One step futher: an explicit solution to Robbins' problem when $n=4$

TL;DR

This paper provides an explicit solution to Robbins' optimal stopping problem for four observations, where the goal is to minimize the expected rank of the selected uniform random variable.

Contribution

The paper derives the first explicit solution for Robbins' problem when the number of observations is four, advancing understanding of optimal stopping rules in this setting.

Findings

Explicit solution for n=4 Robbins' problem

Optimal stopping rule minimizes expected rank

Advances theoretical understanding of sequential decision-making

Abstract

Fix some $n \in \mathbb{N}$ and let $X_1, X_2,\dots, X_n$ be independent random variables drawn from the uniform distribution on $[0,1]$. A decision maker is shown the variables sequentially and, after each observation, must decide whether or not to keep the current one, with payoff the overall rank of the selected observation. Decisions are final: no recall is allowed, no regret is tolerated. The objective is to act in such a way as to minimise the expected payoff. In this note we give the explicit solution to this problem, known as Robbins' problem of optimal stopping, when $n=4$.