A recent line of research has established a novel desideratum for designing approximately-revenue-optimal multi-item mechanisms, namely the buy-many constraint. Under this constraint, prices for different allocations made by the mechanism must be subadditive, implying that the price of a bundle cannot exceed the sum of prices of individual items it contains. This natural constraint has enabled several positive results in multi-item mechanism design bypassing well-established impossibility results. Our work addresses the main open question from this literature of extending the buy-many constraint to multiple buyer settings and developing an approximation. We propose a new revenue benchmark for multi-buyer mechanisms via an ex-ante relaxation that captures several different ways of extending the buy-many constraint to the multi-buyer setting. Our main result is that a simple sequential item pricing mechanism with buyer-specific prices can achieve an $O(\log m)$ approximation to this revenue benchmark when all buyers have unit-demand or additive preferences over m items. This is the best possible as it directly matches the previous results for the single-buyer setting where no simple mechanism can obtain a better approximation. From a technical viewpoint we make two novel contributions. First, we develop a supply-constrained version of buy-many approximation for a single buyer. Second, we develop a multi-dimensional online contention resolution scheme for unit-demand buyers that may be of independent interest in mechanism design.

翻译：近期一系列研究确立了设计近似收益最优的多物品机制的新准则，即"买多"约束。在该约束下，机制对不同配置的定价必须满足次可加性，意味着捆绑商品的价格不能超过其包含的单件商品价格之和。这一自然约束使得多物品机制设计领域能够绕过既有的不可能性结论，取得若干积极成果。本研究解决了该文献中的核心开放问题：将"买多"约束扩展至多买家场景并构建近似机制。我们通过事前松弛方法，为多买家机制提出了新的收益基准，该基准囊括了将"买多"约束扩展至多买家场景的多种不同方式。主要结论是：当所有买家对m件商品具有单位需求或可加偏好时，采用买家特定定价的简单序贯商品定价机制，能够实现该收益基准的$O(\log m)$近似比。这一结果已达到最优，因为它直接匹配了单买家场景中无简单机制可获得更优近似比的已知结论。在技术层面，我们做出两项创新：第一，针对单一买家建立了约束供给条件下的"买多"近似框架；第二，针对单位需求买家开发了多维度在线竞争解决方案，该方案在机制设计中具有独立研究价值。