Benchmark-Tight Approximation Ratio of Simple Mechanism for a Unit-Demand Buyer

2026-06-08

AI Summary

This study addresses revenue maximization in a single-buyer, unit-demand setting, surpassing the previously established 4-approximation barrier for simple mechanisms relative to the ex ante relaxation benchmark. The authors introduce a novel approach that eschews Myerson auctions and instead integrates ironed virtual valuation techniques, the ex ante relaxation benchmark, and prophet inequalities within the single-sample paradigm to directly analyze the Uniform-Ironed-Virtual-Value (UIVV) item pricing mechanism. This work establishes the first tight 3-approximation guarantee for simple multi-item mechanisms and provides a fully constructive proof of a 3-competitive prophet inequality. Moreover, the paper demonstrates that this 3-approximation ratio is unimprovable for UIVV-based mechanisms within the single-sample framework.

Abstract

We study revenue maximization in the unit-demand single-buyer setting. Our main result is that \textsf{Uniform-Ironed-Virtual-Value Item Pricing} guarantees a {\em tight} $3$-approximation to the \textsf{Duality Relaxation Benchmark} [Chawla-Malec-Sivan, EC'10/GEB'15; Cai-Devanur-Weinberg, STOC'16/ SICOMP'21], breaking the barrier of $4$ since [Chawla-Hartline-Malec-Sivan, STOC'10; Chawla-Malec-Sivan, EC'10/GEB'15]. To our knowledge, this is the first {\em benchmark-tight} revenue guarantee of any simple multi-item mechanism. Technically, all previous works employ \textsf{Myerson Auction} as an intermediary. The barrier of $4$ follows as \textsf{Uniform-Ironed-Virtual-Value Item Pricing} achieves a {\em tight} $2$-approximation to \textsf{Myerson Auction}, which then achieves a {\em tight} $2$-approximation to \textsf{Duality Relaxation Benchmark}. Instead, our new approach avoids \textsf{Myerson Auction}, thus enabling the improvement. Central to our work are a {\em benchmark-based} $3$-competitive prophet inequality and its {\em fully constructive} proof. Such variant prophet inequalities shall find future applications, e.g., to Multi-Item Mechanism Design where optimal revenues are relaxed to various more accessible benchmarks. We complement our benchmark-tight ratio with an impossibility result. All previous works and ours follow the {\em single-dimensional representative} approach introduced by [Chawla-Hartline-Kleinberg, EC'07]. Against \textsf{Duality Relaxation Benchmark}, it turns out that this approach cannot beat our bound of $3$ for a large class of \textsf{Item Pricing}'s.

Problem

• revenue maximization
• unit-demand buyer
• approximation ratio
• simple mechanism
• duality relaxation benchmark

Innovation

• benchmark-tight approximation
• unit-demand buyer
• item pricing
• prophet inequality
• duality relaxation benchmark