Smoothness Meets Autobidding: Tight Price of Anarchy Bounds for Simultaneous First-Price Auctions (2506.20908v1)

Abstract: Online advertising systems have recently transitioned to autobidding, enabling advertisers to delegate bidding decisions to automated agents. Each advertiser directs their agent to optimize a valuation-dependent objective subject to return-on-investment (ROI) or budget constraints. Given their relevance, there has been a surge in literature studying the liquid welfare price of anarchy (POA) of core auction formats in autobidding, among which simultaneous first-price auctions (FPA). These models capture a large range of heterogeneous agent behaviors, requiring advanced proofs to derive tight POA bounds. Recently, Deng et al. (NeurIPS 2024) showed that the POA of FPA for mixed autobidders (i.e., value and utility maximizers) under ROI is 2.18 for additive valuations. We extend the smoothness framework of Syrgkanis and Tardos (STOC 2013) to autobidding. A key contribution is a technique to balance smoothness parameters across heterogeneous agent types. Finding the best POA bound reduces to solving a POA-revealing mathematical program. Our approach has three strengths: (1) Simplicity: We prove smoothness for single-item FPA. Results for simultaneous FPA follow via our theorem. For example, by showing smoothness for value and utility maximizers, we obtain the tight POA of 2.18 for mixed autobidding. (2) Extendibility: Our Extension Theorem adapts to simultaneous FPA with reserve prices and agents with fractionally subadditive valuations and heterogeneous payment sensitivities and target ROI parameters. We establish the first (mostly) tight POA bounds for several models beyond the autobidding state of the art. (3) Generality: Our framework bounds the POA of coarse correlated equilibria (CCE), which arise when hybrid agents employ regret-minimizing algorithms. Building on Kolumbus and Nisan (WWW 2022), we show that CCE from such agents have properties that keep their POA low.