We consider a social choice setting in which agents and alternatives are represented by points in a metric space, and the cost of an agent for an alternative is the distance between the corresponding points in the space. The goal is to choose a single alternative to (approximately) minimize the social cost (cost of all agents) or the maximum cost of any agent, when only limited information about the preferences of the agents is given. Previous work has shown that the best possible distortion one can hope to achieve is $3$ when access to the ordinal preferences of the agents is given, even when the distances between alternatives in the metric space are known. We improve upon this bound of $3$ by designing deterministic mechanisms that exploit a bit of cardinal information. We show that it is possible to achieve distortion $1+\sqrt{2}$ by using the ordinal preferences of the agents, the distances between alternatives, and a threshold approval set per agent that contains all alternatives for whom her cost is within an appropriately chosen factor of her cost for her most-preferred alternative. We show that this bound is the best possible for any deterministic mechanism in general metric spaces, and also provide improved bounds for the fundamental case of a line metric.

翻译：我们考虑一个社会选择环境，其中代理和备选方案由度量空间中的点表示，代理对备选方案的成本是空间中对应点之间的距离。目标是在仅知代理偏好的有限信息时，选择单个备选方案以（近似）最小化社会成本（所有代理的总成本）或任何代理的最大成本。先前研究表明，当获得代理的序数偏好信息时（即使已知度量空间中备选方案之间的距离），所能实现的最佳失真度仅为$3$。我们通过设计利用少量基数信息的确定性机制改进了这一$3$的界限。我们证明：通过使用代理的序数偏好、备选方案之间的距离，以及每个代理的阈值批准集（该集合包含其成本在相对于其最偏好备选方案成本适当缩放因子范围内的所有备选方案），可以实现失真度$1+\sqrt{2}$。我们证明这一界限对于一般度量空间中的任何确定性机制都是最优的，并为基础线度量情形提供了改进的界限。