The geometric median of a tuple of vectors is the vector that minimizes the sum of Euclidean distances to the vectors of the tuple. Interestingly, the geometric median can also be viewed as the equilibrium of a process where each vector of the tuple pulls on a common decision point with a unitary force towards them, promoting the "one voter, one unit force" fairness principle. In this paper, we analyze the strategyproofness of the geometric median as a voting system. Assuming that voters want to minimize the Euclidean distance between their preferred vector and the outcome of the vote, we first prove that, in the general case, the geometric median is not even $\alpha$-strategyproof. However, in the limit of a large number of voters, assuming that voters' preferred vectors are drawn i.i.d. from a distribution of preferred vectors, we also prove that the geometric median is asymptotically $\alpha$-strategyproof. The bound $\alpha$ describes what a voter can gain (at most) by deviating from truthfulness. We show how to compute this bound as a function of the distribution followed by the vectors. We then generalize our results to the case where each voter actually cares more about some dimensions rather than others. Roughly, we show that, if some dimensions are more polarized and regarded as more important, then the geometric median becomes less strategyproof. Interestingly, we also show how the skewed geometric medians can be used to improve strategyproofness. Nevertheless, if voters care differently about different dimensions, we prove that no skewed geometric median can achieve strategyproofness for all of them. Overall, our results provide insight into the extent to which the (skewed) geometric median is a suitable approach to aggregate high-dimensional disagreements.

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关于几何中位数的策略证明

向量元组的几何中值是最小化到元组向量的欧几里得距离之和的向量。有趣的是，几何中位数也可以被视为一个过程的平衡，其中元组的每个向量都以统一的力量拉向一个共同的决策点，促进“一个选民，一个单位力量”的公平原则。在本文中，我们分析了几何中位数作为投票系统的策略性。假设选民想要最小化他们的首选向量和投票结果之间的欧几里得距离，我们首先证明，在一般情况下，几何中位数甚至不是 $\alpha$-strategyproof。然而，在大量选民的限制下，假设选民的偏好向量是从偏好向量的分布中抽取的，我们还证明了几何中位数渐近 $\alpha$-strategyproof。界限 $\alpha$ 描述了选民通过偏离真实性可以获得（最多）什么。我们展示了如何将这个界限计算为向量跟随的分布的函数。然后我们将我们的结果推广到每个选民实际上更关心某些维度而不是其他维度的情况。粗略地，我们表明，如果某些维度更加两极分化并被认为更重要，那么几何中位数变得不那么具有策略性。有趣的是，我们还展示了如何使用倾斜的几何中位数来提高策略性。尽管如此，如果选民对不同维度的关注程度不同，我们证明没有任何倾斜的几何中位数可以实现所有维度的策略证明。全面的，