We introduce a privacy-signaling game problem in which a transmitter with privacy concerns observes a pair of correlated random variables which are modeled as jointly Gaussian. The transmitter aims to hide one of these random variables and convey the other one whereas the objective of the receiver is to accurately estimate both of the random variables. 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 for scalar sources a payoff dominant Nash equilibrium among all admissible policies is attained by affine policies. We prove that these Nash equilibria coincide with the Stackelberg equilibria. We also show that there always exists an informative Stackelberg equilibrium for the multidimensional parameter setting. We revisit the information bottleneck problem within our Stackelberg framework where under the information bottleneck setup the sender observes only one of the parameters. We characterize the Stackelberg equilibria under certain conditions and when these conditions are not met we establish the existence of informative equilibria.
We introduce a privacy-signaling game problem in which a transmitter with privacy concerns observes a pair of correlated random variables which are modeled as jointly Gaussian. The transmitter aims to hide one of these random variables and convey the other one whereas the objective of the receiver is to accurately estimate both of the random variables. 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 for scalar sources a payoff dominant Nash equilibrium among all admissible policies is attained by affine policies. We prove that these Nash equilibria coincide with the Stackelberg equilibria. We also show that there always exists an informative Stackelberg equilibrium for the multidimensional parameter setting. We revisit the information bottleneck problem within our Stackelberg framework where under the information bottleneck setup the sender observes only one of the parameters. We characterize the Stackelberg equilibria under certain conditions and when these conditions are not met we establish the existence of informative equilibria.
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.
We introduce a privacy-signaling game problem in which a transmitter with privacy concerns observes a pair of correlated random vectors which are modeled as jointly Gaussian. The transmitter 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 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 parameters 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.
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.
Simultaneous (Nash) and sequential (Stackelberg) equilibria of two-player dynamic quadratic cheap talk and signaling game problems are investigated under a perfect Bayesian formulation. For the dynamic scalar and multi-dimensional cheap talk, the Nash equilibrium cannot be fully revealing whereas the Stackelberg equilibrium is always fully revealing. Further, the final state Nash equilibria have to be essentially quantized when the source is scalar and has a density, and non-revealing for the multi-dimensional case. In the dynamic signaling game where the transmission of a Gauss-Markov source over a memoryless Gaussian channel is considered, affine policies constitute an invariant subspace under best response maps for both scalar and multi-dimensional sources under Nash equilibria; however, the Stackelberg equilibrium policies are always linear for scalar sources but may be non-linear for multi-dimensional sources. Further, under the Stackelberg setup, the conditions under which the equilibrium is non-informative are derived for scalar sources.
This paper studies the decentralized quadratic cheap talk and signaling game problems when an encoder and a decoder, viewed as two decision makers, have misaligned objective functions. The main contributions of this study are the extension of Crawford and Sobel's cheap talk formulation to multi-dimensional sources and to noisy channel setups. We consider both (simultaneous) Nash equilibria and (sequential) Stackelberg equilibria. We show that for arbitrary scalar sources, in the presence of misalignment, the quantized nature of all equilibrium policies holds for Nash equilibria in the sense that all Nash equilibria are equivalent to those achieved by quantized encoder policies. On the other hand, all Stackelberg equilibria policies are fully informative. For multi-dimensional setups, unlike the scalar case, Nash equilibrium policies may be of non-quantized nature, and even linear. In the noisy setup, a Gaussian source is to be transmitted over an additive Gaussian channel. The goals of the encoder and the decoder are misaligned by a bias term and encoder's cost also includes a penalty term on signal power. Conditions for the existence of affine Nash equilibria as well as general informative equilibria are presented. For the noisy setup, the only Stackelberg equilibrium is the linear equilibrium when the variables are scalar. Our findings provide further conditions on when affine policies may be optimal in decentralized multi-criteria control problems and lead to conditions for the presence of active information transmission in strategic environments.
In the classical communication setting, multiple senders having access to the same source of information and transmitting it over channel(s) to a receiver, in general, leads to a decrease in estimation error at the receiver. However, if the objectives of the information providers are different from that of the estimator, this might result in interesting strategic interactions. In this work, we consider a hierarchical signaling game between two senders (information designers) and a single receiver (decision maker) each having their own, possibly misaligned, objectives. The senders lead the game by committing to individual information disclosure policies simultaneously, within the framework of a Nash game among themselves. This is followed by the receiver's action decision. With Gaussian information structure and quadratic objectives (which depend on underlying state and receiver's action) for all the players, we show that in general the equilibrium is not unique. While we show that full revelation of the state is always an equilibrium, we propose an algorithm to achieve non trivial equilibria. Through simulations we show that misalignment between senders' objectives is beneficial for the receiver.