[Paper Review] Efficiency of the First-Price Auction in the Autobidding World

Yuan Deng, Jieming Mao+3 more|arXiv (Cornell University)|Aug 22, 2022

Auction Theory and Applications4 citations

TL;DR

This paper analyzes the efficiency of first-price auctions in the autobidding world, where bidders are either utility maximizers or value maximizers (autobidders). It shows the price of anarchy is 1/2 under full autobidding and degrades to approximately 0.457 in mixed environments with both bidder types, with the bound derived from an optimization. When machine-learned advice improves, the price of anarchy approaches 1, enhancing auction efficiency.

ABSTRACT

We study the price of anarchy of the first-price auction in the autobidding world, where bidders can be either utility maximizers (i.e., traditional bidders) or value maximizers (i.e., autobidders). We show that with autobidders only, the price of anarchy of the first-price auction is $1/2$, and with both kinds of bidders, the price of anarchy degrades to about $0.457$ (the precise number is given by an optimization). These results complement the recent result by Jin and Lu [2022] showing that the price of anarchy of the first-price auction with traditional bidders only is $1 - 1/e^2$. We further investigate a setting where the seller can utilize machine-learned advice to improve the efficiency of the auctions. There, we show that as the accuracy of the advice increases, the price of anarchy improves smoothly from about $0.457$ to $1$.

Motivation & Objective

• To analyze the efficiency of first-price auctions when bidders use automated bidding strategies (autobidding), particularly value maximizers.
• To quantify the worst-case efficiency loss—measured by the price of anarchy (PoA)—in mixed environments with both utility maximizers and value maximizers.
• To evaluate the impact of machine-learned advice on improving auction efficiency in first-price auctions under autobidding.
• To extend prior results on PoA in first-price auctions by characterizing performance in the emerging autobidding ecosystem.

Proposed method

• Models bidders as either utility maximizers (traditional) or value maximizers (autobidders) who optimize for value subject to ROI constraints.
• Analyzes equilibrium outcomes in first-price auctions using game-theoretic tools, focusing on welfare loss relative to social optimum.
• Derives the PoA as the solution to a constrained optimization problem: min_{t∈[0,1]} (1 + t ln t)/(2 - t + t ln t), which evaluates to approximately 0.457.
• Introduces a framework where the seller uses machine-learned advice to set reserves, and proves that PoA improves smoothly to 1 as advice accuracy increases.
• Uses duality and equilibrium condition analysis to show that optimal strategies require equalization of bid incentives across bidder types.
• Employs variational and calculus-based techniques to prove tightness of the PoA bound by constructing matching worst-case instances.

Experimental results

Research questions

• RQ1 What is the price of anarchy in a first-price auction when all bidders are value maximizers (autobidders)?
• RQ2 How does the price of anarchy change when both utility maximizers and value maximizers coexist in a first-price auction?
• RQ3 Can machine-learned advice on bidder values improve the efficiency of first-price auctions in the autobidding setting?
• RQ4 Is the price of anarchy in first-price auctions with mixed bidders strictly worse than in second-price auctions under the same conditions?
• RQ5 How does the PoA behave asymptotically as the accuracy of machine-learned advice approaches perfect prediction?

Key findings

• In a fully autobidding environment, the price of anarchy (PoA) of the first-price auction is exactly 1/2.
• In mixed environments with both utility maximizers and value maximizers, the PoA degrades to approximately 0.457, derived from the optimization min_{t∈[0,1]} (1 + t ln t)/(2 - t + t ln t).
• This bound is tight, as the paper constructs a worst-case instance achieving this PoA.
• When machine-learned advice is used to set reserves, the PoA of the first-price auction approaches 1 as the advice accuracy increases.
• The PoA in the first-price auction with mixed bidders is strictly less than that of the second-price auction (which remains 1/2 under the same conditions).
• The improvement via advice is smooth and continuous, demonstrating that better predictions lead to better efficiency without abrupt thresholds.