Optimal Deterministic Mechanisms for an Additive Buyer

TL;DR

This paper characterizes optimal deterministic revenue-maximizing mechanisms for a single additive buyer with two independent items, revealing submodularity, monotonicity, and symmetry properties, and highlighting limitations in extending these results.

Contribution

It provides the first detailed characterization of optimal deterministic mechanisms in a setting where Myerson's theorem does not apply, especially for two independent items.

Findings

Optimal mechanisms are submodular and monotone.

In IID cases, optimal mechanisms are symmetric.

Characterizations do not extend to correlated distributions or more than two items.

Abstract

We study revenue maximization by deterministic mechanisms for the simplest case for which Myerson's characterization does not hold: a single seller selling two items, with independently distributed values, to a single additive buyer. We prove that optimal mechanisms are submodular and hence monotone. Furthermore, we show that in the IID case, optimal mechanisms are symmetric. Our characterizations are surprisingly non-trivial, and we show that they fail to extend in several natural ways, e.g. for correlated distributions or more than two items. In particular, this shows that the optimality of symmetric mechanisms does not follow from the symmetry of the IID distribution.