The transition to auto-bidding in online advertising has shifted the focus of auction theory from quasi-linear utility maximization to value maximization subject to financial constraints. We study mechanism design for buyers with private budgets and private Return-on-Spend (RoS) constraints, but public valuations, a setting motivated by modern advertising platforms where valuations are predicted via machine learning models. We introduce the extended Eisenberg-Gale program, a convex optimization framework generalized to incorporate RoS constraints. We demonstrate that the solution to this program is unique and characterizes the market's competitive equilibrium. Based on this theoretical analysis, we design a market-clearing mechanism and prove two key properties: (1) it is incentive-compatible with respect to financial constraints, making truthful reporting the optimal strategy; and (2) it achieves a tight 1/2-approximation of the first-best revenue benchmark, the maximum revenue of any feasible mechanism, regardless of IC. Finally, to enable practical implementation, we present a decentralized online algorithm. Ignoring logarithmic factors, we prove that under this algorithm, both the seller's revenue and each buyer's utility converge to the equilibrium benchmarks with a sublinear regret of $\tilde{O}(\sqrt{m})$ over $m$ auctions.

翻译：在线广告向自动竞价模式的转变，已将拍卖理论的研究重心从拟线性效用最大化转向受财务约束的价值最大化。我们研究了买方具有私有预算和私有投资回报率约束、但估值公开的机制设计问题，这一设定源于现代广告平台中估值通常通过机器学习模型预测的现实背景。我们提出了扩展的艾森伯格-盖尔规划，这是一个推广至包含投资回报率约束的凸优化框架。我们证明该规划的解具有唯一性，并刻画了市场的竞争均衡。基于此理论分析，我们设计了一种市场出清机制，并证明了两个关键性质：(1) 该机制在财务约束方面具有激励相容性，使得如实报告成为最优策略；(2) 该机制能够紧密逼近最优收益基准的1/2，即任何可行机制（无论是否满足激励相容）所能实现的最大收益。最后，为实现实际应用，我们提出了一种去中心化的在线算法。忽略对数因子，我们证明在该算法下，卖方的收益和每个买方的效用均以$\tilde{O}(\sqrt{m})$的次线性遗憾度收敛于均衡基准，其中$m$为拍卖次数。