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

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.