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.

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