Windowed Prophet Inequalities

The prophet inequalities problem has received significant study over the past decades and has several applications such as to online auctions. In this paper, we study two variants of the i.i.d. prophet inequalities problem, namely the windowed prophet inequalities problem and the batched prophet inequalities problem. For the windowed prophet inequalities problem, we show that for window size $o(n)$, the optimal competitive ratio is $α\approx 0.745$, the same as in the non-windowed case. In the case where the window size is $n/k$ for some constant $k$, we show that $α_k < WIN_{n/k} \le α_k + o_k(1)$ where $WIN_{n/k}$ is the optimal competitive ratio for the window size $n/k$ prophet inequalities problem and $α_k$ is the optimal competitive ratio for the $k$ sample i.i.d. prophet inequalities problem. Finally, we prove an equivalence between the batched prophet inequalities problem and the i.i.d. prophet inequalities problem.