Prophet inequalities: Separating random order from order selection

preprint Prophet inequalities: Separating random order from order selection submitted Giordano Giambartolomei author Frederik Mallmann-Trenn Frederik Mallmann-Trenn author Raimundo J Saona Urmeneta author BD1DF4C4-D767-11E9-B658-BC13E66974250000-0001-5103-038X KrCh department Formal Methods for Stochastic Models: Algorithms and Applications project 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. In this model a ratio of 0.7251 has been proved to be attainable for any instance. In very recent work, this has been improved up to 0.7258. 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 the recent work aforementioned, through simulations, this ratio has been shown to be 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, more rigorous 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. 2023 eng arXiv 2304.0402410.48550/arXiv.2304.04024 Giambartolomei G, Frederik Mallmann-Trenn FM-T, Saona Urmeneta RJ. Prophet inequalities: Separating random order from order selection. arXiv. doi:<a href="https://doi.org/10.48550/arXiv.2304.04024">10.48550/arXiv.2304.04024</a> Giambartolomei, Giordano, Frederik Mallmann-Trenn Frederik Mallmann-Trenn, and Raimundo J Saona Urmeneta. “Prophet Inequalities: Separating Random Order from Order Selection.” ArXiv, n.d. <a href="https://doi.org/10.48550/arXiv.2304.04024">https://doi.org/10.48550/arXiv.2304.04024</a>. Giambartolomei, G., Frederik Mallmann-Trenn, F. M.-T., & Saona Urmeneta, R. J. (n.d.). Prophet inequalities: Separating random order from order selection. arXiv. <a href="https://doi.org/10.48550/arXiv.2304.04024">https://doi.org/10.48550/arXiv.2304.04024</a> G. Giambartolomei, F.M.-T. Frederik Mallmann-Trenn, R.J. Saona Urmeneta, ArXiv (n.d.). Giambartolomei G, Frederik Mallmann-Trenn FM-T, Saona Urmeneta RJ. Prophet inequalities: Separating random order from order selection. arXiv, 2304.04024. G. Giambartolomei, F. M.-T. Frederik Mallmann-Trenn, and R. J. Saona Urmeneta, “Prophet inequalities: Separating random order from order selection,” arXiv. . Giambartolomei, Giordano, et al. “Prophet Inequalities: Separating Random Order from Order Selection.” ArXiv, 2304.04024, doi:<a href="https://doi.org/10.48550/arXiv.2304.04024">10.48550/arXiv.2304.04024</a>. 171002024-06-03T07:44:54Z2025-04-14T07:52:47Z