The prophet inequality is one of the cornerstone problems in optimal stopping theory and has become a crucial tool for designing sequential algorithms in Bayesian settings. In the i.i.d. $k$-selection prophet inequality problem, we sequentially observe $n$ non-negative random values sampled from a known distribution. Each time, a decision is made to accept or reject the value, and under the constraint of accepting at most $k$. For $k=1$, Hill and Kertz [Ann. Probab. 1982] provided an upper bound on the worst-case approximation ratio that was later matched by an algorithm of Correa et al. [Math. Oper. Res. 2021]. The worst-case tight approximation ratio for $k=1$ is computed by studying a differential equation that naturally appears when analyzing the optimal dynamic programming policy. A similar result for $k>1$ has remained elusive. In this work, we introduce a nonlinear system of differential equations for the i.i.d. $k$-selection prophet inequality that generalizes Hill and Kertz's equation when $k=1$. Our nonlinear system is defined by $k$ constants that determine its functional structure, and their summation provides a lower bound on the optimal policy's asymptotic approximation ratio for the i.i.d. $k$-selection prophet inequality. To obtain this result, we introduce for every $k$ an infinite-dimensional linear programming formulation that fully characterizes the worst-case tight approximation ratio of the $k$-selection prophet inequality problem for every $n$, and then we follow a dual-fitting approach to link with our nonlinear system for sufficiently large values of $n$. As a corollary, we use our provable lower bounds to establish a tight approximation ratio for the stochastic sequential assignment problem in the i.i.d. non-negative regime.

翻译：先知不等式是最优停止理论中的基石问题之一，并已成为贝叶斯环境下设计序列算法的关键工具。在同独立分布 $k$-选择先知不等式问题中，我们顺序观测从已知分布中采样的 $n$ 个非负随机值。每次需决定接受或拒绝该值，且受最多接受 $k$ 个值的约束。当 $k=1$ 时，Hill 和 Kertz [Ann. Probab. 1982] 给出了最坏情况近似比的上界，该上界后来被 Correa 等人 [Math. Oper. Res. 2021] 的算法所匹配。$k=1$ 时的最坏情况紧近似比可通过研究分析最优动态规划策略时自然出现的微分方程来计算。对于 $k>1$ 的类似结果一直未能得到。在本工作中，我们为同独立分布 $k$-选择先知不等式引入了一个非线性微分方程组，该方程组在 $k=1$ 时推广了 Hill 和 Kertz 的方程。我们的非线性系统由 $k$ 个常数定义，这些常数决定了其函数结构，它们的和提供了同独立分布 $k$-选择先知不等式最优策略渐近近似比的下界。为获得此结果，我们针对每个 $k$ 引入了一个无限维线性规划形式化方法，该方法完整刻画了任意 $n$ 下 $k$-选择先知不等式问题的最坏情况紧近似比，随后通过对偶拟合方法在充分大的 $n$ 值下将其与我们的非线性系统建立联系。作为推论，我们利用可证明的下界，为同独立分布非负机制下的随机序列分配问题建立了紧近似比。