The rise of automated bidding strategies in online advertising presents new challenges in designing and analyzing efficient auction mechanisms. In this paper, we focus on proportional mechanisms within the context of auto-bidding and study the efficiency of pure Nash equilibria, specifically the price of anarchy (PoA), under the liquid welfare objective. We first establish a tight PoA bound of 2 for the standard proportional mechanism. Next, we introduce a modified version with an alternative payment scheme that achieves a PoA bound of $1 + \frac{O(1)}{n-1}$ where $n \geq 2$ denotes the number of bidding agents. This improvement surpasses the existing PoA barrier of 2 and approaches full efficiency as the number of agents increases. Our methodology leverages duality and the Karush-Kuhn-Tucker (KKT) conditions from linear and convex programming. Despite its conceptual simplicity, our approach proves powerful and may offer broader applications for establishing PoA bounds.

翻译：在线广告中自动化竞价策略的兴起给设计和分析高效拍卖机制带来了新挑战。本文聚焦自动竞价场景中的比例机制，研究其在流动性福利目标下纯纳什均衡的效率，特别是无政府价格（PoA）。首先，我们为标准比例机制建立了紧致的PoA上界2。其次，我们引入了一种采用替代支付方案的改进版本，该版本实现了$1 + \frac{O(1)}{n-1}$的PoA上界，其中$n \geq 2$表示竞价代理的数量。这一改进超越了现有PoA上界2，并随着代理数量增加趋近于完全效率。我们的方法利用了线性与凸规划中的对偶性以及Karush-Kuhn-Tucker（KKT）条件。尽管概念简单，我们的方法被证明是强有力的，并可能为建立PoA上界提供更广泛的应用前景。