Tight Inapproximability for Welfare-Maximizing Autobidding Equilibria

TL;DR

This work analyzes the complexity of finding welfare- and revenue-maximizing autobidding equilibria under return-on-spend constraints in parallel second-price auctions. Using a Gap Label Cover reduction with carefully designed gadgets, it proves tight inapproximability for welfare (no efficient algorithm can beat a $2-\epsilon$ factor) and substantial hardness for revenue (logarithmic, with conditional polynomial hardness via the Projection Games conjecture). The authors further refine these results in the presence of machine-learning guidance and extend the hardness to learning dynamics, including time-average RoS constraints and responsive learning sequences. The results bridge efficiency guarantees (PoA of at most $2$) with computational intractability, highlighting limits on automated mechanism design and learning dynamics in autobidding ecosystems and suggesting directions for future work in robust auction design and alternate formats.

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 $2 - ε$ is NP-hard for any constant $ε> 0$. Moreover, deciding whether there exists an autobidding equilibrium that attains a $1/2 + ε$ fraction of the optimal welfare -- unfettered by equilibrium constraints -- is NP-hard for any constant $ε> 0$. 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.

Figures (1)

• Figure 1: Overview of our reduction for autobidding equilibria.

Theorems & Definitions (79)

• Theorem 1.1
• Theorem 1.2
• Corollary 1.3: Computing non-trivial autobidding equilibria is -hard
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Theorem 1.8
• Definition 2.1: Autobidding equilibrium; Li24:Vulnerabilities
• Remark 2.2: Reserve prices