Competing Optimally Against An Imperfect Prophet
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Consider a gambler who observes the realizations of n independent, non-negative, distribution-labeled random variables arriving in a uniform random order and can stop the sequence at any time to obtain a reward of the most recent observation. In 2017, Correa et al. showed that when all distributions are identical, it is possible to design a stopping time that achieves a β≈ 0.745 fraction of the maximum value (the “prophet” benchmark), matching an upper bound of Hill and Kertz. In 2019, Correa et al. showed that when the distributions differ, it is no longer possible to achieve this bound: they prove an upper bound of √(3) - 1 < 0.74 on the best approximation ratio. We show that it is possible to asymptotically achieve the β≈ 0.745 bound even in the case of non-identical distributions, as long as we are allowed to remove a small constant number of distributions. Formally, we show that for any ε, there exists a constant C_ε = poly(1/ε) (independent of n) such that after removing C_ε distributions of our choice, we can achieve a (β - ε)-approximation to the resulting maximum. We additionally show it is possible to asymptotically achieve an exact β approximation ratio for several natural classes of problem instances, including small prophets (where each distribution is concentrated near zero) and frequent prophets (where each distribution occurs at least some number of times).
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