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

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.