# Opportunity costs in the game of best choice

## TL;DR

This paper introduces a weighted variant of the secretary problem that accounts for interview costs, showing that even minimal costs significantly reduce the success probability from approximately 37% to 28%.

## Contribution

It models interview costs by weighting candidate permutations and compares this to the classical model, revealing the impact of costs on optimal success probabilities.

## Key findings

- Imposing interview costs lowers success probability from 1/e to about 28%.
- Weighted model reflects real-world costs more accurately.
- Even minimal costs have a substantial effect on decision strategies.

## Abstract

The game of best choice, also known as the secretary problem, is a model for sequential decision making with many variations in the literature. Notably, the classical setup assumes that the sequence of candidate rankings is uniformly distributed over time and that there is no expense associated with the candidate interviews. Here, we weight each ranking permutation according to the position of the best candidate in order to model costs incurred from conducting interviews with candidates that are ultimately not hired. We compare our weighted model with the classical (uniform) model via a limiting process. It turns out that imposing even infinitesimal costs on the interviews results in a probability of success that is about 28%, as opposed to 1/e (about 37%) in the classical case.