发表机构
Adrasteia Labs and UIUC; Zhongnan University of Economics and Law; Tsinghua University; University of Michigan, Ann Arbor; Boston College; University of Wisconsin–Madison（阿德拉西亚实验室和伊利诺伊大学厄巴纳-香槟分校; 中南财经政法大学; 清华大学; 密歇根大学安娜堡分校; 波士顿学院; 威斯康星大学麦迪逊分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文精确确定了固定价格双边交易的最坏福利保证，证明最优标价保证第一最佳福利的0.738024比例，并给出最优性证书及多单位交易更难的结论。
AI 中文摘要
一个具有独立私有价值的卖方和买方只能以标价进行交易。我们精确确定了该机制的最坏情况：最优标价总能保证第一最佳福利的 $\beta_*=0.738024...$ 比例，其中 $\beta_*$ 是一个显式方程的根，最坏情况下的买方在缩放意义下是唯一的，且不存在一对分布达到最坏情况。这填补了从 SODA 2016 到两篇 STOC 2023 论文及 AAAI 2026 所留下的区间 $[0.7292,0.73805]$。证明是一个显式的最优性证书：经过一次变量替换后，最优值与任何买方分布之间的差距是一系列非负积分的和，如同线性规划对偶，等式标识出最坏情况的形状。该证书还给出了交易收益与卖方初始福利之间的完整权衡：当第一最佳交易收益是初始卖方福利的 $\kappa$ 比例时，最优价格获得的收益的精确最坏比例 $\rho(\kappa)$ 满足 $\rho(\kappa)\sim2/\log(1/\kappa)$ 当 $\kappa\to0$。最坏情况的买方有两个常数段，由一条显式的非指数曲线连接，通过一个值趋于无穷的消失原子逼近。同一常数也是占优策略机制在个体理性和强预算平衡下每一实现中的确切保证。对于具有递增次模估值的两个单位，一个显式的有限实例的比率低于 $0.7290804$，因此多单位交易严格难于单单位交易。对于任何有界有序买方对，我们刻画了保证针对每个有序卖方对达到规定比率所需的最小共同价格质量；对这一单侧泛函的一个界将确定精确的两单位常数。在一个显式族内，唯一的最小值是 $0.729080...$，推测为最优。
英文摘要
A seller and a buyer with independent private values can trade only at a posted price. We determine the worst case of this mechanism exactly: the best posted price always guarantees a $β_*=0.738024\ldots$ fraction of first-best welfare, where $β_*$ is given in closed form by the root of an explicit equation, the worst-case buyer is unique up to scaling, and the worst case is never attained. This closes the gap $[0.7292,0.73805]$ left by work from SODA 2016 through two STOC 2023 papers and AAAI 2026. The proof is an explicit certificate of optimality: after one change of variables, the gap between the optimum and the value of any buyer is a sum of nonnegative integrals, as in a linear-programming dual, and equality identifies the worst-case shape. The certificate also gives the complete tradeoff between gains from trade and the seller's initial welfare: when first-best gains from trade are a fraction $κ$ of initial seller welfare, the exact worst-case fraction $ρ(κ)$ of gains obtained by the best price satisfies $ρ(κ)\sim2/\log(1/κ)$ as $κ\to0$. The worst-case buyer's survival function has two constant segments joined by an explicit nonexponential curve, and the worst case is approached through a vanishing atom escaping to infinity. The same constant is the exact guarantee of dominant-strategy mechanisms with individual rationality and strong budget balance in every realization. For two units with increasing submodular valuations, an explicit finite instance has ratio below $0.7290804$, so two units are strictly harder than one. Fixing the buyer, we characterize the least probability a random common price needs to guarantee a given ratio against every seller; bounding this quantity over all buyers would determine the exact two-unit constant. Within an explicit family the worst ratio is $0.729080\ldots$, conjectured to be the two-unit constant.
