# Multiplicative Pacing Equilibria in Auction Markets

arXiv: https://arxiv.org/abs/1706.07151 · HTML: https://mlsanity.com/paper/1706.07151/

## Key facts

- Cited by 16 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 22 Jun 2017.

## Built on by (most cited first, 16 of 16)

- [A Unifying Framework for Online Optimization with Long-Term Constraints](https://mlsanity.com/paper/2209.07454/) (2022)
- [Stochastic Bandits for Multi-platform Budget Optimization in Online Advertising](https://mlsanity.com/paper/2103.10246/) (2021)
- [Autobidders with Budget and ROI Constraints: Efficiency, Regret, and Pacing Dynamics](https://mlsanity.com/paper/2301.13306/) (2023)
- [Optimal Spend Rate Estimation and Pacing for Ad Campaigns with Budgets](https://mlsanity.com/paper/2202.05881/) (2022)
- [Best of Many Worlds Guarantees for Online Learning with Knapsacks](https://mlsanity.com/paper/2202.13710/) (2022)
- [Maximizing the Success Probability of Policy Allocations in Online Systems](https://mlsanity.com/paper/2312.16267/) (2023)
- [Analysis of Dual-Based PID Controllers through Convolutional Mirror Descent](https://mlsanity.com/paper/2202.06152/) (2022)
- [Multi-Platform Budget Management in Ad Markets with Non-IC Auctions](https://mlsanity.com/paper/2306.07352/) (2023)
- [Dynamic Budget Throttling in Repeated Second-Price Auctions](https://mlsanity.com/paper/2207.04690/) (2022)
- [MESOB: Balancing Equilibria & Social Optimality](https://mlsanity.com/paper/2307.07911/) (2023)
- [Delivery Optimized Discovery in Behavioral User Segmentation under Budget Constraint](https://mlsanity.com/paper/2402.03388/) (2024)
- [Large-Scale Contextual Market Equilibrium Computation through Deep Learning](https://mlsanity.com/paper/2406.15459/) (2024)
- [Large-Scale Auto-bidding with Nash Equilibrium Constraints](https://mlsanity.com/paper/2503.10304/) (2025)
- [A Lightweight MPC Bidding Framework for Brand Auction Ads](https://mlsanity.com/paper/2603.07721/) (2026)
- [Attribution Markets: A Fisher-Market Formulation for Fractional Credit Assignment Between Planned Tasks and Performed Actions](https://mlsanity.com/paper/2607.20694/) (2026)
- [When Privacy Moves ML-Mediated Decisions On Device: Information and Incentive Misalignment in Auctions](https://mlsanity.com/paper/2609.33312/) (2026)

## Abstract (arXiv)

Budgets play a significant role in real-world sequential auction markets such as those implemented by internet companies. To maximize the value provided to auction participants, spending is smoothed across auctions so budgets are used for the best opportunities. Motivated by a mechanism used in practice by several companies, this paper considers a smoothing procedure that relies on em pacing multipliers: on behalf of each buyer, the auction market applies a factor between 0 and 1 that uniformly scales the bids across all auctions. Reinterpreting this process as a game between buyers, we introduce the notion of em pacing equilibrium, and prove that they are always guaranteed to exist. We demonstrate through examples that a market can have multiple pacing equilibria with large variations in several natural objectives. We show that pacing equilibria refine another popular solution concept, competitive equilibria, and show further connections between the two solution concepts. Although we show that computing either a social-welfare-maximizing or a revenue-maximizing pacing equilibrium is NP-hard, we present a mixed-integer program (MIP) that can be used to find equilibria optimizing several relevant objectives. We use the MIP to provide evidence that: (1) equilibrium multiplicity occurs very rarely across several families of random instances, (2) static MIP solutions can be used to improve the outcomes achieved by a dynamic pacing algorithm with instances based on a real-world auction market, and (3) for the instances we study, buyers do not have an incentive to misreport bids or budgets provided there are enough participants in the auction.