具有最大值知识的贝叶斯最优停止
Bayesian Optimal Stopping with Maximum Value Knowledge
AI总结：
本文研究仅知最大值分布的相关报价最优停止问题，证明基于垄断价格的阈值策略渐近最优，并为光滑分布给出紧的二次收敛保证，改进了相关值先知不等式的性能界。
摘要：
我们考虑一个具有 n 个相关报价的最优停止问题，目标是设计一种（随机化的）停止策略，使我们在序列中停止时所获得的报价期望值最大化。我们不假设知道完整的相关结构（这在实践中是不现实的），而只假设知道序列最大值的分布，并希望分析最大值服从该分布的最坏情况相关结构。这可以看作是在没有任何分布信息已知的情形与所有单个报价的（可能相关的）分布均已知的贝叶斯情形之间的一种权衡。作为我们的第一个主要结果，我们证明，在最大值的期望值随 n 亚线性增长的假设下，使用最大值分布的垄断价格的确定性阈值策略是渐近最优的。在我们的第二个主要结果中，我们通过为足够光滑的最大值分布推导出紧的二次收敛保证，进一步收紧了这个界。我们的结果还为具有相关值的先知不等式提供了更细粒度的图景，对于相关值而言，无分布界通常只能给出数量级为 1/n 的性能保证。
英文摘要：
We consider an optimal stopping problem with n correlated offers where the goal is to design a (randomized) stopping strategy that maximizes the expected value of the offer in the sequence at which we stop. Instead of assuming to know the complete correlation structure, which is unrealistic in practice, we only assume to have knowledge of the distribution of the maximum value of the sequence, and want to analyze the worst-case correlation structure whose maximum follows this distribution. This can be seen as a trade-off between the setting in which no distributional information is known, and the Bayesian setting in which the (possibly correlated) distributions of all the individual offers are known. As our first main result we show that a deterministic threshold strategy using the monopoly price of the distribution of the maximum value is asymptotically optimal assuming that the expectation of the maximum value grows sublinearly in n. In our second main result, we further tighten this bound by deriving a tight quadratic convergence guarantee for sufficiently smooth distributions of the maximum value. Our results also give rise to a more fine-grained picture regarding prophet inequalities with correlated values, for which distribution-free bounds often only yield a performance guarantee that is of the order 1/n.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
