Prophet inequalities are fundamental optimal stopping problems, where a decision-maker observes sequentially items with values sampled independently from known distributions, and must decide at each new observation to either stop and gain the current value or reject it irrevocably and move to the next step. This model is often too pessimistic and does not adequately represent real-world online selection processes. Potentially, rejected items can be revisited and a fraction of their value can be recovered. To analyze this problem, we consider general decay functions $D_1,D_2,\ldots$, quantifying the value to be recovered from a rejected item, depending on how far it has been observed in the past. We analyze how lookback improves, or not, the competitive ratio in prophet inequalities in different order models. We show that, under mild monotonicity assumptions on the decay functions, the problem can be reduced to the case where all the decay functions are equal to the same function $x \mapsto \gamma x$, where $\gamma = \inf_{x>0} \inf_{j \geq 1} D_j(x)/x$. Consequently, we focus on this setting and refine the analyses of the competitive ratios, with upper and lower bounds expressed as increasing functions of $\gamma$.

翻译：先知不等式是基础的最优停止问题，其中决策者顺序观测从已知分布中独立采样的项目价值，并必须在每次新观测时决定是停止并获得当前价值，还是不可撤销地拒绝该项目并进入下一步。该模型通常过于悲观，未能充分反映现实世界中的在线选择过程。实际上，被拒绝的项目可能被重新考虑，且其价值的一部分可以被回收。为分析此问题，我们考虑一般衰减函数$D_1,D_2,\ldots$，用于量化从被拒绝项目中可回收的价值，该价值取决于该项目在过去被观测的时间距离。我们分析了在不同顺序模型中，回溯如何改善（或不改善）先知不等式中的竞争比。我们证明，在衰减函数满足温和单调性假设的条件下，该问题可简化为所有衰减函数均等于同一函数$x \mapsto \gamma x$的情形，其中$\gamma = \inf_{x>0} \inf_{j \geq 1} D_j(x)/x$。因此，我们聚焦于此设定，并细化了对竞争比的分析，给出了上下界，这些界限表示为$\gamma$的递增函数。