In the prophet inequality problem, a gambler faces a sequence of items arriving online with values drawn independently from known distributions. On seeing an item, the gambler must choose whether to accept its value as her reward and quit the game, or reject it and continue. The gambler's aim is to maximize her expected reward relative to the expected maximum of the values of all items. Since the seventies, a tight bound of 1/2 has been known for this competitive ratio in the setting where the items arrive in an adversarial order (Krengel and Sucheston, 1977, 1978). However, the optimum ratio still remains unknown in the order selection setting, where the gambler selects the arrival order, as well as in prophet secretary, where the items arrive in a random order. Moreover, it is not even known whether a separation exists between the two settings. In this paper, we show that the power of order selection allows the gambler to guarantee a strictly better competitive ratio than if the items arrive randomly. For the order selection setting, we identify an instance for which Peng and Tang's (FOCS'22) state-of-the-art algorithm performs no better than their claimed competitive ratio of (approximately) 0.7251, thus illustrating the need for an improved approach. We therefore extend their design and provide a more general algorithm design framework, using which we show that their ratio can be beaten, by designing a 0.7258-competitive algorithm. For the random order setting, we improve upon Correa, Saona and Ziliotto's (SODA'19) 0.732-hardness result to show a hardness of 0.7254 for general algorithms - even in the setting where the gambler knows the arrival order beforehand, thus establishing a separation between the order selection and random order settings.

翻译：在先知不等式问题中，博弈者面对一系列在线到达的物品，其价值独立地来自已知分布。当看到一件物品时，博弈者必须决定接受其价值作为奖励并退出游戏，或拒绝并继续。博弈者的目标是最大化其期望奖励相对于所有物品价值的期望最大值。自七十年代以来，在物品以对抗顺序到达的设置中，该竞争比的紧密界限1/2便已明确（Krengel和Sucheston，1977，1978）。然而，在博弈者可选择到达顺序的顺序选择设置中，以及在物品以随机顺序到达的先知秘书问题中，最优比率仍未可知。此外，甚至不清楚这两种设置之间是否存在分离。本文表明，顺序选择的能力使博弈者能够保证比随机顺序到达时严格更好的竞争比。对于顺序选择设置，我们识别了一个实例，其中Peng和Tang（FOCS'22）的最先进算法的表现不超过其声称的竞争比（约0.7251），从而说明需要改进方法。因此，我们扩展其设计，提供了一个更通用的算法设计框架，并利用该框架证明其比率可以被超越，设计出一个0.7258-竞争算法。对于随机顺序设置，我们改进了Correa、Saona和Ziliotto（SODA'19）的0.732困难性结果，证明了一般算法的下界为0.7254——即使博弈者事先知晓到达顺序的设置中也是如此，从而建立了顺序选择与随机顺序设置之间的分离。