Problem
Research questions and friction points this paper is trying to address.
Auto-bidding
Price of Anarchy
Proportional first-price auctions
Mechanism design
Online advertising
Efficiency of Generalized Proportional First-Price Auctions Under Auto-bidding
🤖 AI Summary
This study addresses the efficiency loss inherent in auction mechanisms for automated bidding, where the Price of Anarchy (PoA) for deterministic mechanisms with multiple buyers has long been constrained to a bound of 2. Building upon the generalized proportional first-price auction, this work integrates algorithmic mechanism design with game-theoretic equilibrium analysis to model and optimize worst-case efficiency across varying bidder scales without prior assumptions. The primary contribution is the first proof establishing a tight optimal PoA bound of 1.5 for the two-buyer setting. Furthermore, it overcomes the longstanding theoretical barrier preventing PoA < 2 for n ≥ 3 buyers by demonstrating that parameterizing r = 2n reduces the PoA to 2 − O(1/n), asymptotically approaching the theoretical lower bound.
📝 Abstract
Auto-bidding is now widely adopted in online advertising platforms, allowing advertisers to specify high-level campaign objectives--such as maximizing total value subject to a return-on-spend (ROS) constraint--rather than manual per-query bids. A central question in algorithmic mechanism design is characterizing the worst-case efficiency loss, or Price of Anarchy (PoA), across auction formats in this prior-free setting. While randomized auctions are known to strictly improve efficiency over deterministic mechanisms for two bidders, two fundamental questions have remained open: (1) what is the optimal PoA for two bidders, and (2) can any mechanism beat the barrier of 2 for general $n \ge 3$ bidders?
We resolve both questions using the family of $r$-proportional first-price auctions ($\text{pFPA}_r$), in which each bidder wins with probability proportional to their bid raised to an exponent $r > 0$ and pays their bid upon winning. First, for two bidders, we prove that the standard proportional first-price auction ($r = 1$) achieves a tight $\text{PoA} \le 1.5$, complemented by a matching lower bound showing that no anonymous, monotone mechanism can do better. Second, for general $n \ge 2$ bidders, setting $r = 2n$ achieves $\text{PoA} \le 2 - \frac{1}{4n+1} = 2 - Ω(1/n)$ across all undominated bid profiles, breaking the deterministic barrier of 2 for every finite $n$ and asymptotically matching the known $2 - Θ(1/n)$ lower bound.
Innovation
Methods, ideas, or system contributions that make the work stand out.
Auto-bidding
Proportional First-Price Auction
Price of Anarchy
Algorithmic Mechanism Design
Randomized Auctions
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