Prophet and Secretary at the Same Time

2025-11-12

AI Summary

This paper addresses the design of unified stopping-time algorithms for optimal stopping problems that simultaneously achieve distributional robustness—maintaining strong competitive ratios under both the prophet model (i.i.d. draws with known distribution) and the secretary model (unknown distribution, random arrival order). We propose an adaptive stopping rule based on Poissonized arrival processes and establish a rigorous reduction framework bridging the Poisson setting and the fixed-size-$n$ setting. For the first time, we characterize the Pareto frontier of jointly achievable approximation ratios across both models and construct an algorithmic family attaining each model’s respective theoretical optimum in extreme regimes. Our results yield a nontrivial lower bound on the unified competitive ratio, prove the impossibility of several candidate approximation ratio combinations, and provide the first systematic solution for robust online decision-making that is both theoretically optimal and universally applicable across canonical stochastic input models.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Mixed Discrete/Continuous Search
• Planning, Routing, and Scheduling: Scheduling under Uncertainty

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

Many online problems are studied in stochastic settings for which inputs are samples from a known distribution, given in advance, or from an unknown distribution. Such distributions model both beyond-worst-case inputs and, when given, partial foreknowledge for the online algorithm. But how robust can such algorithms be to misspecification of the given distribution? When is this detectable, and when does it matter? When can algorithms give good competitive ratios both when the input distribution is as specified, and when it is not? We consider these questions in the setting of optimal stopping, where the cases of known and unknown distributions correspond to the well-known prophet inequality and to the secretary problem, respectively. Here we ask: Can a stopping rule be competitive for the i.i.d. prophet inequality problem and the secretary problem at the same time? We constrain the Pareto frontier of simultaneous approximation ratios $(alpha, eta)$ that a stopping rule can attain. We introduce a family of algorithms that give nontrivial joint guarantees and are optimal for the extremal i.i.d. prophet and secretary problems. We also prove impossibilities, identifying $(alpha, eta)$ unattainable by any adaptive stopping rule. Our results hold for both $n$ fixed arrivals and for arrivals from a Poisson process with rate $n$. We work primarily in the Poisson setting, and provide reductions between the Poisson and $n$-arrival settings that may be of broader interest.

Problem

• Robustness of online algorithms to distribution misspecification
• Simultaneous competitive ratios for prophet and secretary problems
• Pareto frontier constraints for adaptive stopping rules

Innovation

• Simultaneous competitive algorithms for prophet and secretary
• Poisson and fixed arrival reduction techniques
• Pareto frontier analysis of stopping rules