Lower Bounds on Revenue of Approximately Optimal Auctions

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 v _max = max{V ₁,V ₂,…,V _n } achieves a revenue of at least a $\frac{1}{e}$ fraction of the geometric expectation of v _max. This generic bound is a measure of how revenue improves/degrades as a function of the concentration/spread of v _max.

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 v _max.