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. Due to its conceptual simplicity, our approach 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为设计与分析高效拍卖机制带来了新挑战。本文聚焦于自动竞价场景下的比例机制，以流动福利为目标，研究纯纳什均衡的效率，特别是无政府状态代价（PoA）。首先，我们为标准比例机制建立了严格的2倍PoA界。其次，我们引入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界，其中$n \geq 2$表示竞价代理数量。这一改进超越了现有2倍PoA的界限，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及Karush-Kuhn-Tucker（KKT）条件。由于其概念简洁性，该方法可能在建立PoA界方面具有更广泛的应用前景。