The Complexity of Pacing for Second-Price Auctions

TL;DR

This paper proves that computing approximate pacing equilibria in second-price auctions is PPAD-complete, highlighting the computational complexity and implications for budget management dynamics in online ad auctions.

Contribution

It establishes the PPAD-completeness of finding approximate pacing equilibria in second-price auctions, resolving an open problem and challenging existing dynamic convergence assumptions.

Findings

PPAD-completeness of approximate pacing equilibrium computation

Tatonnement dynamics unlikely to converge efficiently in second-price auctions

Existence of complex supply-aware market equilibria with linear utilities

Abstract

Budget constraints are ubiquitous in online advertisement auctions. To manage these constraints and smooth out the expenditure across auctions, the bidders (or the platform on behalf of them) often employ pacing: each bidder is assigned a pacing multiplier between zero and one, and her bid on each item is multiplicatively scaled down by the pacing multiplier. This naturally gives rise to a game in which each bidder strategically selects a multiplier. The appropriate notion of equilibrium in this game is known as a pacing equilibrium. In this work, we show that the problem of finding an approximate pacing equilibrium is PPAD-complete for second-price auctions. This resolves an open question of Conitzer et al. [2021]. As a consequence of our hardness result, we show that the tatonnement-style budget-management dynamics introduced by Borgs et al. [2007] are unlikely to converge efficiently for repeated second-price auctions. This disproves a conjecture by Borgs et al. [2007], under the assumption that the complexity class PPAD is not equal to P. Our hardness result also implies the existence of a refinement of supply-aware market equilibria which is hard to compute with simple linear utilities.