  We study a collaborative learning problem where $m$ agents estimate a vector
$\mu\in\mathbb{R}^d$ by collecting samples from normal distributions, with each
agent $i$ incurring a cost $c_{i,k} \in (0, \infty]$ to sample from the
$k^{\text{th}}$ distribution $\mathcal{N}(\mu_k, \sigma^2)$. Instead of working
on their own, agents can collect data that is cheap to them, and share it with
others in exchange for data that is expensive or even inaccessible to them,
thereby simultaneously reducing data collection costs and estimation error.
However, when agents have different collection costs, we need to first decide
how to fairly divide the work of data collection so as to benefit all agents.
Moreover, in naive sharing protocols, strategic agents may under-collect and/or
fabricate data, leading to socially undesirable outcomes. Our mechanism
addresses these challenges by combining ideas from cooperative and
non-cooperative game theory. We use ideas from axiomatic bargaining to divide
the cost of data collection. Given such a solution, we develop a Nash
incentive-compatible (NIC) mechanism to enforce truthful reporting. We achieve
a $\mathcal{O}(\sqrt{m})$ approximation to the minimum social penalty (sum of
agent estimation errors and data collection costs) in the worst case, and a
$\mathcal{O}(1)$ approximation under favorable conditions. We complement this
with a hardness result, showing that $\Omega(\sqrt{m})$ is unavoidable in any
NIC mechanism.
