Splitting Guarantees for Prophet Inequalities via Nonlinear Systems

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 items. 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 non-linear 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. nonnegative regime.