Motivated by autobidding systems in online advertising, we study revenue maximization in markets with divisible goods and budget-constrained buyers with linear valuations. Our aim is to compute a single price for each good and an allocation that maximizes total revenue. We show that the First-Price Pacing Equilibrium (FPPE) guarantees at least half of the optimal revenue, even when compared to the maximal revenue of buyer-specific prices. This guarantee is particularly striking in light of our hardness result: we prove that revenue maximization under individual rationality and single-price-per-good constraints is APX-hard. We further extend our analysis in two directions: first, we introduce an online analogue of FPPE and show that it achieves a constant-factor revenue guarantee, specifically a $1/4$-approximation; second, we consider buyers with concave valuation functions, characterizing an FPPE-type outcome as the solution to an Eisenberg-Gale-style convex program and showing that the revenue approximation degrades gracefully with the degree of nonlinearity of the valuations.

翻译：受在线广告中自动竞价系统的启发，我们研究了具有可分商品、预算约束且具有线性估值的买方市场中的收益最大化问题。我们的目标是计算每种商品的单一价格以及能够最大化总收益的分配方案。我们证明，即使与买方特定价格下的最大收益相比，一价步调均衡（FPPE）也能保证至少一半的最优收益。鉴于我们的硬度结果，这一保证尤为显著：我们证明了在个体理性与每商品单一价格约束下的收益最大化问题是APX-难的。我们进一步从两个方向扩展了分析：首先，我们引入了FPPE的在线类比，并证明其实现了常数倍的收益保证，具体为$1/4$近似比；其次，我们考虑了具有凹估值函数的买方，将FPPE类型的结果刻画为Eisenberg-Gale风格凸规划的解，并证明收益近似比随估值非线性程度的增加而优雅地下降。