  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 $\gamma$.
