On the Strategyproofness of the Geometric Median

Abstract

The geometric median of a tuple of vectors is the vector that minimizes the sum of Euclidean distances to the vectors of the tuple. Also called the Fermat-Weber problem and often applied to facility location, the geometric median is also an appealing tool for voting in high dimension, which may be applied, e.g., to collaborative content moderation on social medias. 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 α-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 α-strategyproof. The bound α 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, and we determine how this impacts the strategyproofness bound α. 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.

Moral Machine or Tyranny of the Majority?

“…Median-based approaches: geometric medians and coordinate-wise medians. El-Mhamdi et al (2021) mentions that average-based mechanisms are susceptible to manipulation even by an individual. In response, they describe two types of median-based aggregation approaches that generalize to d dimensions: geometric medians and coordinate-wise medians.…”