Selling Multiple Correlated Goods: Revenue Maximization and Menu-Size Complexity (old title: "The Menu-Size Complexity of Auctions")

TL;DR

This paper demonstrates that simple selling mechanisms are often highly suboptimal for multiple correlated goods, introduces menu size as a complexity measure, and shows revenue can grow polynomially with menu size, with no finite menu size guaranteeing a positive revenue fraction.

Contribution

It proves that simple mechanisms are nearly ineffective for multiple goods, introduces menu size as a key complexity measure, and establishes polynomial revenue growth with menu size.

Findings

Simple mechanisms yield negligible revenue for multiple goods.

Revenue can grow polynomially with menu size.

No finite menu size guarantees a positive revenue fraction.

Abstract

We consider the well known, and notoriously difficult, problem of a single revenue-maximizing seller selling two or more heterogeneous goods to a single buyer whose private values for the goods are drawn from a (possibly correlated) known distribution, and whose valuation is additive over the goods. We show that when there are two (or more) goods, _simple mechanisms_ -- such as selling the goods separately or as a bundle -- _may yield only a negligible fraction of the optimal revenue_. This resolves the open problem of Briest, Chawla, Kleinberg, and Weinberg (JET 2015) who prove the result for at least three goods in the related setup of a unit-demand buyer. We also introduce the menu size as a simple measure of the complexity of mechanisms, and show that the revenue may increase polynomially with _menu size_ and that no bounded menu size can ensure any positive fraction of the optimal revenue. The menu size also turns out to "pin down" the revenue properties of deterministic mechanisms.