  Budget constraints are ubiquitous in online advertisement auctions. To manage
these constraints and smooth out the expenditure across auctions, the bidders
(or the platform on behalf of them) often employ pacing: each bidder is
assigned a pacing multiplier between zero and one, and her bid on each item is
multiplicatively scaled down by the pacing multiplier. This naturally gives
rise to a game in which each bidder strategically selects a multiplier. The
appropriate notion of equilibrium in this game is known as a pacing
equilibrium. In this work, we show that the problem of finding an approximate
pacing equilibrium is PPAD-complete for second-price auctions. This resolves an
open question of Conitzer et al. [2021]. As a consequence of our hardness
result, we show that the tatonnement-style budget-management dynamics
introduced by Borgs et al. [2007] are unlikely to converge efficiently for
repeated second-price auctions. This disproves a conjecture by Borgs et al.
[2007], under the assumption that the complexity class PPAD is not equal to P.
Our hardness result also implies the existence of a refinement of supply-aware
market equilibria which is hard to compute with simple linear utilities.
