Algorithms for Robbins' Problem using Markov Decision Processes

2026-08-27

AI Summary

本文针对Robbins问题，通过建模为马尔可夫决策过程并提出有限状态抽象方法，以近似解决在已知全部信息下最小化选择候选人期望排名的问题。

Abstract

In this paper, we consider Robbins' problem, which is a full information variant of the well-known secretary selection problem. In this version of the problem, the goal is to minimize the expected rank of the selected candidate among $n$ that are interviewed sequentially, and a decision to select or not the $m^{th}$ candidate needs to be taken right after the interview (so without seeing the last $n-m$ candidates and without recall). We first show how to model instances of Robbins' problem as infinite Markov Decision Processes (MDPs). Then we propose several finite-state abstractions of these MDPs that allow us to approximate the value of the problem for fixed $n$. While it is known that the full memory of past candidates' values is necessary for optimal expected rank minimization, making the analysis of the problem challenging, we highlight simple memory structures that are sufficient for obtaining near-optimal selection strategies. Additionally, we provide approximate values for Robbins' problem for numbers of candidates $n$ up to 100 for which no good approximations were previously known (the exact value is only known for instances where $n \leq 4$ and numerical approximations were for small values of $n$ not exceeding one digit), for all $n : 5 \leq n \leq 100$, we give better approximation than what was previously known.

Problem

• Robbins' Problem
• Markov Decision Processes
• Expected Rank Minimization

Innovation

• Markov Decision Processes
• finite-state abstractions
• near-optimal selection strategies
• approximate values