Submultiplicative Glivenko-Cantelli and Uniform Convergence of Revenues

TL;DR

This paper introduces a new variant of the Glivenko-Cantelli Theorem with tighter bounds for extreme values, and applies it to derive sample complexity bounds for revenue convergence in economic models.

Contribution

It develops a submultiplicative version of the Glivenko-Cantelli Theorem and applies it to establish revenue convergence bounds based on valuation moments.

Findings

Tighter convergence bounds for extreme values of the CDF.

Characterization of conditions for almost sure uniform revenue convergence.

A zero-one law based on the finiteness of the first valuation moment.

Abstract

In this work we derive a variant of the classic Glivenko-Cantelli Theorem, which asserts uniform convergence of the empirical Cumulative Distribution Function (CDF) to the CDF of the underlying distribution. Our variant allows for tighter convergence bounds for extreme values of the CDF. We apply our bound in the context of revenue learning, which is a well-studied problem in economics and algorithmic game theory. We derive sample-complexity bounds on the uniform convergence rate of the empirical revenues to the true revenues, assuming a bound on the $k$th moment of the valuations, for any (possibly fractional) $k>1$. For uniform convergence in the limit, we give a complete characterization and a zero-one law: if the first moment of the valuations is finite, then uniform convergence almost surely occurs; conversely, if the first moment is infinite, then uniform convergence almost never occurs.