  We study collaborative normal mean estimation, where $m$ strategic agents
collect i.i.d samples from a normal distribution $\mathcal{N}(\mu, \sigma^2)$
at a cost. They all wish to estimate the mean $\mu$. By sharing data with each
other, agents can obtain better estimates while keeping the cost of data
collection small. To facilitate this collaboration, we wish to design
mechanisms that encourage agents to collect a sufficient amount of data and
share it truthfully, so that they are all better off than working alone. In
naive mechanisms, such as simply pooling and sharing all the data, an
individual agent might find it beneficial to under-collect and/or fabricate
data, which can lead to poor social outcomes. We design a novel mechanism that
overcomes these challenges via two key techniques: first, when sharing the
others' data with an agent, the mechanism corrupts this dataset proportional to
how much the data reported by the agent differs from the others; second, we
design minimax optimal estimators for the corrupted dataset. Our mechanism,
which is Nash incentive compatible and individually rational, achieves a social
penalty (sum of all agents' estimation errors and data collection costs) that
is at most a factor 2 of the global minimum. When applied to high dimensional
(non-Gaussian) distributions with bounded variance, this mechanism retains
these three properties, but with slightly weaker results. Finally, in two
special cases where we restrict the strategy space of the agents, we design
mechanisms that essentially achieve the global minimum.
