Prophet Inequalities: Separating Random Order from Order Selection

Prophet inequalities are a central object of study in optimal stopping theory. A gambler is sent values in an online fashion, 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. In this model a ratio of $0.7251$ is attainable for any instance. Recently, this has been improved up to $0.7258$ by Bubna and Chiplunkar (2023). 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. In a computer-assisted proof, Bubna and Chiplunkar (2023) also showed that this ratio is at most $0.7254$ for the Random Order model, thus establishing for the first time that carefully choosing the order, instead of simply taking it at random, benefits the gambler. We give an alternative, non-simulation-assisted proof of this fact, by showing mathematically that in the Random Order model, no algorithm can achieve a ratio larger than $0.7235$. This sets a new state-of-the-art hardness for this model, and establishes more formally that there is a real benefit in choosing the order.