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.

翻译：[摘要] 我们研究了两个经典序列决策问题——潘多拉魔盒与先知不等式——的条件风险价值变体。对于潘多拉魔盒问题，风险感知版本经一维变分约简后，仍保留精确的魏茨曼式指标解。而先知不等式问题则呈现不同景象：对任意风险水平$α\in(0,1)$而言，若缺乏分布结构约束，则无法保证存在统一的正常数近似比——这与风险中性情形$α=1$形成鲜明对比。我们刻画了严格依赖于实例的紧致保证。即使在双物品困难实例中，先知的CVaR基准可任意大，而所有在线策略的CVaR均有界。这种不可能性源于CVaR目标函数的本质特性：它仅测度最差$α$分位数结果，因此在线策略为保留获得上$(1-α)$分位数高收益机会所做的任何牺牲，均无法改善其CVaR表现。进一步研究表明，额外分布结构能恢复统一性结果：在满足重中心化递增失效率平均条件的连续收益分布下，阈值策略可实现显式常数界。