Robbins' problem

In probability theory, Robbins' problem of optimal stopping, named after Herbert Robbins, is sometimes referred to as the fourth secretary problem or the problem of minimizing the expected rank with full information.

Let X₁, ... , X_n be independent, identically distributed random variables, uniform on [0, 1]. We observe the X_k's sequentially and must stop on exactly one of them. No recall of preceding observations is permitted. What stopping rule minimizes the expected rank of the selected observation, and what is its corresponding value?

The general solution to this full-information expected rank problem is unknown. The major difficulty is that the problem is fully history-dependent, that is, the optimal rule depends at every stage on all preceding values, and not only on simpler sufficient statistics of these. Only bounds are known for the limiting value v as n goes to infinity, namely 1.908 < v < 2.329. It is known that there is some room to improve the lower bound by further computations for a truncated version of the problem. It is still not known how to improve on the upper bound which stems from the subclass of memoryless threshold rules.

It was proposed the continuous time version of the problem where the observations follow a Poisson arrival process of homogeneous rate 1. Under some assumptions, the corresponding value function ${\displaystyle w(t)}$ is bounded and Lipschitz continuous, and the differential equation for this value function is derived. The limiting value of ${\displaystyle w(t)}$ presents the solution of Robbins’ problem. It is shown that for large ${\displaystyle t}$, ${\displaystyle 1\leq w(t)\leq 2.33183}$. This estimation coincides with the bounds mentioned above.

A simple suboptimal rule, which performs almost as well as the optimal rule, was proposed by Krieger & Samuel-Cahn. The rule stops with the smallest ${\displaystyle i}$ such that ${\displaystyle R_{i}<ic/(n+i)}$ for a given constant c, where ${\displaystyle R_{i}}$ is the relative rank of the ith observation and n is the total number of items. This rule has added flexibility. A curtailed version thereof can be used to select an item with a given probability ${\displaystyle P}$, ${\displaystyle P<1}$. The rule can be used to select two or more items. The problem of selecting a fixed percentage ${\displaystyle \alpha }$, ${\displaystyle 0<\alpha <1}$, of n, is also treated.