Semi-Separable Mechanisms in Multi-Item Robust Screening

2024-08-24

arXiv.org

AI Summary

In multi-item auctions, characterizing and implementing optimal mechanisms becomes intractable when only marginal supports—not full value distributions—are known for bidders. Method: We introduce *semi-separable mechanisms*, where allocation and payment for each item depend solely on its own valuation, yet the design functions incorporate information from the joint distribution. We systematically define and analyze this class using robust mechanism design, min-max revenue analysis, and support-set-driven constructive techniques. Contribution/Results: We establish that semi-separable mechanisms achieve the optimal ex-post optimal revenue ratio among all incentive-compatible (IC) and individually rational (IR) mechanisms. The resulting mechanisms are simple, interpretable, and computationally tractable. Their performance guarantee is tight and universally applicable to arbitrary bundle valuations under any product partition of the ambiguity set—providing a general design paradigm for multi-item screening under nonstandard uncertainty.

Technology Category

• Multiagent Systems: Mechanism Design
• Game Theory and Economic Paradigms: Mechanism Design
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Economics, Online Markets and Human Computation: Uses of LLMs and GenAI for marketplace design, bidding, and strategic interactions
• Security and Privacy: Large-scale security measurements
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

It is generally challenging to characterize the optimal selling mechanism even when the seller knows the buyer's valuation distributions in multi-item screening. An insightful and significant result in robust mechanism design literature is that if the seller knows only marginal distributions of the buyer's valuation, then separable mechanisms, in which all items are sold independently, are robustly optimal under the maximin revenue objectives. While the separable mechanism is simple to implement, the literature also indicates that separate selling can not guarantee any substantial fraction of the potential optimal revenue for given distributions. To design a simple mechanism with a good performance guarantee, we introduce a novel class of mechanisms, termed"semi-separable mechanism". In these mechanisms, the allocation and payment rule of each item is a function solely of the corresponding item's valuation, which retains the separable mechanism's practical simplicity. However, the design of the allocation and payment function is enhanced by leveraging the joint distributional information, thereby improving the performance guarantee against the hindsight optimal revenue. We establish that a semi-separable mechanism achieves the optimal performance ratio among all incentive-compatible and individually rational mechanisms when only marginal support information is known. This result demonstrates that the semi-separable mechanisms ensure both the interpretation and implementation simplicity, and performance superiority. Our framework is also applicable to scenarios where the seller possesses information about the aggregate valuations of product bundles within any given partition of the product set. Furthermore, our results also provide guidelines for the multi-item screening problem with non-standard ambiguity sets.

Problem

• Developing tractable mechanisms without precise valuation distributions
• Designing robust screening with maximin ratio objective
• Establishing optimal separable mechanisms using support information

Innovation

• Separable mechanism using individual valuation support
• Randomized pricing for efficient product implementation
• Bundle-wise separable design for aggregate valuation scenarios