Pacing Equilibria in Second-Price Auctions with Few Goods

TL;DR

This paper introduces a polynomial-time algorithm for computing second-price pacing equilibria in online advertising auctions with a limited number of goods, extending to large markets with aggregated valuation types.

Contribution

It presents a novel geometric and linear programming approach to efficiently compute SPPEs in markets with few goods or aggregated valuation types.

Findings

Efficient polynomial-time algorithm for constant goods case.

Mapping bidders' pacing to bid maxima simplifies equilibrium computation.

Tractability extends to large markets with aggregated valuation types.

Abstract

In this paper, we investigate the computation of second-price pacing equilibria (SPPEs), a foundational model in online advertising auctions. We present a polynomial-time algorithm for computing exact SPPEs in instances with a constant number of goods. Our core technique maps buyers' pacing multipliers to the highest bids on each good, effectively partitioning the parameter space into a set of distinct geometric cells. By enumerating these cells, we fix the relative ordering of the bids and reduce the problem of equilibrium computation to a linear feasibility program. Finally, we demonstrate that this tractability extends to large-scale markets with an arbitrary number of goods, provided the goods can be aggregated into a constant number of valuation types.