Robbins' problem

Robbins' problem is an optimal stopping problem, also known as the fourth secretary problem. Its statement is as follows.

Let $X_{1},...,X_{n}$ be independent, identically distributed random variables, uniform on $[0,1]$. We observe the $X_{k}$ 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, and only some bounds are known for the limiting value as $n$ goes to infinity.

References

• Minimizing the expected rank with full information, F. T. Bruss and T. S. Ferguson, J. Appl. Probab. Volume 30, #1 (1993), pp. 616-626
• Half-Prophets and Robbins' Problem of Minimizing the expected rank, F. T. Bruss and T. S. Ferguson, Springer Lecture Notes in Stat. Volume 1 in honor of J.M. Gani, (1996), pp. 1-17
• The secretary problem; minimizing the expected rank with i.i.d. random variables, D. Assaf and E. Samuel-Cahn, Adv. Appl. Prob. Volume 28, (1996), pp. 828-852 Cat.Inist
• What is known about Robbins' Problem? F. T. Bruss, J. Appl. Probab. Volume 42, #1 (2005), pp. 108-120 Euclid