Nash Equilibria in Auctions with Pacing Strategies: Complexity and Inefficiency

2026-09-28

AI Summary

This study investigates the equilibrium properties and efficiency loss of pacing strategies in games with multiple simultaneous first-price auctions. Focusing on a model where bidders participate by uniformly scaling their valuations, we rigorously analyze the problem by integrating game-theoretic modeling, computational complexity analysis, and Price of Anarchy theory. Our primary contributions are threefold. First, we demonstrate that approximate pure Nash equilibria may not exist in this setting. Second, we prove that the equilibrium decision problem is NP-complete, although polynomial-time algorithms exist when specific parameters are fixed. Finally, we precisely characterize the system's efficiency loss, establishing that both the Price of Anarchy and the Price of Stability are exactly e/(e−1).

Abstract

We introduce and study Auctions with Pacing Strategies (APS) games, a full-information model in which utility-maximizing bidders compete across many simultaneous first-price auctions, each choosing a single pacing multiplier that uniformly scales their values into bids. We settle three central questions. First, we show that there are instances that admit no approximate pure Nash equilibria. Then, we prove that the problem of deciding whether an APS game admits an (approximate) equilibrium is NP-complete in general, but can be solved in polynomial time if either the number of bidders or the number of items is fixed. Finally, when an equilibrium does exist, we characterize its inefficiency exactly, showing that both the Price of Anarchy and the Price of Stability equal $\frac{e}{e-1}$.

Problem

• Nash Equilibria
• Pacing Strategies
• Auction Games
• Computational Complexity
• Price of Anarchy

Innovation

• Pacing Strategies
• Nash Equilibria
• First-price Auctions
• Computational Complexity
• Price of Anarchy