Splitting Guarantees for Prophet Inequalities via Nonlinear Systems
Johannes Brustle, Sebastian Perez-Salazar, Victor Verdugo
TL;DR
The paper tackles the i.i.d. $k$-selection prophet inequality by extending the Hill–Kertz ODE framework to general $k$ and formulating an infinite-dimensional LP in quantile space that exactly characterizes the worst-case approximation ratio. A central contribution is a closed-form nonlinear system of differential equations (NLS$_k$) parameterized by $\theta_1,\ldots,\theta_k$, whose solutions yield provable lower bounds on the asymptotic ratio $\gamma_{n,k}$ for large $n$, via a dual-fitting approach that links the LP to the nonlinear system. The authors prove that for sufficiently large $n$, $\gamma_{n,k} \ge (1-24k\frac{\ln(n)^2}{n})\sum_{j=1}^k \theta_j^*$, where $\theta^*=(\theta_1^*,\ldots,\theta_k^*)$ solves NLS$_k(\theta)$. As a corollary, their bounds provide a tight asymptotic value for the stochastic sequential assignment problem (SSAP), yielding a precise characterization of its optimal competitive ratio in the iid non-negative regime. Overall, the work advances provable guarantees for multi-selection prophet inequalities and connects LP duality, nonlinear ODE analysis, and quantile-based policies in a unified framework.
Abstract
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.