Prophet inequalities are a central object of study in optimal stopping theory. A gambler is sent values online, sampled from an instance of independent distributions, in an adversarial, random or selected order, depending on the model. When observing each value, the gambler either accepts it as a reward or irrevocably rejects it and proceeds to observe the next value. The goal of the gambler, who cannot see the future, is maximising the expected value of the reward while competing against the expectation of a prophet (the offline maximum). In other words, one seeks to maximise the gambler-to-prophet ratio of the expectations. The model, in which the gambler selects the arrival order first, and then observes the values, is known as Order Selection. Recently it has been shown that in this model a ratio of $0.7251$ can be attained for any instance. If the gambler chooses the arrival order (uniformly) at random, we obtain the Random Order model. The worst case ratio over all possible instances has been extensively studied for at least $40$ years. Still, it is not known if carefully choosing the order, or simply taking it at random, benefits the gambler. We prove that, in the Random Order model, no algorithm can achieve a ratio larger than $0.7235$, thus showing for the first time that there is a real benefit in choosing the order.

翻译：先知不等式是最优停止理论中的核心研究对象。一位赌徒在不同模型下以对抗性、随机或选定顺序依次接收来自独立分布实例的值。观察每个值时，赌徒要么将其接受为奖励，要么不可逆地拒绝并继续观察下一个值。无法预知未来的赌徒目标是最大化预期奖励值，同时与先知（离线最大值）的期望值竞争。换言之，人们寻求最大化赌徒与先知期望值之比。在赌徒首先选择到达顺序再观察值的模型中，这被称为顺序选择。最近研究表明，此模型中对任何实例均可达到0.7251的比值。若赌徒（均匀）随机选择到达顺序，则得到随机顺序模型。所有可能实例上的最坏情况比值已被广泛研究了至少40年。然而，尚不清楚精心选择顺序或随机选取是否对赌徒有利。我们证明，在随机顺序模型中，任何算法都无法实现大于0.7235的比值，从而首次表明选择顺序确实能带来实际收益。