We study budget feasible procurement auctions, in which $n$ agents, each with a privately held service cost, offer their services to an employer. The employer seeks to maximize a public submodular valuation function over the set of hired agents, while facing a hard budget constraint. We consider an online posted-price setting, in which agents arrive in a uniformly random order (a.k.a. \emph{secretary arrivals}) and the employer must make irrevocable take-it-or-leave-it offers upon their arrival. The employer does not get any feedback about the agent service costs other than whether they accept the offer or not. We introduce Repeated Descent (a.k.a. \RED), a deterministic framework based on adaptive linear posted pricing. \RED enforces budget feasibility by adaptively adjusting its pricing and balancing each pricing level with the number of agents considered in it. Using \RED as the main building block, we obtain a $1046$-competitive posted-price mechanism for online budget feasible auctions with secretary agent arrivals and submodular valuations, thus improving on the previously best known ratio of (Charalampopoulos et al., EC 2025) by several orders of magnitude. Combining \RED with random subsampling, we obtain the first constant-competitive posted-price budget feasible mechanism for non-monotone submodular valuations. On the negative side, we show that every online budget feasible mechanism with XOS valuations has a competitive ratio of $Ω\!\left(\tfrac{\log n}{(\log\log n)^2}\right)$.

翻译：我们研究预算可行的采购拍卖，其中 $n$ 个代理人各自拥有私人服务成本，向雇主提供其服务。雇主希望在严格预算约束下，最大化关于雇佣代理人集合的公开次模估价函数。我们考虑在线定价设置，代理人以均匀随机顺序到达（即\textit{秘书到达}），雇主必须在代理人到达时做出不可撤销的"要么接受要么放弃"报价。雇主除了知道代理人是否接受报价外，无法获得关于其服务成本的任何反馈。我们提出重复下降（即 \RED），一种基于自适应线性定价的确定性框架。\RED 通过自适应调整定价并平衡每个定价水平与其所考虑的代理人数量来强制执行预算可行性。以 \RED 为主要构建模块，我们为具有秘书到达和次模估价的在线预算可行拍卖获得了 $1046$-竞争比定价机制，从而将 (Charalampopoulos et al., EC 2025) 中先前已知的最佳比率提升了几个数量级。将 \RED 与随机子采样相结合，我们为非线性次模估价首次得到了常数竞争比的定价预算可行机制。在消极方面，我们证明每个具有 XOS 估价的在线预算可行机制都有 $Ω\!\left(\tfrac{\log n}{(\log\log n)^2}\right)$ 的竞争比。