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We introduce a new measure for the performance of online algorithms inBayesian settings, where the input is drawn from a known prior, but therealizations are revealed one-by-one in an online fashion. Our new measure iscalled order-competitive ratio. It is defined as the worst case (over alldistribution sequences) ratio between the performance of the best order-unawareand order-aware algorithms, and quantifies the loss that is incurred due tolack of knowledge of the arrival order. Despite the growing interest in therole of the arrival order on the performance of online algorithms, this losshas been overlooked thus far.  We study the order-competitive ratio in the paradigmatic prophet inequalityproblem, for the two common objective functions of (i) maximizing the expectedvalue, and (ii) maximizing the probability of obtaining the largest value; andwith respect to two families of algorithms, namely (i) adaptive algorithms, and(ii) single-threshold algorithms. We provide tight bounds for all fourcombinations, with respect to deterministic algorithms. Our analysis requiresnew ideas and departs from standard techniques. In particular, our adaptivealgorithms inevitably go beyond single-threshold algorithms. The results withrespect to the order-competitive ratio measure capture the intuition thatadaptive algorithms are stronger than single-threshold ones, and may lead to abetter algorithmic advice than the classical competitive ratio measure.

"Who Is Next in Line?'' On the Significance of Knowing the Arrival Order in Bayesian Online Settings