Optimal Type-Dependent Liquid Welfare Guarantees for Autobidding Agents with Budgets

Online advertising systems have recently transitioned to autobidding, allowing advertisers to delegate bidding decisions to automated agents. Each advertiser directs their agent to optimize an objective function subject to return-on-investment (ROI) and budget constraints. Given their practical relevance, this shift has spurred a surge of research on the liquid welfare price of anarchy (POA) of fundamental auction formats under autobidding, most notably simultaneous first-price auctions (FPA). One of the main challenges is to understand the efficiency of FPA in the presence of heterogeneous agent types. We introduce a type-dependent smoothness framework that enables a unified analysis of the POA in such complex autobidding environments. In our approach, we derive type-dependent smoothness parameters which we carefully balance to obtain POA bounds. This balancing gives rise to a POA-revealing mathematical program, which we use to determine tight bounds on the POA of coarse correlated equilibria (CCE). Our framework is versatile enough to handle heterogeneous agent types and extends to the general class of fractionally subadditive valuations. Additionally, we develop a novel reduction technique that transforms budget-constrained agents into budget-unconstrained ones. Combining this reduction technique with our smoothness framework enables us to derive tight bounds on the POA of CCE in the general hybrid agent model with both ROI and budget constraints. Among other results, our bounds uncover an intriguing threshold phenomenon showing that the POA depends intricately on the smallest and largest agent types. We also extend our study to FPAs with reserve prices, which can be interpreted as predictions of agents’ values, to further improve efficiency guarantees.