A lower bound on seller revenue in single buyer monopoly auctions

TL;DR

This paper establishes a tight lower bound on seller revenue in single-buyer auctions, linking it to the geometric expectation of valuations, and characterizes the conditions for near-optimal revenue.

Contribution

It introduces a precise lower bound on revenue based on valuation distribution and identifies the equal revenue distribution as the unique case of equality.

Findings

Lower bound on revenue is 1/e times the geometric expectation.

Equal revenue distribution uniquely achieves the lower bound.

Revenue approaches valuation expectation when valuation and geometric expectation are close.

Abstract

We consider a monopoly seller who optimally auctions a single object to a single potential buyer, with a known distribution of valuations. We show that a tight lower bound on the seller's expected revenue is $1/e$ times the geometric expectation of the buyer's valuation, and that this bound is uniquely achieved for the equal revenue distribution. We show also that when the valuation's expectation and geometric expectation are close, then the seller's expected revenue is close to the expected valuation.