Efficiency of Proportional Mechanisms in Online Auto-Bidding Advertising

TL;DR

This paper analyzes the efficiency of proportional mechanisms in online auto-bidding advertising, establishing tight bounds on the price of anarchy and proposing improvements that approach full efficiency as the number of agents grows.

Contribution

It introduces a modified proportional mechanism with a new payment scheme that achieves a better PoA bound, surpassing the previous barrier of 2.

Findings

Tight PoA bound of 2 for standard proportional mechanism.

Modified mechanism achieves PoA approaching 1 as agents increase.

Methodology uses duality and KKT conditions for analysis.

Abstract

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.