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 group-oblivious full-information mechanisms that do not know the groups but have access to the exact distances between agents and alternatives in the metric space, group-oblivious ordinal-information 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.

翻译：我们考虑一个投票问题，其中一组智能体对一组备选方案具有度量偏好，并且被划分为互不相交的群体。给定智能体的偏好及其所属群体的信息，我们的目标是选择一个备选方案，以近似最小化一个考虑智能体群体分组的客观函数。我们研究了两个自然的分组公平目标，即最大均值与均值最大，它们是组内与组间最大成本与平均成本的不同组合。我们展示了根据机制可获取信息量，各类机制所能实现的最佳失真的紧界。具体而言，我们考虑了三种机制：不掌握分组信息但能获取度量空间中智能体与备选方案精确距离的群体不知情全信息机制；同样不掌握分组信息但仅获知智能体序数偏好的群体不知情序数信息机制；以及完全掌握智能体分组结构并同时具备度量空间序数信息的群体知情机制。