Optimal Competitive Auctions

TL;DR

This paper characterizes and constructs optimal truthful auctions for digital goods, achieving the best worst-case revenue ratios for key benchmarks and confirming longstanding conjectures in auction theory.

Contribution

It provides a necessary and sufficient condition for competitive ratios, constructs optimal auctions for two benchmarks, and settles a major open problem in digital goods auction design.

Findings

Optimal auctions match known lower bounds for the benchmark ^{(2)}.

Confirmed the conjecture that existing bounds are tight.

Derived the optimal competitive ratio for limited-supply Vickrey auctions as (n/(n-1))^{n-1}-1.

Abstract

We study the design of truthful auctions for selling identical items in unlimited supply (e.g., digital goods) to n unit demand buyers. This classic problem stands out from profit-maximizing auction design literature as it requires no probabilistic assumptions on buyers' valuations and employs the framework of competitive analysis. Our objective is to optimize the worst-case performance of an auction, measured by the ratio between a given benchmark and revenue generated by the auction. We establish a sufficient and necessary condition that characterizes competitive ratios for all monotone benchmarks. The characterization identifies the worst-case distribution of instances and reveals intrinsic relations between competitive ratios and benchmarks in the competitive analysis. With the characterization at hand, we show optimal competitive auctions for two natural benchmarks. The most well-studied benchmark $\mathcal{F}^{(2)}(\cdot)$ measures the envy-free optimal revenue where at least two buyers win. Goldberg et al. [13] showed a sequence of lower bounds on the competitive ratio for each number of buyers n. They conjectured that all these bounds are tight. We show that optimal competitive auctions match these bounds. Thus, we confirm the conjecture and settle a central open problem in the design of digital goods auctions. As one more application we examine another economically meaningful benchmark, which measures the optimal revenue across all limited-supply Vickrey auctions. We identify the optimal competitive ratios to be $(\frac{n}{n-1})^{n-1}-1$ for each number of buyers n, that is $e-1$ as $n$ approaches infinity.