  We consider online variations of the Pandora's box problem (Weitzman. 1979),
a standard model for understanding issues related to the cost of acquiring
information for decision-making. Our problem generalizes both the classic
Pandora's box problem and the prophet inequality framework. Boxes are presented
online, each with a random value and cost drew jointly from some known
distribution. Pandora chooses online whether to open each box given its cost,
and then chooses irrevocably whether to keep the revealed prize or pass on it.
We aim for approximation algorithms against adversaries that can choose the
largest prize over any opened box, and use optimal offline policies to decide
which boxes to open (without knowledge of the value inside). We consider
variations where Pandora can collect multiple prizes subject to feasibility
constraints, such as cardinality, matroid, or knapsack constraints. We also
consider variations related to classic multi-armed bandit problems from
reinforcement learning. Our results use a reduction-based framework where we
separate the issues of the cost of acquiring information from the online
decision process of which prizes to keep. Our work shows that in many
scenarios, Pandora can achieve a good approximation to the best possible
performance.
