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

2026-05-18

AI Summary

This study addresses the Pandora’s box problem and prophet inequalities under the Conditional Value-at-Risk (CVaR) risk measure. By employing a variational reduction, the risk-aware Pandora’s box problem is transformed into a one-dimensional index policy, for which an exact Weitzman-type solution is established. Regarding prophet inequalities, the work demonstrates that no constant-factor approximation can be guaranteed without distributional assumptions; however, under continuous distributions satisfying the increasing failure rate average (IFRA) condition, a threshold policy yields an explicit constant-factor approximation guarantee. This paper provides the first systematic characterization of theoretical limits for these two sequential decision problems within the CVaR framework, revealing that risk-sensitive objectives can undermine classical approximation guarantees and identifying new feasibility conditions that depend critically on distributional structure.

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 $α\in(0,1)$, no positive constant approximation guarantee can hold without distributional structure, in sharp contrast with the risk-neutral case $α=1$, 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 $α$-fraction of outcomes, so any compromise an online policy makes to preserve the chance of a large payoff in the upper $(1-α)$-fraction does not help its CVaR. It turns out that additional distributional structure restores a uniform result: under continuous reward distributions satisfying a recentered increasing-failure-rate-average (IFRA) condition, a threshold policy achieves an explicit constant bound.

Problem

• Conditional Value-at-Risk
• Pandora's Box
• Prophet Inequality
• Risk-Aware Decision Making
• Sequential Decision Problems

Innovation

• Conditional Value-at-Risk
• Pandora's Box
• Prophet Inequality
• Risk-Aware Sequential Decision
• IFRA Condition