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

Zhengyang Liu

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arXiv · 2608.16682

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

Zhengyang Liu

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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 $\gamma$-approximate equilibrium is $\mathsf{PPAD}$-hard for every constant $\gamma\in[0,1)$. For throttling, computing a $\delta$-approximate equilibrium is $\mathsf{PPAD}$-hard for every constant $\delta\in(0,1)$. 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.

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Zhengyang Liu. 2026-08-17. Tight Inapproximability of Pacing and Throttling Equilibria in Second-Price Auctions. https://arxiv.org/abs/2608.16682

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