Tight Inapproximability for Welfare-Maximizing Autobidding Equilibria

TL;DR

This paper proves that computing welfare- and revenue-maximizing equilibria in autobidding second-price auctions with RoS constraints is NP-hard to approximate within certain factors, highlighting fundamental computational limitations.

Contribution

The paper establishes tight inapproximability bounds for welfare and revenue maximization in autobidding equilibria, extending previous hardness results and analyzing learning algorithms.

Findings

NP-hardness of welfare approximation within factor 2 - ε

NP-hardness of deciding existence of better-than-worst-case equilibria

Logarithmic inapproximability for revenue, under conjectures

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.