A Continuous-Time Approach to Robbins' Problem of Minimizing the Expected Rank

Abstract

Let X1, X2, …, Xn be independent random variables uniformly distributed on [0,1]. We observe these sequentially and have to stop on exactly one of them. No recall of preceding observations is permitted. What stopping rule minimizes the expected rank of the selected observation? What is the value of the expected rank (as a function of n) and what is the limit of this value when n goes to ∞? This full-information expected selected-rank problem is known as Robbins' problem of minimizing the expected rank, and its general solution is unknown. In this paper we provide an alternative approach to Robbins' problem. Our model is similar to that of Gnedin (2007). For this, we consider a continuous-time version of the problem in which the observations follow a Poisson arrival process on ℝ+ × [0,1] of homogeneous rate 1. Translating the previous optimal selection problem in this setting, we prove that, under reasonable assumptions, the corresponding value function w(t) is bounded and Lipschitz continuous. Our main result is that the limiting value of the Poisson embedded problem exists and is equal to that of Robbins' problem. We prove that w(t) is differentiable and also derive a differential equation for this function. Although we have not succeeded in using this equation to improve on bounds on the optimal limiting value, we argue that it has this potential.