  We investigate a privacy-signaling game problem in which a sender with
privacy concerns observes a pair of correlated random vectors which are modeled
as jointly Gaussian. The sender aims to hide one of these random vectors and
convey the other one whereas the objective of the receiver is to accurately
estimate both of the random vectors. We analyze these conflicting objectives in
a game theoretic framework with quadratic costs where depending on the
commitment conditions (of the sender), we consider Nash or Stackelberg
(Bayesian persuasion) equilibria. We show that a payoff dominant Nash
equilibrium among all admissible policies is attained by a set of explicitly
characterized linear policies. We also show that a payoff dominant Nash
equilibrium coincides with a Stackelberg equilibrium. We formulate the
information bottleneck problem within our Stackelberg framework under the mean
squared error distortion criterion where the information bottleneck setup has a
further restriction that only one of the random variables is observed at the
sender. We show that this MMSE Gaussian Information Bottleneck Problem admits a
linear solution which is explicitly characterized in the paper. We provide
explicit conditions on when the optimal solutions, or equilibrium solutions in
the Nash setup, are informative or noninformative.
