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

TL;DR

The paper strengthens Bulow–Klemperer-style results for multi-unit auctions by showing that under stronger distributional assumptions, such as MHR and lambda-regularity, far fewer additional buyers are needed for the VCG auction to match or approximate the Bayesian-optimal revenue. It introduces a unified worst-case reduction to one-parameter families of truncated generalized Pareto distributions, enabling precise finite-market and asymptotic competition-complexity bounds and tractable numerical computation. It also analyzes a prior-independent supply-limiting variant of VCG, demonstrating significant revenue improvements over standard VCG when aiming for near-optimal revenue and providing closed-form asymptotics in large markets. Together, these results reveal that both distributional strength and simple prior-independent refinements can substantially reduce the required level of competition to achieve (near-)optimal revenue in multi-unit auctions, with practical implications for mechanism design in markets with homogeneous goods and limited prior information.

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.

Figures (8)

• Figure 1: The asymptotic competition complexity (CC) (to achieve 99.99% of the Bayesian optimal revenue) as a function of supply-demand ratio $\alpha\in[0, 1]$. The black, dark-gray, and light-gray curves correspond to regular $\mathcal{F}_{\text{Reg}}$, $0.5$-regular $\mathcal{F}_{\text{0.5-Reg}}$, and MHR distributions $\mathcal{F}_{\text{MHR}}$, respectively.
• Figure 2: The asymptotic competition complexity (CC) of the VCG Auction (solid curve) and the asymptotic optimal competition complexity (CC) of the Supply-Limiting VCG Auction (dashed curve) as a function of target approximation ratio $\Gamma\in(0, 1)$ in the balanced market (i.e., $\alpha=1$). The black, dark-gray, and light-gray curves correspond to regular $\mathcal{F}_{\text{Reg}}$, $0.5$-regular $\mathcal{F}_{\text{0.5-Reg}}$, and MHR distributions $\mathcal{F}_{\text{MHR}}$, respectively.
• Figure 3: Graphical illustration of the analysis for \ref{['prop:bk vcg:worst case']}. The black (resp. gray) solid curve is the revenue curve $R^\dagger$ (resp. $R_{(r^*,\lambda)}$) induced by distribution $F^\dagger$ (resp. $F_{(r^*,\lambda)}$). The black dashed line in Case (ii) is $r^*/v^\dagger\cdot R^\dagger(q)$. Quantile $q^*$ is the monopoly quantile and $q^\dagger$ is the quantile threshold defined in \ref{['lem:bk vcg:osdensity single crossing']}.
• Figure 4: Illustration of \ref{['example:ratio monotonicity']}. The black (gray) curve is revenue curve $R_1$ ($R_2$).
• Figure 5: The expected revenue of the VCG Auction (of selling 10 units to 15 buyers whose values are drawn i.i.d. from $1/q^*$-truncated exponential distribution) as a function of monopoly quantile $q^*\in\left[{1}/{e},1\right]$. The local-minimum expected revenue of the VCG Auction is not attained at $q^*={1}/{e}$; instead, it is attained at some $q^*$ in a right neighborhood of ${1}/{e}$ and at $q^*=1$.

Theorems & Definitions (54)

• Example 1.1
• Lemma 2.1: mye-81
• Definition 2.1
• Theorem 2.2: BK-96
• Definition 2.2: Competition complexity
• Definition 2.3: Revenue curve, BR-89
• Lemma 2.3: BR-89
• Definition 3.1: Truncated exponential distribution
• Proposition 3.1: Characterization of a subclass of worst-case MHR distributions
• Remark 3.2