In today's online advertising markets, a crucial requirement for an advertiser is to control her total expenditure within a time horizon under some budget. Among various budget control methods, throttling has emerged as a popular choice, managing an advertiser's total expenditure by selecting only a subset of auctions to participate in. This paper provides a theoretical panorama of a single advertiser's dynamic budget throttling process in repeated second-price auctions. We first establish a lower bound on the regret and an upper bound on the asymptotic competitive ratio for any throttling algorithm, respectively, when the advertiser's values are stochastic and adversarial. Regarding the algorithmic side, we propose the OGD-CB algorithm, which guarantees a near-optimal expected regret with stochastic values. On the other hand, when values are adversarial, we prove that this algorithm also reaches the upper bound on the asymptotic competitive ratio. We further compare throttling with pacing, another widely adopted budget control method, in repeated second-price auctions. In the stochastic case, we demonstrate that pacing is generally superior to throttling for the advertiser, supporting the well-known result that pacing is asymptotically optimal in this scenario. However, in the adversarial case, we give an exciting result indicating that throttling is also an asymptotically optimal dynamic bidding strategy. Our results bridge the gaps in theoretical research of throttling in repeated auctions and comprehensively reveal the ability of this popular budget-smoothing strategy.

翻译：在当今在线广告市场中，广告主的一个关键需求是在时间范围内将总支出控制在预算限额内。在各种预算控制方法中，节流机制已成为主流选择，它通过仅选择部分拍卖参与来管理广告主的总支出。本文为单一广告主在重复第二价格拍卖中的动态预算节流过程提供了理论全景分析。我们首先分别针对广告主价值随机和对抗性场景，建立了任何节流算法遗憾值下界与渐近竞争比上界。在算法层面，我们提出OGD-CB算法，该算法在价值随机情况下能保证近最优期望遗憾值。另一方面，当价值具有对抗性时，我们证明该算法同样能达到渐近竞争比的上界。我们进一步比较了节流与另一种广泛使用的预算控制方法——节奏控制——在重复第二价格拍卖中的表现。在随机情形下，我们证明节奏控制通常优于节流，这印证了该场景下节奏控制渐近最优的已知结论。然而在对抗性情形下，我们得出一个令人振奋的结果：节流同样是一种渐近最优的动态竞价策略。我们的研究填补了重复拍卖中节流机制理论研究的空白，并全面揭示了这种流行预算平滑策略的能力。