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原文摘要

We study prophet inequalities for $n$ i.i.d. base rewards subject to multiplicative discounts across $S$ phases. For equal-length phases and a common discount factor $\widetildeγ$, we characterize the optimal competitive ratio of single-quantile policies as $n\to\infty$ with $S$ and $\widetildeγ$ fixed. The resulting sharp guarantee describes the dependence on the phase parameters: it recovers $1-1/e$ without discounting and approaches $1/2$ in the worst-case regime. We also establish a distribution-free comparison that bounds the discounted prophet by a fractional i.i.d. maximum with effective horizon $B$, the sum of the discount weights across all observations. The comparison yields acceptance probability $1/B$, determined only by the phase profile. This calibration attains the canonical optimum even among single-quantile policies that may choose their acceptance probability using the known base distribution. The comparison also gives guarantees for heterogeneous discounts and unequal phase lengths. A complementary optimal-stopping upper bound establishes the same limiting optimum of $1/2$ for arbitrary stopping rules in the canonical model. Finally, the same effective-horizon calibration sets acceptance probability $1-\widetildeγ$ under infinite-horizon geometric discounting. This explicit quantile rule attains Boshuizen's (1994) known optimal worst-case ratio $1/(1+\widetildeγ)$ without solving a Bellman equation.