Quadratic Privacy-Signaling Games and Payoff Dominant Equilibria

Abstract

We consider a privacy-signaling game problem in which a transmitter with privacy concerns and a receiver, which does not pay attention to these privacy concerns, communicate. In this communication scenario, the transmitter observes a pair of correlated random variables which are modeled as jointly Gaussian. The transmitter constructs its message based on these random variables with the aim to hide one of them and convey the other one. In contrast, the objective of the receiver is to accurately estimate both of the random variables so as to gather as much information as possible. These conflicting objectives are analyzed in a game theoretic framework where depending on the commitment conditions (of the sender), we consider Nash or Stackelberg equilibria. We show that a payoff dominant (i.e., most desirable for both players) Nash equilibrium is attained by affine policies and we explicitly characterize these policies. In addition, the strategies at the characterized Nash equilibrium is shown to form also a Stackelberg equilibrium. Furthermore, we show that there always exists an informative Stackelberg equilibrium for the multidimensional parameter setup. We also revisit 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 parameters is observed at the sender. We fully characterize the Stackelberg equilibria under certain conditions and when these conditions are not met we establish the existence of informative equilibria.

Privacy signaling games with binary alphabets

“…For discrete RVs they provide a numerical algorithm to find an equilibrium whereas for Gaussian RVs, a bound on the estimation error is provided and affine policies are shown to achieve this bound. In [9], a privacy-signaling game problem for the setup in Fig. 1 is considered in which a transmitter with privacy concerns observes a pair of correlated random vectors which are modeled as jointly Gaussian.…”

“…Remark 3.2: (Connection to similar work) In [9], a similar setup is considered in which the random sources are jointly Gaussian, and the squared error is utilized as privacy and information exchange metric. Similar to our result, it is shown that Stackelberg and payoff dominant Nash equilibria coincide.…”

“…From the information structure of the stochastic encoder and decoder in ( 6)-( 8) the constraint sets in (9) and (10) are clear. Hence it suffices to characterize only the mutual information and the loss function.…”

“…In this paper, we consider the scenario illustrated in Fig. 1 that was first introduced in [9]. In this setup, a transmitter e ( , ) g x y ( , ) x y z ( )…”

“…For this setup, we study the interactions between the transmitter and the receiver whose objectives are not-aligned due to the privacy concerns of the transmitter in a game theoretic framework. However, in contrast to [9] that deals with jointly Gaussian random vectors and linear policies, here we deal with finite alphabets and consider different objectives for our transmitter and receiver.…”