We investigate the role of commitment in optimal stopping by studying all the variants between Prophet Inequality (PI) and Pandora's Box (PB). Both problems deal with a set of variables drawn from known distributions. In PI the gambler observes an adversarial order of these variables with the goal of selecting one that maximizes the expected value against a prophet who knows the exact values realized. The gambler has to irrevocably decide at each step whether to select the value or discard it (commitment). On the other hand, in PB the gambler selects the order of inspecting the variables and for each pays an observation cost to see the actual value realized, aiming to choose one to maximize the net cost of the value chosen minus the observation cost paid. The gambler in PB can return and select any variable already seen (no commitment). For all the variants between these problems that arise by changing parameters such as (1) commitment (2) observation cost (3) order selection, we concisely summarize the known results and fill the gaps of variants not yet studied. We also uncover connections to Ski-Rental, a classic online algorithm problem.

翻译：我们通过研究先知不等式（PI）与潘多拉魔盒（PB）之间的所有变体，探讨承诺在最优停止中的作用。这两个问题都涉及一组从已知分布中抽取的随机变量。在PI中，博弈者按对抗顺序观察这些变量，目标是在与知晓所有变量真实取值的先知对抗时，选择一个变量以最大化期望收益。博弈者必须在每一步不可撤销地决定是选择当前值还是将其丢弃（承诺）。另一方面，在PB中，博弈者自行选择检查变量的顺序，并为每个变量支付观察成本以获知其真实取值，旨在最大化所选变量的价值减去已支付观察成本的净收益。PB中的博弈者可以返回并选择任何已观察过的变量（无承诺）。对于通过改变以下参数产生的所有问题变体：（1）承诺（2）观察成本（3）顺序选择，我们简明总结了已知结果，并填补了尚未研究的变体空白。我们还揭示了其与经典在线算法问题——滑雪租赁问题之间的联系。