Iterated full information secretary problem

Małgorzata Kuchta

doi:10.1007/s00186-017-0594-0

Abstract

We consider a full information best-choice problem where an administrator who has only one on-line choice in m consecutive searches has to choose the best candidate in one of them.

Highlights

In the full information best-choice problem Gilbert and Mosteller (1966) we deal with a discrete time stochastic process (X1, . . . , Xn) where X1, . . . , Xn are i.i.d. random variables with known continuous cumulative distribution function F
The best-choice problem consists of finding a strategy of stopping the process that maximizes the probability P [Xτ = max {X1, . . . , Xn}] over all stopping times τ ≤ n. [see Gnedin (1996)]
The optimal stopping time is given by the following formula: τn∗ = min {t : Xt = max {X1, . . . , Xt }, 1 ≤ t ≤ n and F(Xt ) ≥ dn−t }, if the set under minimum is nonempty, otherwise τn∗ = n

Summary

Introduction and notation

In the full information best-choice problem Gilbert and Mosteller (1966) we deal with a discrete time stochastic process (X1, . . . , Xn) where X1, . . . , Xn are i.i.d. random variables with known continuous cumulative distribution function F. The best-choice problem consists of finding a strategy of stopping the process that maximizes the probability P [Xτ = max {X1, . The optimal stopping time is given by the following formula: τn∗ = min {t : Xt = max {X1, . The maximal probability (using the optimal stopping time) vn = P Xτn∗ = max {X1, . The problem considered here is related to real life situations where contests may be repeated several times but once in one of them the choice is made the procedure ends. The goal of the selector is to stop the search at a time t maximizing the probability that Xt ∈ Max Y (m). Our aim is to find a stopping time τm with respect to the filtration (Ft ) maximizing the probability P Xτm ∈ Max Y (m)

Optimal stopping time

Asymptotics