# An Optimal Algorithm for Stopping on the Element Closest to the Center of an Interval

## TL;DR

This paper develops an optimal stopping algorithm for selecting the number closest to 0.5 from a sequence of uniformly distributed random numbers, achieving an asymptotic success probability of approximately .8/.7.8/.7.8/.7.

## Contribution

The paper introduces a new optimal stopping strategy for selecting the element closest to the interval's center based solely on relative ranks.

## Key findings

- Success probability asymptotically .8/.7
- Optimal stopping rule derived and proven effective
- Performance matches theoretical asymptotic analysis

## Abstract

Real numbers from the interval [0, 1] are randomly selected with uniform distribution. There are $n$ of them and they are revealed one by one. However, we do not know their values but only their relative ranks. We want to stop on recently revealed number maximizing the probability that that number is closest to $\frac{1}{2}$. We design an optimal stopping algorithm achieving our goal and prove that its probability of success is asymptotically equivalent to $\frac{1}{\sqrt{n}}\sqrt{\frac{2}{\pi}}$.