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. The presence of private budgets complicates the classic single-product monopoly problem, making optimal mechanisms difficult to analyze. To overcome this, we evaluate the robust performance of approximation mechanisms relative to optimal mechanisms. We work with three measures of performance: the guaranteed fraction of optimal revenue (GFOR) for restricted classes of mechanisms, the maximal value of relaxation (MVR) for relaxed classes, and a revenue non-monotonicity gap for either relaxed or restricted classes. Our analysis reveals sharp contrasts. On the positive side, we show that for distributions with bounded support, simple mechanisms with poly-logarithmic menu size can approximate optimal revenue arbitrarily well, regardless of correlation between valuations and budgets. On the negative side, we establish strong impossibility results: for distributions with unbounded support, or even bounded distributions concentrated in the unit square, no simple mechanism - or indeed any mechanism with a finite or sublinear menu - can guarantee a positive fraction of the optimal revenue. We also demonstrate unbounded revenue gains from certain relaxations when valuations and budgets are negatively correlated, and highlight cases of revenue non-monotonicity. Taken together, our results underscore the fragility of approximation approaches in the presence of private budgets: except for a narrow set of conditions, approximation mechanisms incur large revenue losses, pointing to fundamental limits of simplicity and robustness in mechanism design. Our analysis highlights that approximation results are highly sensitive to details of the design environment.

翻译：我们研究买方-卖方情境下的收益最大化问题，其中卖方拥有单一物品，买方同时具有私人估值和私人预算。私人预算的存在使得经典单一产品垄断问题复杂化，导致最优机制难以分析。为克服此困难，我们评估近似机制相对于最优机制的稳健性能。我们采用三种性能度量指标：针对受限机制类的保证最优收益比例（GFOR）、针对松弛机制类的松弛最大值（MVR），以及针对松弛或受限机制类的收益非单调性缺口。我们的分析揭示了鲜明对比。积极方面表明：对于有界支撑分布，无论估值与预算之间是否存在相关性，具有多对数级菜单规模的简单机制均可任意逼近最优收益。消极方面则建立了强不可能性结果：对于无界支撑分布，甚至集中于单位正方形内的有界分布，任何简单机制——乃至具有有限或次线性菜单的任何机制——均无法保证获得正比例的最优收益。我们还证明了当估值与预算负相关时，特定松弛机制可产生无界收益增益，并重点揭示了收益非单调性的案例。综合而言，我们的研究结果凸显了私人预算存在时近似方法的脆弱性：除狭窄的条件集合外，近似机制会导致显著收益损失，这揭示了机制设计中简洁性与稳健性的根本局限。我们的分析表明，近似结果对设计环境的具体细节具有高度敏感性。