基于连续时间先知不等式的基于库存阈值的竞争分析
Competitive Analysis of Stock-based Thresholds via Prophet Inequalities in Continuous Time
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中文总结 AI 辅助
该研究针对带非齐次泊松到达的连续时间K单元在线资源分配问题，提出基于库存阈值策略的竞争分析框架，获两阈值K=2时竞争比0.6269、三阈值K=3时0.6816的强保证。
中文摘要 AI 辅助
我们研究了具有非齐次泊松到达和时变估值分布的连续时间K单元在线资源分配问题。虽然最优动态策略通常取决于剩余库存和时间范围内剩余的时间，但我们聚焦于更简单且具有实际吸引力的基于库存阈值策略类，其有限数量的阈值仅取决于剩余单元数量。我们针对多单元先知基准评估这些策略，该基准事后选择K个最佳实现值。我们的主要贡献是一种新的连续时间基于库存阈值的竞争分析框架。我们首先通过类型覆盖对偶重新表述问题。分析对偶的核心挑战在于对偶既是无限维的又是非凸的：对手可以选择时变到达和估值过程，而策略性能非线性地依赖于随机库存轨迹。我们通过将连续时间对抗问题简化为泊松优化问题PoisOPTRe_K，然后证明其最坏情况解的清晰结构特性，克服了这些挑战。特别是，对抗性到达具有截止和延迟填充结构，这为两个阈值产生了精确的四参数公式，并为一般阈值产生了有限嵌套区间表示。这些简化使得保证可直接计算。例如，对于两个阈值，当K=2时，我们获得竞争比0.6269；对于三个阈值，当K=3时，我们获得竞争比0.6816。由此，我们证明了简单的基于库存阈值策略尽管忽略了日历时间，仍能实现强先知不等式保证。
英文摘要
We study a continuous-time $K$-unit online resource allocation problem with nonhomogeneous Poisson arrivals and time-varying valuation distributions. While the optimal dynamic policy generally depends on both the remaining inventory and the time left in the horizon, we focus on a simpler and practically appealing class of stock-based threshold policies, whose limited number of thresholds depend only on the number of units remaining. We evaluate these policies against the multi-unit prophet benchmark, which selects the best $K$ realized values in hindsight. Our main contribution is a new competitive-analysis framework for stock-based thresholds in continuous time. We first reformulate the problem through a type-covering dual. The central challenge for analyzing the dual is that the dual is both infinite-dimensional and non-convex: the adversary can choose time-varying arrival and valuation processes, while the policy performance depends nonlinearly on the stochastic inventory trajectory. We overcome these challenges by reducing the continuous-time adversarial problem to a Poisson optimization $PoisOPTRe_K$, and then proving sharp structural properties of its worst-case solutions. In particular, adversarial arrivals admit cutoff and late-filling structures, which yield an exact four-parameter formulation for two thresholds and a finite nested-interval representation for general thresholds. These reductions make the guarantees directly computable. For example, for two thresholds, we obtain a competitive ratio $0.6269$ for $K=2$; with three thresholds, we obtain the ratio $0.6816$ for $K=3$. In this way, we show that simple stock-based thresholds achieve strong prophet-inequality guarantees despite ignoring calendar time.
