发表机构
University of Glasgow; University of Technology, Nuremberg（格拉斯哥大学; 纽伦堡工业大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对单加性买家的多物品垄断拍卖场景，通过构建拉格朗日对偶证明与新的收益不等式，推导了三种简单确定性拍卖的更紧近似比，大幅改进了已有最优收益上界结果。
AI 中文摘要
我们研究经典的多物品垄断场景：存在单个加性买家以及$m$件异质物品，物品的估值相互独立但不一定同分布。最优的诚实拍卖可能是随机化的且结构复杂。我们分析三种简单确定性拍卖的近似比：分别出售所有物品、将所有物品作为一个大捆绑包出售，以及在两者中选择更优的方案。\n我们的技术核心是针对分别出售策略的最坏情况近似比的非线性数学规划公式，适用于估值落在网格$\{0,1/K,2/K,\dots,1\}$中的离散拍卖场景。针对两件同分布（iid）物品，我们构建了新颖的紧拉格朗日对偶证明，可对任意离散化参数$K$精确确定该比值。令$K\to\infty$，我们得到连续估值场景下的紧界$1+W(1/e)\approx1.278$，其中$W$表示Lambert-W函数，填补了Hart和Nisan[EC'12, JET 2017]工作中留下的$[1.278,1.368]$区间缺口。\n对于$m\geq2$件独立物品，另一种对偶构造给出了分别出售策略近似比的上界，该上界由物品估值的基础统计量表示。将此界与关联最优收益（REV）、分别出售收益（SREV）和大捆绑包收益（BREV）的新不等式相结合，我们推导了所有三种拍卖的改进保证。最值得注意的是，我们证明了$REV\leq 3.5 \max\{SREV,BREV\}$，优于Ma和Simchi-Levi[AISTATS'21]的5.2系数以及Babaioff、Immorlica、Lucier和Weinberg[FOCS'14, JACM 2020]的6系数。对于同分布物品，我们还证明了$REV\leq 4.4534 BREV$。
英文摘要
We study the classical multi-item monopoly setting with a single additive buyer and $m$ heterogeneous items whose values are independent but not necessarily identically distributed. Optimal truthful auctions may be randomized and complicated. We analyze the approximation ratios of three simple deterministic auctions: selling all items separately, selling them as a single grand bundle, and choosing the better of the two. Our technical cornerstone is a nonlinear mathematical programming formulation of the worst-case approximation ratio of selling separately, in discrete auctions where values lie in the grid $\{0,1/K,2/K,\dots ,1\}$. For two iid items, we construct novel tight Lagrangian dual certificates that determine this ratio exactly for any discretization parameter $K$. Taking $K\to\infty$, we obtain the tight bound $1+W(1/e)\approx 1.278$ in the continuous-valued setting, where $W$ denotes the Lambert-W function, closing the $[1.278,1.368]$ gap from the work of Hart and Nisan [EC'12, JET 2017]. For $m\geq2$ independent items, a different dual construction gives an upper bound on the approximation ratio of selling separately in terms of basic statistics of the item values. Combining this bound with new inequalities relating optimal revenue (REV), separate-selling revenue (SREV), and grand-bundle revenue (BREV), we derive improved guarantees for all three auctions. Most notably, we prove \[REV\leq 3 \max\{SREV,BREV\},\] improving upon the long-standing $5.2$ factor of Ma and Simchi-Levi [arXiv 2015, AISTATS'21] and the $6$ factor of Babaioff, Immorlica, Lucier and Weinberg [FOCS'14, JACM 2020]. For iid items, we also prove $REV\leq 4.18 BREV$.
CommentsWe improved the approximation ratio of REV/max{SREV,BREV} to 3. For iid items we improved the approximation ratio of REV/BREV to 4.18
