Tight Inapproximability of Pacing and Throttling Equilibria in Second-Price Auctions

2026-08-17

AI Summary

This study addresses the computational challenge of equilibrium computation for pacing and throttling under budget constraints in second-price auctions. By integrating PPAD complexity theory with reduction constructions, this work performs an approximate analysis demonstrating that both mechanisms share an identical PPAD-hard approximation threshold. This finding establishes a tight boundary proving that no non-trivial constant-factor approximation can circumvent the inherent computational barrier. Consequently, the research reveals the fundamental irremovability of fixed-point obstacles under non-trivial parameters, providing a unified theoretical framework for understanding the computational complexity of budget control mechanisms and definitively characterizing the limits of approximate equilibrium computation.

Abstract

Budget-constrained advertisers commonly rely on two control mechanisms: pacing scales bids, whereas throttling randomizes participation. We prove that, in second-price auctions, these two different mechanisms share the same sharp approximation-hardness threshold. For pacing, computing a $γ$-approximate equilibrium is $\mathsf{PPAD}$-hard for every constant $γ\in[0,1)$. For throttling, computing a $δ$-approximate equilibrium is $\mathsf{PPAD}$-hard for every constant $δ\in(0,1)$. At parameter $1$, the complementarity requirement becomes vacuous and the all-zero solution is feasible. That is, approximation does not eliminate the fixed-point barrier at any nontrivial parameter value.

Problem

• Second-Price Auctions
• Pacing Equilibrium
• Throttling Equilibrium
• Inapproximability
• PPAD-hard

Innovation

• Second-Price Auctions
• PPAD-hardness
• Pacing Equilibrium
• Throttling Equilibrium
• Approximation Hardness