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.

翻译：Bulow与Klemperer（1996）的经典结果表明：在具有$m$单位商品且$n\geq m$个买家的多单位拍卖中，若买家估值从正则分布中独立同分布采样，则增加$m$个额外买家后的VCG拍卖所得收益至少不低于最优收益。然而对于正则分布，增加$m$个额外买家有时确属必要，这意味着VCG拍卖的"竞争复杂度"为$m$。本研究旨在从两个维度改进竞争复杂度的理论结果。首先，在更强的分布假设下，VCG拍卖的竞争复杂度显著下降。在具有MHR分布的平衡市场（$m=n$）中，仅需增加$(e^{1/e} - 1 + o(1))n \approx 0.4447n$个额外买家即可达到最优收益——该数值不足正则分布条件下所需数量的一半——且该渐近界限是紧致的。我们既给出了小规模$n$值市场的精确有限结果，也推导了任意$m\leq n$的通用市场及任意目标收益比例下的闭式渐近公式。其次，我们分析了一种供应限制型VCG拍卖变体，该机制以先验独立的方式限制售出商品数量。当目标为实现近乎最优收益时，此机制严格优于标准VCG拍卖，所需额外买家数量显著减少。综合而言，我们的研究表明：更强的分布假设以及对VCG拍卖的简单先验独立改进，均能大幅减少实现（接近）最优收益所需的额外买家数量。本研究的分析核心在于建立了截断广义帕累托分布的统⼀最坏情况约化框架，从而同时实现了数值计算与解析处理的可行性。