  We study the value of data privacy in a game-theoretic model of trading
private data, where a data collector purchases private data from strategic data
subjects (individuals) through an incentive mechanism. The private data of each
individual represents her knowledge about an underlying state, which is the
information that the data collector desires to learn. Different from most of
the existing work on privacy-aware surveys, our model does not assume the data
collector to be trustworthy. Then, an individual takes full control of its own
data privacy and reports only a privacy-preserving version of her data.
  In this paper, the value of $\epsilon$ units of privacy is measured by the
minimum payment of all nonnegative payment mechanisms, under which an
individual's best response at a Nash equilibrium is to report the data with a
privacy level of $\epsilon$. The higher $\epsilon$ is, the less private the
reported data is. We derive lower and upper bounds on the value of privacy
which are asymptotically tight as the number of data subjects becomes large.
Specifically, the lower bound assures that it is impossible to use less amount
of payment to buy $\epsilon$ units of privacy, and the upper bound is given by
an achievable payment mechanism that we designed. Based on these fundamental
limits, we further derive lower and upper bounds on the minimum total payment
for the data collector to achieve a given learning accuracy target, and show
that the total payment of the designed mechanism is at most one individual's
payment away from the minimum.
