  Most of the literature on online algorithms and sequential decision-making
focuses on settings with "irrevocable decisions" where the algorithm's decision
upon arrival of the new input is set in stone and can never change in the
future. One canonical example is the classic prophet inequality problem, where
realizations of a sequence of independent random variables $X_1, X_2,\ldots$
with known distributions are drawn one by one and a decision maker decides when
to stop and accept the arriving random variable, with the goal of maximizing
the expected value of their pick. We consider "prophet inequalities with
recourse" in the linear buyback cost setting, where after accepting a variable
$X_i$, we can still discard $X_i$ later and accept another variable $X_j$, at a
\textit{buyback cost} of $f \times X_i$. The goal is to maximize the expected
net reward, which is the value of the final accepted variable minus the total
buyback cost. Our first main result is an optimal prophet inequality in the
regime of $f \geq 1$, where we prove that we can achieve an expected reward
$\frac{1+f}{1+2f}$ times the expected offline optimum. The problem is still
open for $0<f<1$ and we give some partial results in this regime. In
particular, as our second main result, we characterize the asymptotic behavior
of the competitive ratio for small $f$ and provide almost matching upper and
lower bounds that show a factor of $1-\Theta\left(f\log(\frac{1}{f})\right)$.
Our results are obtained by two fundamentally different approaches: One is
inspired by various proofs of the classical prophet inequality, while the
second is based on combinatorial optimization techniques involving LP duality,
flows, and cuts.
