  We study a natural application of contract design to search problems with
probabilistic prior and exploration costs. These problems have a plethora of
applications and are expressed concisely within the Pandora's Box model. Its
optimal solution is the ingenious index policy proposed originally by Weitzman
in 1979.
  In our principal-agent setting, the search task is delegated to an agent. The
agent performs a sequential exploration of $n$ boxes, suffers the exploration
cost for each inspected box, and selects the content (called the prize) of one
inspected box as outcome. Agent and principal obtain an individual value based
on the selected prize. To influence the search, the principal a-priori designs
a contract with a non-negative payment to the agent for each potential prize.
The goal of the principal to maximize her expected reward, i.e., value minus
payment. We show how to compute optimal contracts for the principal in several
scenarios.
  A popular and important subclass are linear contracts, and we show how to
compute optimal linear contracts in polynomial time. For general contracts, we
consider the standard assumption that the agent suffers cost but obtains value
only from the transfers by the principal. Interestingly, a suitable adaptation
of the index policy results in an optimal contract here. More generally, for
general contracts with non-zero agent values for outcomes we show how to
compute an optimal contract in two cases: (1) when each box has only one prize
with non-zero value for principal and agent, (2) for i.i.d. boxes with a single
prize with positive value for the principal. These results show that optimal
contracts can be highly non-trivial, and their design goes significantly beyond
the application or re-interpretation of the index policy.
