A Multi-Dimensional Online Contention Resolution Scheme for Revenue Maximization

TL;DR

This paper introduces a new multi-dimensional online contention resolution scheme (OCRS) that approximates revenue in multi-buyer, multi-item settings with subadditive valuations, extending prior work beyond unit-demand buyers.

Contribution

It develops a novel OCRS for revenue maximization applicable to many subadditive buyers, achieving an $O( ext{log}^2 m)$ approximation for sequential item pricing.

Findings

Approximate ex ante buy-many revenue within $O( ext{log}^2 m)$ factor.

Constructed a multi-dimensional OCRS for revenue maximization.

Logarithmic dependence on number of items is necessary.

Abstract

We study multi-buyer multi-item sequential item pricing mechanisms for revenue maximization with the goal of approximating a natural fractional relaxation -- the ex ante optimal revenue. We assume that buyers' values are subadditive but make no assumptions on the value distributions. While the optimal revenue, and therefore also the ex ante benchmark, is inapproximable by any simple mechanism in this context, previous work has shown that a weaker benchmark that optimizes over so-called ``buy-many" mechanisms can be approximable. Approximations are known, in particular, for settings with either a single buyer or many unit-demand buyers. We extend these results to the much broader setting of many subadditive buyers. We show that the ex ante buy-many revenue can be approximated via sequential item pricings to within an $O(\log^2 m)$ factor, where $m$ is the number of items. We also show that a logarithmic dependence on $m$ is necessary. Our approximation is achieved through the construction of a new multi-dimensional Online Contention Resolution Scheme (OCRS), that provides an online rounding of the optimal ex ante solution. Chawla et al. arXiv:2204.01962 previously constructed an OCRS for revenue for unit-demand buyers, but their construction relied heavily on the ``almost single dimensional" nature of unit-demand values. Prior to that work, OCRSes have only been studied in the context of social welfare maximization for single-parameter buyers. For the welfare objective, constant-factor approximations have been demonstrated for a wide range of combinatorial constraints on item allocations and classes of buyer valuation functions. Our work opens up the possibility of a similar success story for revenue maximization.