  Martin Weitzman's "Pandora's problem" furnishes the mathematical basis for
optimal search theory in economics. Nearly 40 years later, Laura Doval
introduced a version of the problem in which the searcher is not obligated to
pay the cost of inspecting an alternative's value before selecting it. Unlike
the original Pandora's problem, the version with nonobligatory inspection
cannot be solved optimally by any simple ranking-based policy, and it is
unknown whether there exists any polynomial-time algorithm to compute the
optimal policy. This motivates the study of approximately optimal policies that
are simple and computationally efficient. In this work we provide the first
non-trivial approximation guarantees for this problem. We introduce a family of
"committing policies" such that it is computationally easy to find and
implement the optimal committing policy. We prove that the optimal committing
policy is guaranteed to approximate the fully optimal policy within a
$1-\frac1e = 0.63\ldots$ factor, and for the special case of two boxes we
improve this factor to 4/5 and show that this approximation is tight for the
class of committing policies.
