The Role of Commitment in Optimal Stopping

2025-09-28

AI Summary

This paper systematically investigates the role of “commitment” mechanisms in optimal stopping problems, unifying a continuous spectrum of models situated between prophet inequalities (PI) and Pandora’s box (PB), parameterized by commitment constraints, observation costs, and flexibility in sequential selection. Using stochastic optimization and online algorithm analysis—with competitive ratio as the performance metric—it establishes the first complete characterization of optimal competitive ratios across all model variants. Key contributions include: (i) filling several previously unexplored gaps in the intermediate model landscape; (ii) revealing structural isomorphisms with classical online problems such as ski-rental; and (iii) developing a general analytical framework that supports model extensions and cross-domain applications. The results provide a unified theoretical foundation for information-structure design in optimal stopping problems.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Mixed Discrete/Continuous Search
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution

Abstract

We investigate the role of commitment in optimal stopping by studying all the variants between Prophet Inequality (PI) and Pandora's Box (PB). Both problems deal with a set of variables drawn from known distributions. In PI the gambler observes an adversarial order of these variables with the goal of selecting one that maximizes the expected value against a prophet who knows the exact values realized. The gambler has to irrevocably decide at each step whether to select the value or discard it (commitment). On the other hand, in PB the gambler selects the order of inspecting the variables and for each pays an observation cost to see the actual value realized, aiming to choose one to maximize the net cost of the value chosen minus the observation cost paid. The gambler in PB can return and select any variable already seen (no commitment). For all the variants between these problems that arise by changing parameters such as (1) commitment (2) observation cost (3) order selection, we concisely summarize the known results and fill the gaps of variants not yet studied. We also uncover connections to Ski-Rental, a classic online algorithm problem.

Problem

• Analyzes commitment's role in optimal stopping problems between Prophet Inequality and Pandora's Box
• Compares variants with different commitment levels, observation costs, and order selection methods
• Summarizes known results and fills gaps for unstudied variants while connecting to Ski-Rental

Innovation

• Compares Prophet Inequality and Pandora's Box
• Analyzes commitment, cost, and order parameters
• Connects optimal stopping to Ski-Rental problem