Optimal Stopping with Rank-Dependent Loss

TL;DR

This paper studies optimal stopping rules based on rank-dependent loss functions for iid observations, analyzing their asymptotic behavior and revealing persistent history dependence in the limit.

Contribution

It introduces a limiting framework for rank-dependent stopping problems and demonstrates that history dependence remains in the asymptotic regime, answering a question by Bruss.

Findings

Established bounds on the stopping value in the limit

Revealed persistent history dependence in the optimal rule

Connected the problem to planar Poisson processes

Abstract

For $\tau$ a stopping rule adapted to a sequence of $n$ iid observations, we define the loss to be $\ex [ q(R_\tau)]$, where $R_j$ is the rank of the $j$th observation, and $q$ is a nondecreasing function of the rank. This setting covers both the best choice problem with $q(r)={\bf 1}(r>1)$, and Robbins' problem with $q(r)=r$. As $n\to\infty$ the stopping problem acquires a limiting form which is associated with the planar Poisson process. Inspecting the limit we establish bounds on the stopping value and reveal qualitative features of the optimal rule. In particular, we show that the complete history dependence persists in the limit, thus answering a question asked by Bruss in the context of Robbins' problem.