# Weighted games of best choice

## TL;DR

This paper extends the classical secretary problem by analyzing weighted distributions of candidate rankings, specifically Ewens and Mallows models, and introduces a class of permutation statistics that simplify optimal strategy analysis.

## Contribution

It provides the optimal strategies and winning probabilities for secretary problems under Ewens and Mallows distributions, and introduces a class of permutation statistics that ensure positional strategies.

## Key findings

- Optimal strategies for Ewens and Mallows distributions derived.
- Winning probabilities computed for these weighted models.
- A new class of permutation statistics ensures positional strategies.

## Abstract

The game of best choice (also known as the secretary problem) is a model for sequential decision making with a long history and many variations. The classical setup assumes that the sequence of candidate rankings are uniformly distributed. Given a statistic on the symmetric group, one can instead weight each permutation according to an exponential function in the statistic. We play the game of best choice on the Ewens and Mallows distributions that are obtained in this way from the number of left-to-right maxima and number of inversions in the permutation, respectively. For each of these, we give the optimal strategy and probability of winning. Moreover, we introduce a general class of permutation statistics that always produces games of best choice whose optimal strategies are positional, which simplifies their analysis considerably.