A more general Pandora's rule?

TL;DR

This paper explores the generalization of Pandora's rule beyond the classic model, identifying conditions under which it remains optimal for broader objectives and connecting it to multi-armed bandit problems.

Contribution

It characterizes the class of problems where a generalized Pandora's rule is optimal and links the rule's optimality to the Gittins index theorem in multi-armed bandit settings.

Findings

Generalized Pandora's rule is optimal only for specific objective functions.

Gittins index theorem can be used to prove Pandora's rule optimality in certain cases.

Some non-bandit problems also admit Pandora's rule as optimal.

Abstract

In a classic model analysed by Weitzman an agent is presented with boxes containing prizes. She may open boxes in any order, discover prizes within, and optimally stop. She wishes to maximize the expected value of the greatest prize found, minus costs of opening boxes. The problem is solved by a so-called Pandora's rule, and has applications to searching for a house or job. However, this does not model the problem of a student who searches for the subject to choose as her major and benefits from all courses she takes while searching. So motivated, we ask whether there are any problems for which a generalized Pandora's rule is optimal when the objective is a more general function of all the discovered prizes. We show that if a generalized Pandora's rule is optimal for all specifications of costs and prize distributions, then the objective function must take a special form. We also explain how the Gittins index theorem can be applied to an equivalent multi-armed bandit problem to prove optimality of Pandora's rule for the student's problem. However, we also show that there do exist some problems which are not of multi-armed bandit type for which Pandora's rule is optimal.