v}_1 = \overline{v}_2=1$
  • Figure 2: Performance Ratios of $M_{\gamma^*}$ and Decomposed Separable Mechanism in \ref{['eq:separate']} for $\underline{v}_2=0.5$, $\overline{v}_1 = \overline{v}_2=1$ with Different $\underline{v}_1$
  • Figure 3: Support of the Joint Distribution of $(v_1,v_2)$ when $\underline{v}_1=2,\, \underline{v}_2=4,\, \overline{v}_1=\overline{v}_2=12,\, \omega_1 = 3$
  • Figure 4: Support for Some Examples of $\widetilde{\mathcal{F}}\subseteq \mathcal{F}$ with $\mathbb{F}_{\gamma^*}\in\widetilde{\mathcal{F}}$
  • Figure 5: Feasible Valuation Set $\mathcal{V}_{\mathcal{B}} = \{\boldsymbol{v}: v_1+v_2 \in [2,4],\, v_3\in[1,2 ]\}$
  • ...and 3 more figures

Theorems & Definitions (40)

  • Lemma 1
  • Definition 1: Separable Mechanism
  • Lemma 2
  • Lemma 3
  • Proposition 1
  • Lemma 4
  • Corollary 1: Two Products
  • Proposition 2
  • Corollary 2
  • Definition 2
  • ...and 30 more