Risk of Bad Tails: CVaR-Aware Pandora's Box and Prophet Inequality
Jingwei Ji

TL;DR
This paper explores CVaR variants of Pandora's box and prophet inequality, revealing exact solutions for Pandora's box and fundamental limitations for prophet inequalities without distributional assumptions.
Contribution
It provides a CVaR-aware analysis of these problems, characterizes the limitations in the general case, and identifies conditions under which effective policies exist.
Findings
Exact Weitzman-style index solution for CVaR Pandora's box.
No constant approximation guarantee for CVaR prophet inequality without structure.
Threshold policies achieve bounds under continuous IFRA distributions.
Abstract
We study Conditional Value-at-Risk (CVaR) variants of two canonical sequential decision problems: Pandora's box and the prophet inequality. For Pandora's box, the risk-aware problem retains an exact Weitzman-style index solution after a one-dimensional variational reduction. For the prophet inequality, the picture is different: for every CVaR level , no positive constant approximation guarantee can hold without distributional structure, in sharp contrast with the risk-neutral case , and we characterize the tight instance-dependent guarantee. Already in two-item hard instances, the prophet's CVaR benchmark can be made arbitrarily large while every online policy's CVaR remains bounded. This impossibility is due to the nature of CVaR objective: it measures only the worst -fraction of outcomes, so any compromise an online policy makes to preserve the chance…
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