Windowed Prophet Inequalities

TL;DR

This paper investigates windowed and batched variants of prophet inequalities, establishing optimal ratios for small windows and showing equivalence between batched and i.i.d. cases, advancing understanding of online selection problems.

Contribution

It introduces new bounds for windowed prophet inequalities and proves an equivalence with the batched variant, extending the theoretical framework of prophet inequalities.

Findings

Optimal ratio for small window size is approximately 0.745.

For window size n/k, the ratio approaches that of the k-sample i.i.d. problem.

Batched prophet inequalities are equivalent to the i.i.d. prophet inequalities problem.

Abstract

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 $\alpha \approx 0.745$, 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 $\alpha_k < WIN_{n/k} \le \alpha_k + o_k(1)$ where $WIN_{n/k}$ is the optimal competitive ratio for the window size $n/k$ prophet inequalities problem and $\alpha_k$ 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.