The Group Perspective on Fairness in Multi-Winner Voting

In computational social choice theory, many studies use the setting of spatial or metric voting, where both candidates and voters are embedded in a common metric space $X$ representing positions on issues. The “cost” to a voter $v$ of candidate $c$ is their distance $d(v,c)$ in the space $X$, and it is presumed that voters asked to rank their preferences would do so in order of proximity. Many authors have searched for voting rules that tend to elect “optimal” (lowest-cost) winners, or where rules cost ratios have some guarantee in the form of an upper bound. Most work has focused on optimizing cost in scenarios with worst-case embeddings.

In this paper we study fairness for multi-winner voting rules in a metric setting, focusing our attention on a novel definition we call {\bf group inefficiency} to evaluate representation for select groups of voters. We show that depending upon \emph{who} fairness is measured for, the findings can change significantly and, that overall or worst-off definitions generally do not succeed in identifying the perspectives of the most important voter groups. Using simple metric settings, we evaluate this notion of fairness for many common multi-winner election mechanisms, some of which are currently being used in real elections and have yet to be considered under a similar lens by related work.