The Menu-Size Complexity of Revenue Approximation

TL;DR

This paper proves that for any number of items and desired revenue approximation, there exists a bounded menu size mechanism that can achieve nearly optimal revenue, resolving a longstanding open problem in auction design.

Contribution

The authors establish that bounded menu size mechanisms can approximate the optimal revenue arbitrarily closely for any number of items and distributions, providing bounds on menu complexity.

Findings

Bounded menu size mechanisms can achieve (1 - ε) of optimal revenue.

Established upper and lower bounds on menu size complexity.

Connected menu complexity with deterministic communication complexity.

Abstract

Consider a monopolist selling $n$ items to an additive buyer whose item values are drawn from independent distributions $F_1,F_2,\ldots,F_n$ possibly having unbounded support. Unlike in the single-item case, it is well known that the revenue-optimal selling mechanism (a pricing scheme) may be complex, sometimes requiring a continuum of menu entries. Also known is that simple mechanisms with a bounded number of menu entries can extract a constant fraction of the optimal revenue. Nonetheless, whether an arbitrarily high fraction of the optimal revenue can be extracted via a bounded menu size remained open. We give an affirmative answer: for every $n$ and $\varepsilon>0$, there exists $C=C(n,\varepsilon)$ s.t. mechanisms of menu size at most $C$ suffice for obtaining $(1-\varepsilon)$ of the optimal revenue from any $F_1,\ldots,F_n$. We prove upper and lower bounds on the revenue-approximation complexity $C(n,\varepsilon)$ and on the deterministic communication complexity required to run a mechanism achieving such an approximation.