Tight Inapproximability for Welfare-Maximizing Autobidding Equilibria

2026-02-09

AI Summary

This work investigates the computational complexity of computing equilibria that maximize welfare and revenue in second-price auctions with automated bidding under return-on-investment constraints. Leveraging a reduction based on the Projection Games Conjecture, it establishes the first tight inapproximability bounds: achieving a welfare approximation factor better than \(2 - \varepsilon\) is NP-hard, and revenue exhibits logarithmic inapproximability. These hardness results persist even when incorporating value predictions or restricting to simple learning algorithms, preserving constant-factor inapproximability. The findings imply that determining whether any nontrivial equilibrium exists that improves upon worst-case guarantees is NP-hard. Moreover, the study tightly connects these inapproximability results to the price of anarchy, thereby characterizing the fundamental theoretical limits of computing such equilibria.

Technology Category

• Multiagent Systems: Mechanism Design
• Game Theory and Economic Paradigms: Mechanism Design
• Search and Optimization: Learning to Search

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

We examine the complexity of computing welfare- and revenue-maximizing equilibria in autobidding second-price auctions subject to return-on-spend (RoS) constraints. We show that computing an autobidding equilibrium that approximates the welfare-optimal one within a factor of $2 - \epsilon$ is NP-hard for any constant $\epsilon>0$. Moreover, deciding whether there exists an autobidding equilibrium that attains a $1/2 + \epsilon$ fraction of the optimal welfare -- unfettered by equilibrium constraints -- is NP-hard for any constant $\epsilon>0$. This hardness result is tight in view of the fact that the price of anarchy (PoA) is at most $2$, and shows that deciding whether a non-trivial autobidding equilibrium exists -- one that is even marginally better than the worst-case guarantee -- is intractable. For revenue, we establish a stronger logarithmic inapproximability, while under the projection games conjecture, our reduction rules out even a polynomial approximation factor. These results significantly strengthen the APX-hardness of Li and Tang (AAAI'24). Furthermore, we refine our reduction in the presence of ML advice concerning the buyers'valuations, revealing again a close connection between the inapproximability threshold and PoA bounds. Finally, we examine relaxed notions of equilibrium attained by simple learning algorithms, establishing constant inapproximability for both revenue and welfare.

Problem

• autobidding
• welfare maximization
• revenue maximization
• inapproximability
• second-price auctions

Innovation

• autobidding
• inapproximability
• price of anarchy
• second-price auctions
• return-on-spend constraints