Simple Mechanisms for a Subadditive Buyer and Applications to Revenue Monotonicity

TL;DR

This paper demonstrates that simple pricing strategies can approximate the optimal revenue for sellers with heterogeneous items and subadditive buyers, and explores the relationship between simple mechanisms and revenue monotonicity.

Contribution

It introduces simple mechanisms that achieve constant-factor revenue approximation for subadditive buyers and connects this to approximate revenue monotonicity.

Findings

Simple pricing strategies approximate optimal revenue within a constant factor.

The core-tail decomposition framework extends to subadditive valuations.

Bounded degradation in revenue with increasing buyer valuations.

Abstract

We study the revenue maximization problem of a seller with n heterogeneous items for sale to a single buyer whose valuation function for sets of items is unknown and drawn from some distribution D. We show that if D is a distribution over subadditive valuations with independent items, then the better of pricing each item separately or pricing only the grand bundle achieves a constant-factor approximation to the revenue of the optimal mechanism. This includes buyers who are k-demand, additive up to a matroid constraint, or additive up to constraints of any downwards-closed set system (and whose values for the individual items are sampled independently), as well as buyers who are fractionally subadditive with item multipliers drawn independently. Our proof makes use of the core-tail decomposition framework developed in prior work showing similar results for the significantly simpler class of additive buyers [LY13, BILW14]. In the second part of the paper, we develop a connection between approximately optimal simple mechanisms and approximate revenue monotonicity with respect to buyers' valuations. Revenue non-monotonicity is the phenomenon that sometimes strictly increasing buyers' values for every set can strictly decrease the revenue of the optimal mechanism [HR12]. Using our main result, we derive a bound on how bad this degradation can be (and dub such a bound a proof of approximate revenue monotonicity); we further show that better bounds on approximate monotonicity imply a better analysis of our simple mechanisms.