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

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.

💡 Research Summary

The paper revisits the classic Bulow‑Klemperer result for multi‑unit auctions, which states that when m identical items are sold to n ≥ m buyers whose values are i.i.d. from a regular distribution, running the VCG (Vickrey‑Clarke‑Groves) auction with m additional buyers guarantees revenue at least as high as the Bayesian optimal mechanism. That result is tight: for regular distributions one sometimes needs exactly m extra buyers. The authors ask whether (i) stronger distributional assumptions can reduce the required number of extra buyers, and (ii) a prior‑independent modification of VCG can further improve the guarantee.

Main contributions

1. Competition complexity under stronger distributional assumptions.
   • For monotone hazard rate (MHR) distributions, the authors prove that in a balanced market (m = n) it suffices to add only (e^{1/e} − 1 + o(1)) n ≈ 0.4447 n extra buyers to make VCG’s expected revenue dominate the optimal revenue. This bound is asymptotically tight.
   • They obtain exact competition‑complexity numbers for all n ≤ 593 by a computer‑aided grid search over a single truncation parameter r of a truncated exponential distribution, which they show (Proposition 3.1) captures the worst‑case among all MHR distributions. For n ≥ 594 they prove a uniform upper bound of 0.456 n, converging to the 0.4447 n limit as n grows.
   • The analysis is extended to the broader class of λ‑regular distributions (λ ∈