The Unified Approach for the Best Choice Problem Applied to Alternative-Choice Selection Problems

TL;DR

This paper extends the unified approach for stopping problems with unknown size to a two-sided secretary problem, providing sharp success probability bounds and solutions for multiple independent streams.

Contribution

It demonstrates the effectiveness of the unified approach for a new class of best-choice problems, including two-sided secretary problems and multiple streams, with elementary methods.

Findings

Sharp lower bound of 1/2 for success probability in the unknown cardinality case.

Solution for k independent streams of arrivals.

Elementary and self-contained approach.

Abstract

The objective of this paper is to show that the so-called unified approach to stopping problems with unknown cardinality introduced in Bruss (1984) proves to be efficient for solving other types of best-choice problems. We show that what we will call the alternative-choice stopping problem, which will be exemplified right away in Section 1, can be seen as a "two-sided" Secretary problem. This problem is instigated by a former problem of R. R. Weber (Cambridge University). Our approach yields for unknown cardinality the sharp lower bound $1/2$ for the probability of success. This problem is, at the same time, a special case of a model more generally based on $k \ge 2$ linearly ordered subsets. We shall also give the solution for such problems for k independent streams of arrivals. Our approach is elementary and self-contained.