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

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 $\alpha\in(0,1)$, no positive constant approximation guarantee can hold without distributional structure, in sharp contrast with the risk-neutral case $\alpha=1$, 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 $\alpha$-fraction of outcomes, so any compromise an online policy makes to preserve the chance of a large payoff in the upper $(1-\alpha)$-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.