Metric distortion under group-fair objectives

We consider a voting problem in which a set of agents have metric preferences over a set of alternatives, and are also partitioned into disjoint groups. Given information about the preferences of the agents and their groups, our goal is to decide an alternative to approximately minimize an objective function that takes the groups of agents into account. We consider two natural group-fair objectives known as Max-of-Avg and Avg-of-Max which are different combinations of the max and the average cost in and out of the groups. We show tight bounds on the best possible distortion that can be achieved by various classes of mechanisms depending on the amount of information they have access to. In particular, we consider full-information group-oblivious mechanisms that do not know the groups but have access to the exact distances between agents and alternatives in the metric space, ordinal-information group-oblivious mechanisms that again do not know the groups but are given the ordinal preferences of the agents, and group-aware mechanisms that have full knowledge of the structure of the agent groups and also ordinal information about the metric space.