On the Soft k-Prophet Problem
29 Pages Posted: 2 Apr 2020
Date Written: February 15, 2020
In the k-prophet problem, a decision maker observes n items sequentially and selects up to k < n items without recourse to maximize the total expected reward of the selected items. We consider a soft version of the problem where the decision maker can select up to k items in expectation rather than almost surely. We show that an optimal solution is a randomized threshold policy that does not benefit from dynamic updates as the stochastic process unfolds. The solution to the soft version requires only the marginal distributions, and is invariant to the dependence structure of the rewards. The total optimal expected reward is independent of the order of the items and is at least as large as that of a clairvoyant prophet that selects the k items with the highest rewards. We show that under mild conditions that there is a joint distribution such that the k-prophet problem and its soft version have the same solution and the same total expected reward. An easy-to-compute method to find such a joint distribution is provided. We extend the results to multi-dimensional constraints, endogenize k and incorporate search cost into the soft k-prophet problem. We also investigate a model that penalizes the variance of the number of selected items.
Keywords: Prophet Inequality, Dynamic Thresholds, Information Signals, Order Statistics, Marginal Distributions
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