arXiv 2010.097052020-10-19cs.DS

约束阶Prophet不等式

Constrained-Order Prophet Inequalities

Makis Arsenis, Odysseas Drosis, Robert Kleinberg

备注：: To appear in ACM-SIAM Symposium on Discrete Algorithms (SODA) 2021, 19 pages

摘要：

自由序先知不等式约束了双方从一组独立随机变量中选择一个值所获得的期望值之间的比率：一个“先知”知道每个变量的值，可以选择最大值，一个“赌徒”可以自由选择观察值的顺序，但必须在观察后立即选择其中一个，而不知道对于未观察到的变量将采样什么值。众所周知，赌徒总是能确保预期收益至少是预言家的$0.669\dots$倍。事实上，存在一个阈值停止规则，它保证赌徒与先知的比率至少为$1-\frac1e=0.632\dots$。相反，如果赌徒必须按照预定的顺序观察这些值，则赌徒与预言家比率的紧边界是$1/2$。在这项工作中，我们研究了这两个极端之间的模型插值。我们假设有一个预定义的排列组合，赌徒可以自由选择观察的顺序是这些预定义的排列组合中的任何一个。令人惊讶的是，我们发现，即使只有两个排序是允许的-即，正向和反向排序-赌徒预言家的比例提高到$φ^{-1}=0.618\dots$，黄金比例的倒数。当允许的排列数超过2时，出现了一个显著的“双平台”现象：从$0.5$增加到$φ^{-1}$后，通过阈值停止规则可实现的赌徒与预言者的比率不会超过$φ^{-1}+o(1)$，直到允许的排列数增加到$O(\log n)$。对于适当选择的一组$O(\text{poly}(\varepsilon^{-1})\cdot\log n)$排列，该比率达到$1-\frac1e-\varepsilon$，并且即使在允许全部$n!$排列时也不超过$1-\frac1e$。

英文摘要：

Free order prophet inequalities bound the ratio between the expected value obtained by two parties each selecting a value from a set of independent random variables: a "prophet" who knows the value of each variable and may select the maximum one, and a "gambler" who is free to choose the order in which to observe the values but must select one of them immediately after observing it, without knowing what values will be sampled for the unobserved variables. It is known that the gambler can always ensure an expected payoff at least $0.669\dots$ times as great as that of the prophet. In fact, there exists a threshold stopping rule which guarantees a gambler-to-prophet ratio of at least $1-\frac1e=0.632\dots$. In contrast, if the gambler must observe the values in a predetermined order, the tight bound for the gambler-to-prophet ratio is $1/2$. In this work we investigate a model that interpolates between these two extremes. We assume there is a predefined set of permutations, and the gambler is free to choose the order of observation to be any one of these predefined permutations. Surprisingly, we show that even when only two orderings are allowed---namely, the forward and reverse orderings---the gambler-to-prophet ratio improves to $φ^{-1}=0.618\dots$, the inverse of the golden ratio. As the number of allowed permutations grows beyond 2, a striking "double plateau" phenomenon emerges: after increasing from $0.5$ to $φ^{-1}$, the gambler-to-prophet ratio achievable by threshold stopping rules does not exceed $φ^{-1}+o(1)$ until the number of allowed permutations grows to $O(\log n)$. The ratio reaches $1-\frac1e-\varepsilon$ for a suitably chosen set of $O(\text{poly}(\varepsilon^{-1})\cdot\log n)$ permutations and does not exceed $1-\frac1e$ even when the full set of $n!$ permutations is allowed.

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。

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英文摘要：

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