The Root of Revenue Continuity

TL;DR

This paper establishes a mathematical relationship showing that small changes in buyer valuation distributions lead to small changes in revenue, using Wasserstein distance, and demonstrates that simple modifications to mechanisms remain nearly optimal under such changes.

Contribution

The paper proves a general inequality linking revenue differences to Wasserstein distance between valuation distributions, enabling near-optimal mechanism adjustments for similar distributions.

Findings

Revenue variation is bounded by Wasserstein distance.

Uniform discounting mechanism remains nearly optimal for close distributions.

Provides a simple, general statement connecting valuation changes to revenue impact.

Abstract

In the setup of selling one or more goods, various papers have shown, in various forms and for various purposes, that a small change in the distribution of a buyer's valuations may cause only a small change in the possible revenue that can be extracted. We prove a simple, clean, convenient, and general statement to this effect: let $X$ and $Y$ be random valuations on $k$ additive goods, and let $W(X,Y)$ be the Wasserstein (or "earth mover's") distance between them; then $$\left\vert \sqrt{Rev(X)}-\sqrt{Rev(Y)}\right\vert \le \sqrt{W(X,Y)}.$$ This further implies that a simple explicit modification of any optimal mechanism for $X$, namely, "uniform discounting," is guaranteed to be almost optimal for any $Y$ that is close to $X$ in the Wasserstein distance.