  The geometric median, an instrumental component of the secure machine
learning toolbox, is known to be effective when robustly aggregating models (or
gradients), gathered from potentially malicious (or strategic) users. What is
less known is the extent to which the geometric median incentivizes dishonest
behaviors. This paper addresses this fundamental question by quantifying its
strategyproofness. While we observe that the geometric median is not even
approximately strategyproof, we prove that it is asymptotically
$\alpha$-strategyproof: when the number of users is large enough, a user that
misbehaves can gain at most a multiplicative factor $\alpha$, which we compute
as a function of the distribution followed by the users. We then generalize our
results to the case where users actually care more about specific dimensions,
determining how this impacts $\alpha$. We also show how the skewed geometric
medians can be used to improve strategyproofness.
