We study revenue maximization in a buyer-seller setting where the seller has a single object and the buyer has both a private valuation and a private budget. Private budgets complicate the classic single-product monopoly problem, making optimal mechanisms difficult to characterize. To address this, we evaluate the robust performance of approximation mechanisms relative to optimal mechanisms using three performance measures: the guaranteed fraction of optimal revenue, the maximal value of relaxation, and a revenue non-monotonicity gap. Our analysis reveals sharp contrasts. For distributions with bounded support, simple mechanisms with polylogarithmic menu size can approximate optimal revenue arbitrarily well, even when valuations and budgets are correlated. By contrast, for distributions with unbounded support, and even for bounded distributions concentrated in the unit square, no simple mechanism -- or any mechanism with a finite or sublinear menu -- can guarantee a positive fraction of optimal revenue. In particular, no finite-menu mechanism guarantees any positive fraction even under independence. We also show unbounded revenue gains from certain relaxations under negative correlation and identify cases of revenue non-monotonicity. Overall, our results show that approximation guarantees with private budgets are fragile, revealing fundamental limits to simplicity and robustness in mechanism design.

翻译：我们研究买方-卖方环境中的收益最大化问题，其中卖方拥有单一物品，买方同时拥有私人估值和私人预算。私人预算使经典单产品垄断问题复杂化，导致最优机制难以刻画。为解决此问题，我们通过三种性能度量指标评估近似机制相对于最优机制的稳健性能：最优收益的保证比例、松弛的最大化价值以及收益非单调性缺口。我们的分析揭示了鲜明对比。对于有界支撑的分布，即使估值与预算存在相关性，具有多对数规模菜单的简单机制也能任意逼近最优收益。相反，对于无界支撑的分布，甚至对于集中在单位正方形内的有界分布，任何简单机制——或任何具有有限或次线性菜单的机制——都无法保证最优收益的正比例。特别地，即使在独立性条件下，任何有限菜单机制也无法保证任何正比例收益。我们还证明了在负相关性下某些松弛机制能带来无界收益增长，并识别出收益非单调性的情形。总体而言，我们的结果表明，在私人预算下近似保证是脆弱的，揭示了机制设计中简洁性与稳健性的根本极限。