The metric distortion of multiwinner voting

We extend the recently introduced framework of metric distortion to multiwinner voting. In this framework, agents and alternatives are located in an underlying metric space. The exact distances between agents and alternatives are unknown. Instead, each agent provides a ranking of the alternatives, ordered from the closest to the farthest. Typically, the goal is to select a single alternative that approximately minimizes the total distance from the agents, and the worst-case approximation ratio is termed distortion. In the case of multiwinner voting, the goal is to select a committee of alternatives that (approximately) minimizes the total cost to all agents. We consider the scenario where the cost of an agent for a committee is her distance from the -th closest alternative in the committee. We reveal a surprising trichotomy on the distortion of multiwinner voting rules in terms of and : The distortion is unbounded when , asymptotically linear in the number of agents when , and constant when .