Running minimum in the best-choice problem

TL;DR

This paper analyzes the best-choice problem for independent observations, establishing optimal stopping rules based on monotone thresholds, deriving success probability bounds, and illustrating the approach with explicit models and the full-information game.

Contribution

It generalizes the best-choice problem to independent, non-i.i.d. observations, providing a variational formulation and new success probability representations.

Findings

Optimal stopping rules are characterized by monotone thresholds.

Universal lower bounds for success probability in the iid case.

Explicit solutions for models with discrete uniform distributions.

Abstract

We consider the best-choice problem for independent (not necessarily iid) observations $X_1, \cdots, X_n$ with the aim of selecting the sample minimum. We show that in this full generality the monotone case of optimal stopping holds and the stopping domain may be defined by the sequence of monotone thresholds. In the iid case we get the universal lower bounds for the success probability. We cast the general problem with independent observations as a variational first-passage problem for the running minimum process which simplifies obtaining the formula for success probability. We illustrate this approach by revisiting the full-information game (where $X_j$'s are iid uniform-$[0,1]$), in particular deriving new representations for the success probability and its limit by $n \rightarrow \infty$. Two explicitly solvable models with discrete $X_j$'s are presented: in the first the distribution is uniform on $\{j,\cdots,n\}$, and in the second the distribution is uniform on $\{1,\cdots, n\}$. These examples are chosen to contrast two situations where the ties vanish or persist in the large-$n$ Poisson limit.