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.

翻译：预算约束在在线广告拍卖中普遍存在。为管理这些约束并平滑跨拍卖的支出，竞拍者（或其代理平台）常采用定价策略：为每个竞拍者分配一个介于0到1之间的定价乘数，其对每个物品的报价按此乘数进行比例缩减。这自然形成了一种博弈，其中每个竞拍者策略性地选择乘数。该博弈的适当均衡概念被称为定价均衡。本文证明，对于二阶拍卖，寻找近似定价均衡的问题是PPAD-完全的。这解决了Conitzer等人[2021]提出的一个开放性问题。基于该硬性结果，我们进一步证明Borgs等人[2007]提出的基于tâtonnement过程的预算管理动态机制，在重复二阶拍卖场景中难以高效收敛。这一结论推翻了Borgs等人[2007]的猜想（前提是假设复杂度类PPAD不等于P）。此外，我们的硬性结果还表明，存在一类难以通过简单线性效用计算的供给感知市场均衡的精炼形式。