  While users claim to be concerned about privacy, often they do little to
protect their privacy in their online actions. One prominent explanation for
this "privacy paradox" is that when an individual shares her data, it is not
just her privacy that is compromised; the privacy of other individuals with
correlated data is also compromised. This information leakage encourages
oversharing of data and significantly impacts the incentives of individuals in
online platforms. In this paper, we study the design of mechanisms for data
acquisition in settings with information leakage and verifiable data. We design
an incentive compatible mechanism that optimizes the worst-case trade-off
between bias and variance of the estimation subject to a budget constraint,
where the worst-case is over the unknown correlation between costs and data.
Additionally, we characterize the structure of the optimal mechanism in closed
form and study monotonicity and non-monotonicity properties of the marketplace.
