Pacing Equilibria in Second-Price Auctions with Few Goods

2026-05-10

AI Summary

This work addresses the challenge of precisely computing budget-smoothed pacing equilibria (SPPE) under second-price auctions in online advertising when the number of items is small. To overcome this, the authors propose a novel approach based on geometric partitioning of the parameter space: by mapping each buyer’s pacing multiplier to their highest bids across items, the space is divided into a finite set of geometric cells, within each of which equilibrium computation reduces to a linear feasibility problem. This method yields the first polynomial-time algorithm for exactly solving SPPE with a constant number of items and extends to large-scale markets featuring a constant number of valuation types. Empirical evaluations demonstrate the algorithm’s efficiency, scalability, and practical applicability in real-world ad auction settings.

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.

Problem

• second-price auctions
• pacing equilibria
• online advertising
• equilibrium computation
• valuation types

Innovation

• second-price pacing equilibria
• polynomial-time algorithm
• geometric cell enumeration
• linear feasibility program
• valuation type aggregation