[Paper Review] Efficiency of non-truthful auctions under auto-bidding

Christopher Liaw, Aranyak Mehta+1 more|arXiv (Cornell University)|Jul 8, 2022

Auction Theory and Applications4 citations

TL;DR

This paper investigates the efficiency of non-truthful auctions in auto-bidding settings, where advertisers use algorithms to optimize for goals like target return on spend. It shows that while deterministic non-truthful auctions cannot improve upon the price of anarchy (PoA) of 2, a novel randomized non-truthful auction achieves a PoA of 1.8—better than any known truthful mechanism—demonstrating that both randomization and non-truthfulness are essential for improved efficiency in auto-bidding systems.

ABSTRACT

Auto-bidding is now widely adopted as an interface between advertisers and internet advertising as it allows advertisers to specify high-level goals, such as maximizing value subject to a value-per-spend constraint. Prior research has mostly focused on auctions which are truthful (such as SPA) since uniform bidding is optimal in such auctions, which makes it manageable to reason about equilibria. A tantalizing question is whether one can obtain more efficient outcomes by leaving the realm of truthful auctions. This is the first paper to study non-truthful auctions in the prior-free auto-bidding setting. Our first result is that non-truthfulness provides no benefit when one considers deterministic auctions. Any deterministic mechanism has a price of anarchy (PoA) of at least $2$, even for $2$ bidders; this matches what can be achieved by deterministic truthful mechanisms. In particular, we prove that the first price auction has PoA of exactly $2$. For our second result, we construct a randomized non-truthful auction that achieves a PoA of $1.8$ for $2$ bidders. This is the best-known PoA for this problem. The previously best-known PoA for this problem was $1.9$ and was achieved with a truthful mechanism. Moreover, we demonstrate the benefit of non-truthfulness in this setting by showing that the truthful version of this randomized auction also has a PoA of $1.9$. Finally, we show that no auction (even randomized, non-truthful) can improve upon a PoA bound of $2$ as the number of advertisers grow to infinity.

Motivation & Objective

• To understand whether non-truthful auctions can achieve better welfare efficiency than truthful auctions in auto-bidding settings.
• To analyze the price of anarchy (PoA) in deterministic and randomized non-truthful auctions under prior-free conditions.
• To evaluate whether combining randomization and non-truthfulness can yield better efficiency than truthful mechanisms.
• To establish fundamental limits on efficiency improvements as the number of bidders grows.

Proposed method

• Proposes a randomized first price auction (rFPA) where the winner is determined based on bid ratios relative to a threshold α, with full payment by the winner.
• Uses a threshold-based allocation rule: if the higher bid exceeds the lower by factor α, the higher bid wins outright; otherwise, allocation is randomized based on bid ratio.
• Analyzes equilibrium behavior using convex optimization to compute best responses, especially for non-uniform bidding in rFPA.
• Compares rFPA with a truthful variant (rTruth) that uses the same allocation rule but a different pricing function, to isolate the effect of non-truthfulness.
• Employs theoretical analysis and simulations to evaluate PoA across multiple synthetic bidder value distributions and query configurations.
• Derives analytical bounds on PoA using logarithmic functions and optimization over bid ratios to prove worst-case efficiency guarantees.

Experimental results

Research questions

• RQ1 Can non-truthful auctions achieve better price of anarchy (PoA) than truthful auctions in auto-bidding?
• RQ2 Does randomization improve welfare efficiency in non-truthful auctions when bidders use auto-bidding agents?
• RQ3 What is the fundamental limit on PoA improvement in non-truthful, randomized auctions as the number of bidders increases?
• RQ4 How does the combination of non-truthfulness and randomization affect equilibrium outcomes compared to truthful mechanisms?

Key findings

• For any deterministic non-truthful auction, the price of anarchy (PoA) is at least 2, even with only two bidders, matching the worst-case PoA of truthful mechanisms.
• The first price auction (FPA) has a PoA of exactly 2, showing that non-uniform bidding does not improve efficiency in deterministic settings.
• A novel randomized non-truthful auction, rFPA, achieves a PoA of at most 1.8, which is the best-known PoA for this problem setting.
• The truthful version of rFPA (rTruth) has a PoA of at least 1.98, demonstrating that non-truthfulness is essential for the improved efficiency.
• As the number of bidders grows to infinity, no auction—whether randomized or non-truthful—can achieve a PoA better than 2, establishing a fundamental limit.
• Simulations confirm that rFPA outperforms both SPA and rTruth in terms of PoA across diverse synthetic bidder value distributions, especially in asymmetric settings.