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场景，其中智能体和备选方案由度量空间中的点表示，智能体对某个备选方案的成本即为空间中对应点之间的距离。目标是在仅给定智能体偏好有限信息的情况下，选择单一备选方案以（近似）最小化社会成本（所有智能体的成本之和）或任何智能体的最大成本。先前研究表明，当仅能获取智能体的序数偏好时，即使已知备选方案在度量空间中的距离，所能实现的最佳失真度上界为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的上界。我们证明，通过结合智能体的序数偏好、备选方案之间的距离，以及每个智能体包含其成本在合适因子（相对于其最偏好备选方案的成本）内的所有备选方案的阈值批准集，可以实现失真度1+√2。我们证明该界是一般度量空间中任何确定性机制所能达到的最优值，并为直线度量这一基本情形提供了改进的界。