Strengthening Bulow-Klemperer-Style Results for Multi-Unit Auctions

2026-02-12

AI Summary

This work addresses the high competition complexity in multi-unit auctions, where the classical Bulow–Klemperer result requires adding $m$ buyers to ensure that the VCG mechanism’s revenue matches that of the optimal auction. Under the monotone hazard rate (MHR) distribution assumption, the paper proposes a prior-independent, supply-limited variant of VCG. By reducing the worst-case analysis to a truncated generalized Pareto distribution and combining numerical and analytical techniques, the authors establish—for the first time in balanced markets—that approximately $0.4447n$ additional buyers suffice to achieve (approximately) optimal revenue. This mechanism strictly outperforms standard VCG, reducing the required number of extra bidders to less than half of that in the original Bulow–Klemperer bound, thereby significantly lowering competition complexity.

Technology Category

• Game Theory and Economic Paradigms: Mechanism Design
• Multiagent Systems: Mechanism Design
• Search and Optimization: Mixed Discrete/Continuous Search

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
• User Modeling, Personalization and Recommendation: Attacks and countermeasures in recommendation systems

Abstract

The classic result of Bulow and Klemperer (1996) shows that in multi-unit auctions with $m$ units and $n\geq m$ buyers whose values are sampled i.i.d. from a regular distribution, the revenue of the VCG auction with $m$ additional buyers is at least as large as the optimal revenue. Unfortunately, for regular distributions, adding $m$ additional buyers is sometimes indeed necessary, so the"competition complexity"of the VCG auction is $m$. We seek proving better competition complexity results in two dimensions. First, under stronger distributional assumptions, the competition complexity of VCG auction drops dramatically. In balanced markets (where $m=n$) with MHR distributions, it is sufficient to only add $(e^{1/e} - 1 + o(1))n \approx 0.4447n$ additional buyers to match the optimal revenue -- less than half the number that is necessary under regularity -- and this bound is asymptotically tight. We provide both exact finite-market results for small value of $n$, and closed-form asymptotic formulas for general market with any $m\leq n$, and any target fraction of the optimal revenue. Second, we analyze a supply-limiting variant of VCG auction that caps the number of units sold in a prior-independent way. Whenever the goal is to achieve almost the optimal revenue, this mechanism strictly improves upon standard VCG auction, requiring significantly fewer additional buyers. Together, our results show that both stronger distributional assumptions, as well as a simple prior-independent refinement to the VCG auction, can each substantially reduce the number of additional buyers that is sufficient to achieve (near-)optimal revenue. Our analysis hinges on a unified worst-case reduction to truncated generalized Pareto distributions, enabling both numerical computation and analytical tractability.

Problem

• multi-unit auctions
• competition complexity
• VCG auction
• optimal revenue
• distributional assumptions

Innovation

• competition complexity
• VCG auction
• MHR distributions
• supply-limiting mechanism
• revenue maximization