阈值动力学与相关先知不等式
Threshold Dynamics and Correlated Prophet Inequalities
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中文总结 AI 辅助
研究由世界变量\(Z\)诱导的两种相关模型下的先知不等式，提出新分析框架刻画单阈值算法奖励动态，给出公共基模型相关结果，还研究公共尺度模型得出无算法能超\(1/n\)竞争比的结论。
中文摘要 AI 辅助
先知不等式已成为分析在线算法性能的核心工具。但多数现有结果假设输入随机变量独立，限制了其适用性。本文研究由世界变量\(Z\)的潜在状态诱导的两种相关模型下的先知不等式。在公共基模型中，分析始终接受最终项的单阈值算法，给出了确定性算法的竞争比及随机化改进后的结果，还通过极小极大论证得到\(Z\)为随机时的相同比例。建立了更强的上下界，排除了该类算法达到独立输入时\(1/2\)比例的可能性。核心技术贡献是新分析框架，刻画了单阈值算法的奖励动态。最后研究了公共尺度模型，表明这种最小乘法相关性导致强不可能性结果，即无算法能达到超过\(1/n\)的竞争比。
英文摘要
Prophet inequalities have become a central tool for analyzing the performance of online algorithms. However, most existing results assume that input random variables are independent, which limits their applicability. Motivated by this gap, we study prophet inequalities under two correlation models induced by a latent state of the world variable $Z$. In the common-base model, the algorithm observes the sequence $Z+X_1,\dots,Z+X_n$. We analyze single-threshold algorithms with the constraint that they always accept the final item, guaranteeing a reward of at least $Z$. When $Z$ is chosen adversarially, we characterize the optimal deterministic algorithm of this form, achieving a competitive ratio of $0.381$. We then show that randomizing improves the guarantee to $0.4$. By a minimax argument, the same ratio is achievable when $Z$ is random. We depart from standard techniques by establishing a stronger lower bound of $0.41$ and an upper bound of $0.475$, ruling out the possibility that this class of algorithms attains the $1/2$ ratio known for independent inputs. The core technical contribution is a new analytical framework that captures the reward dynamics of single-threshold algorithms. We introduce a differential equation characterizing the expected reward of a threshold in the worst-case instance, parameterized by the distribution of the maximum. This equation admits a closed-form and unifies known single-threshold prophet inequalities, yielding a simple threshold-optimality condition applicable to the common-base model. Finally, we study the common-scale model, where inputs take the form $Z\cdot X_1,\dots,Z\cdot X_n$. We show that this minimal multiplicative correlation yields strong impossibility results: no algorithm can achieve a competitive ratio exceeding $1/n$.
