We study the aggregate welfare and individual regret guarantees of dynamic \emph{pacing algorithms} in the context of repeated auctions with budgets. Such algorithms are commonly used as bidding agents in Internet advertising platforms, adaptively learning to shade bids by a tunable linear multiplier in order to match a specified budget. We show that when agents simultaneously apply a natural form of gradient-based pacing, the liquid welfare obtained over the course of the learning dynamics is at least half the optimal expected liquid welfare obtainable by any allocation rule. Crucially, this result holds \emph{without requiring convergence of the dynamics}, allowing us to circumvent known complexity-theoretic obstacles of finding equilibria. This result is also robust to the correlation structure between agent valuations and holds for any \emph{core auction}, a broad class of auctions that includes first-price, second-price, and generalized second-price auctions as special cases. For individual guarantees, we further show such pacing algorithms enjoy \emph{dynamic regret} bounds for individual value maximization, with respect to the sequence of budget-pacing bids, for any auction satisfying a monotone bang-for-buck property. To complement our theoretical findings, we provide semi-synthetic numerical simulations based on auction data from the Bing Advertising platform.

翻译：本研究探讨了在预算约束的重复拍卖场景下，动态预算调节算法的总体福利与个体遗憾保证。此类算法通常作为互联网广告平台中的竞价代理，通过可调线性乘数自适应地学习调整出价（即"出价衰减"），以达到指定预算目标。我们证明，当所有参与者同时采用一种基于梯度的自然调节策略时，学习动态过程中获得的流动性福利至少能达到任何分配规则所能获得的最优预期流动性福利的一半。关键在于，该结论成立无需动态过程收敛，从而能够规避寻找均衡点时已知的计算复杂性理论障碍。此结果对参与者估值间的相关性结构具有鲁棒性，且适用于任何核心拍卖——这是一类广泛的拍卖形式，包含第一价格拍卖、第二价格拍卖和广义第二价格拍卖等特例。在个体保证方面，我们进一步证明此类调节算法在满足单调性价比特性的任何拍卖中，针对预算调节出价序列具有个体价值最大化的动态遗憾界。为补充理论发现，我们基于必应广告平台的拍卖数据进行了半合成数值模拟。