The Competition Complexity of Prophet Inequalities with Correlations
This work addresses a fundamental limitation in online decision-making for resource allocation under uncertainty, extending prior independent-case results to correlated settings, which is incremental but important for applications like auctions and scheduling.
The paper tackles the prophet inequality problem with correlated rewards by determining how many additional rewards an online algorithm needs to approximate the maximum value, showing that the required number depends on the original rewards and that optimal independent-case algorithms fail with correlations. It develops asymptotically optimal algorithms for three scenarios involving block arrivals, shuffled rewards, and pairwise independence within blocks.
We initiate the study of the prophet inequality problem through the resource augmentation framework in scenarios when the values of the rewards are correlated. Our goal is to determine the number of additional rewards an online algorithm requires to approximate the maximum value of the original instance. While the independent reward case is well understood, we extend this research to account for correlations among rewards. Our results demonstrate that, unlike in the independent case, the required number of additional rewards for approximation depends on the number of original rewards, and that block-threshold algorithms, which are optimal in the independent case, may require an infinite number of additional rewards when correlations are present. We develop asymptotically optimal algorithms for the following three scenarios: (1) where rewards arrive in blocks corresponding to the different copies of the original instance; (2) where rewards across all copies are arbitrarily shuffled; and (3) where rewards arrive in blocks corresponding to the different copies of the original instance, and values within each block are pairwise independent rather than fully correlated.