AI 中文总结
本文提出抽象机制中节奏均衡的统一理论，证明在出价最大化按报价支付机制下均衡存在唯一且可凸规划刻画，并推广至满足支付单调性的机制，确保计算效率、帕累托最优及收入最大化。
AI 中文摘要
数字平台日益依赖自动化预算管理系统来调节参与者在各种分配机会中的参与度。一种常见工具是乘性节奏调整，它通过缩放买家的出价，使活动级预算得以逐步消耗。尽管节奏调整在操作上简单，但其总体行为取决于底层机制。在一价单物品拍卖市场中，节奏调整具有强大的结构和计算性质，而这些性质在二价拍卖中往往失效。目前尚不清楚这些性质是否能扩展到更丰富的平台机制。我们开发了一个跨机制层级的节奏均衡统一理论。我们首先表明，即使在一价位置拍卖中，单物品基准的市场均衡解释也可能失效。然后，我们识别出出价最大化的按报价支付机制作为一个广泛类别，在该类别中，节奏均衡存在、唯一，并具有Eisenberg-Gale型凸规划刻画。这保证了高效计算、帕累托效率和液体福利保障，使得节奏调整能够集成到大规模基于优化的分配流程中。最后，对于满足支付单调性条件的抽象机制，我们通过平滑不连续性证明了均衡的存在性。在附加条件和适当的平局打破规则下，我们获得了唯一性、在预算可行节奏向量中的收入最大化性以及防托儿可实施性，并为近似均衡提供了收敛的预算调整动态。总体而言，在适当的机制级条件下，一价节奏调整的许多有利性质超越了单物品拍卖的范畴。
英文摘要
Digital platforms increasingly rely on automated budget-management systems to regulate participation across allocation opportunities. A common tool is multiplicative pacing, which scales buyers' bids so that campaign-level budgets are spent gradually. Although pacing is operationally simple, its aggregate behavior depends on the underlying mechanism. In first-price single-item auction markets, pacing enjoys strong structural and computational properties that often fail in second-price auctions. It is unclear whether these properties extend to richer platform mechanisms. We develop a unified theory of pacing equilibria across a hierarchy of mechanisms. We first show that, even in first-price position auctions, the market-equilibrium interpretation of the single-item benchmark can fail. We then identify bid-maximizing pay-your-bid mechanisms as a broad class in which pacing equilibria exist, are unique, and admit an Eisenberg-Gale-type convex-program characterization. This implies efficient computation, Pareto-efficiency, and liquid-welfare guarantees, enabling pacing integration into large-scale optimization-based allocation routines. Finally, for abstract mechanisms satisfying a payment monotonicity condition, we prove equilibrium existence by smoothing discontinuities. With additional conditions and appropriate tie-breaking, we obtain uniqueness, revenue maximality among budget-feasible pacing vectors, and shill-proof implementability, and provide convergent budget-adjustment dynamics for approximate equilibria. Overall, many favorable properties of first-price pacing extend beyond single-item auctions under appropriate mechanism-level conditions.
