Prophet Inequalities over Time

In this paper, we introduce an over-time variant of the well-known prophet inequality with i.i.d. random variables. Instead of stopping with one realized value at some point in the process, we decide for each step how long we select the value. Then we cannot select another value until this period is over. The goal is to maximize the expectation of the sum of selected values. We describe the structure of the optimal stopping rule and give upper and lower bounds on the prophet inequality. In online algorithms terminology, this corresponds to bounds on the competitive ratio of an online algorithm. We give a surprisingly simple algorithm with a single threshold that results in a prophet inequality of $\approx 0.396$ for all input lengths $n$. Additionally, as our main result, we present a more advanced algorithm resulting in a prophet inequality of $\approx 0.598$ when the number of steps tends to infinity. We complement our results by an upper bound that shows that the best possible prophet inequality is at most $1/\varphi \approx 0.618$, where $\varphi$ denotes the golden ratio.