Lower Bounds on Revenue of Approximately Optimal Auctions

TL;DR

This paper establishes fundamental revenue bounds for simple auction mechanisms in single-item, multi-buyer settings, linking revenue guarantees to the distribution of maximum buyer valuations and demonstrating robustness with additional bidders.

Contribution

It provides a novel, distribution-agnostic revenue guarantee for single posted price auctions based on the maximum valuation distribution, extending to additional bidders and welfare approximation.

Findings

A single posted price achieves at least 1/e of the geometric mean of maximum valuations.

Adding identical bidders improves revenue guarantees similarly to distribution-based bounds.

The bounds relate revenue and welfare approximation to the spread of maximum valuations.

Abstract

We obtain revenue guarantees for the simple pricing mechanism of a single posted price, in terms of a natural parameter of the distribution of buyers' valuations. Our revenue guarantee applies to the single item n buyers setting, with values drawn from an arbitrary joint distribution. Specifically, we show that a single price drawn from the distribution of the maximum valuation Vmax = max {V_1, V_2, ...,V_n} achieves a revenue of at least a 1/e fraction of the geometric expecation of Vmax. This generic bound is a measure of how revenue improves/degrades as a function of the concentration/spread of Vmax. We further show that in absence of buyers' valuation distributions, recruiting an additional set of identical bidders will yield a similar guarantee on revenue. Finally, our bound also gives a measure of the extent to which one can simultaneously approximate welfare and revenue in terms of the concentration/spread of Vmax.