关于交易收益的最优先知不等式
Optimal Prophet Inequalities for Gain from Trade
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- SKLMS, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）
- School of Mathematical Sciences, University of Chinese Academy of Sciences（中国科学院大学数学科学学院）
- School of Mathematical Sciences, Key Laboratory of MEA (Ministry of Education), Shanghai Key Laboratory of PMMP, Nantong Institute for Applied Mathematics and Artificial Intelligence, East China Normal University（华东师范大学数学科学学院）
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中文总结 AI 辅助
本研究提出先知交易模型，以先知为基准，通过阈值算法实现最优竞争比2，并针对平衡、单卖方等情形改进至1.76或1.58，精确刻画库存过程。
中文摘要 AI 辅助
我们首次研究了先知交易（prophet trading）问题，这是一种在线交易模型，其中交易者与以均匀随机顺序到达的卖方和买方进行交互，且他们的价格独立地取自一个共同的已知分布。交易者的目标是最大化期望交易收益（GFT），其性能以全知的先知（prophet）为基准来衡量，该先知事先知道实际的到达顺序和所有价格。与经典先知不等式问题不同，交易者必须在管理库存的同时做出购买和出售决策，这使得先知基准和在线算法的分析都变得相当复杂。对于任意数量的买方和卖方，我们提出了一种基于阈值的简单算法，并证明了其分布无关的竞争比为2，这是最优的。我们的主要技术贡献是对阈值交易引发的库存过程进行了精确刻画。通过将算法的期望GFT表示为买方到达时累计持有概率的函数，我们推导出该库存项的精确公式，这构成了我们分析的基础，并在几个重要的特殊情形下给出了更精确的保证。对于平衡交易（有$n$个卖方和$n$个买方），我们将竞争比改进为$(2n^2-n)/((n+4^{-n}-1)(n+1))$，对于每个$n>1$，该值严格小于2。对于单卖方情形，我们优化了固定阈值算法的分析，得到了渐近竞争比$2e^2/(e^2+1)\approx1.76$。此外，通过利用该设置的额外结构，我们设计了一种自适应阈值算法，其竞争比趋近于$e/(e-1)\approx 1.58$。由于我们一般算法思想的对称性，相同的$1.76$竞争比保证也适用于单买方情形。
英文摘要
We initiate the study of prophet trading, an online trading model in which a trader interacts with sellers and buyers who arrive in a uniformly random order and whose prices are drawn independently from a common known distribution. The trader aims to maximize the expected gain from trade (GFT), with performance measured against an omniscient prophet that knows the realized arrival order and all prices in advance. Unlike the classical prophet inequality problem, the trader must make both purchasing and selling decisions while managing the inventory, which makes both the prophet benchmark and the analysis of online algorithms substantially more intricate. For arbitrary numbers of buyers and sellers, we propose a simple threshold-based algorithm and prove a distribution-free competitive ratio of~2, which is the best possible. Our main technical contribution is an exact characterization of the inventory process induced by threshold trading. By expressing the algorithm's expected GFT in terms of the cumulative holding probability at buyer arrivals, we derive an exact formula for this inventory term, which serves as the foundation of our analysis and yields sharper guarantees in several important special cases. For balanced trading (with $n$ sellers and $n$ buyers), we improve the competitive ratio to $(2n^2-n)/((n+4^{-n}-1)(n+1))$, which is strictly less than 2 for every $n>1$. For the single-seller case, we sharpen the analysis of the fixed-threshold algorithm to obtain an asymptotic competitive ratio of $2e^2/(e^2+1)\approx1.76$. Furthermore, by exploiting the additional structure of this setting, we design an adaptive-threshold algorithm whose competitive ratio approaches $e/(e-1)\approx 1.58$. Due to a symmetry property of our general algorithmic idea, the same $1.76$-competitive guarantee also holds for the single-buyer case.
