# Group Fairness in Committee Selection

## TL;DR

This paper introduces a novel fairness concept in committee selection by extending stability to distributions over committees, demonstrating the existence of stable lotteries across different voter preference models using probabilistic methods.

## Contribution

It extends the notion of stability to distributions over committees and proves the existence of stable lotteries for canonical voter preference models.

## Key findings

- Stable lotteries always exist for approval set and ranking models.
- A unified procedure computes approximately stable lotteries across models.
- The proof employs a new large deviation inequality.

## Abstract

In this paper, we study fairness in committee selection problems. We consider a general notion of fairness via stability: A committee is stable if no coalition of voters can deviate and choose a committee of proportional size, so that all these voters strictly prefer the new committee to the existing one. Our main contribution is to extend this definition to stability of a distribution (or lottery) over committees. We consider two canonical voter preference models: the Approval Set setting where each voter approves a set of candidates and prefers committees with larger intersection with this set; and the Ranking setting where each voter ranks committees based on how much she likes her favorite candidate in a committee. Our main result is to show that stable lotteries always exist for these canonical preference models. Interestingly, given preferences of voters over committees, the procedure for computing an approximately stable lottery is the same for both models and therefore extends to the setting where some voters have the former preference structure and others have the latter. Our existence proof uses the probabilistic method and a new large deviation inequality that may be of independent interest.