  In the classical optimal stopping problem, a player is given a sequence of
random variables $X_1\ldots X_n$ with known distributions. After observing the
realization of $X_i$, the player can either accept the observed reward from
$X_i$ and stop, or reject the observed reward from $X_i$ and continue to
observe the next variable $X_{i+1}$ in the sequence. Under any fixed ordering
of the random variables, an optimal stopping policy, one that maximizes the
player's expected reward, is given by the solution of a simple dynamic program.
In this paper, we investigate the relatively less studied question of selecting
the order in which the random variables should be observed so as to maximize
the expected reward at the stopping time. To demonstrate the benefits of order
selection, we prove a novel prophet inequality showing that, when the support
of each random variable has size at most 2, the optimal ordering can achieve an
expected reward that is within a factor of 1.25 of the expected hindsight
maximum; this is an improvement over the corresponding factor of 2 for the
worst-case ordering. We also provide a simple $O(n^2)$ algorithm for finding an
optimal ordering in this case. Perhaps surprisingly, we demonstrate that a
slightly more general case - each random variable $X_i$ is restricted to have
3-point support of form $\{0, m_i, 1\}$ - is NP-hard, and provide an FPTAS for
that case.
