Optimal Stopping with Rank-Dependent Loss

Abstract

For τ, a stopping rule adapted to a sequence of n independent and identically distributed observations, we define the loss to be E[q(Rτ)], where Rj is the rank of the jth observation and q is a nondecreasing function of the rank. This setting covers both the best-choice problem, with q(r) = 1(r > 1), and Robbins' problem, with q(r) = r. As n tends to ∞, 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 (2005) in the context of Robbins' problem.