Scaling buyer bids achieves fair and efficient budget pacing
Pacing Equilibria in Abstract Mechanisms
Computer Science and Game Theory
Summary
Online platforms often need to manage how buyers spend their budgets across many opportunities, using tools like scaling bids to spread budget use evenly. This paper shows that the success and fairness of this bid scaling depend on the auction rules used. The authors find that for a broad class of auction systems, bid scaling works well, creating unique, stable outcomes that can be computed efficiently. Their work helps extend practical budget management methods beyond simple auctions to more complex platform settings.
What this means in practice
- •For online advertising teams: Improve automated budget pacing across diverse auction formats to maintain efficient spend and bidder fairness.
- •For marketplace platform engineers: Incorporate pacing equilibria with guarantees into complex allocation algorithms supporting multi-slot or position auctions.
A theory result. No direct application yet.
Authors
Salam Afiouni, Christian Kroer
Abstract
Digital platforms increasingly rely on automated budget-management systems to regulate participation across allocation opportunities. A common tool is multiplicative pacing, which scales buyers' bids so that campaign-level budgets are spent gradually. Although pacing is operationally simple, its aggregate behavior depends on the underlying mechanism. In first-price single-item auction markets, pacing enjoys strong structural and computational properties that often fail in second-price auctions. It is unclear whether these properties extend to richer platform mechanisms. We develop a unified theory of pacing equilibria across a hierarchy of mechanisms. We first show that, even in first-price position auctions, the market-equilibrium interpretation of the single-item benchmark can fail. We then identify bid-maximizing pay-your-bid mechanisms as a broad class in which pacing equilibria exist, are unique, and admit an Eisenberg-Gale-type convex-program characterization. This implies efficient computation, Pareto-efficiency, and liquid-welfare guarantees, enabling pacing integration into large-scale optimization-based allocation routines. Finally, for abstract mechanisms satisfying a payment monotonicity condition, we prove equilibrium existence by smoothing discontinuities. With additional conditions and appropriate tie-breaking, we obtain uniqueness, revenue maximality among budget-feasible pacing vectors, and shill-proof implementability, and provide convergent budget-adjustment dynamics for approximate equilibria. Overall, many favorable properties of first-price pacing extend beyond single-item auctions under appropriate mechanism-level conditions.