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Optimal Hierarchical Signaling for Quadratic Cost Measures and General Distributions: A Copositive Program Characterization.

Muhammed O. Sayin, Tamer Başar

Published Jul 22, 2019• 1 citation
OpenAlex

Abstract

In this paper, we address the problem of optimal hierarchical signaling between a sender and a receiver for a general class of square integrable multivariate distributions. The receiver seeks to learn a certain information of interest that is known to the sender while the sender seeks to induce the receiver to perceive that information as a certain private information. For the setting where the players have quadratic cost measures, we analyze the Stackelberg equilibrium, where the sender leads the game by committing his/her strategies beforehand. We show that when the underlying state space is finite, the optimal signaling strategies can be computed through an equivalent linear optimization problem over the cone of completely positive matrices. The equivalence established enables us to use the existing computational tools to solve this class of cone programs approximately with any error rate. For continuous distributions, we also analyze the error of approximation, if the optimal signaling strategy is computed for a discretized version obtained through a quantization scheme, and we provide an upper bound in terms of the quantization error.

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In this paper, we address the problem of optimal hierarchical signaling between a sender and a receiver for a general class of square integrable multivariate distributions. The receiver seeks to learn a certain information of interest that is known to the sender while the sender seeks to induce the receiver to perceive that information as a certain private information. For the setting where the players have quadratic cost measures, we analyze the Stackelberg equilibrium, where the sender leads the game by committing his/her strategies beforehand. We show that when the underlying state space is finite, the optimal signaling strategies can be computed through an equivalent linear optimization problem over the cone of completely positive matrices. The equivalence established enables us to use the existing computational tools to solve this class of cone programs approximately with any error rate. For continuous distributions, we also analyze the error of approximation, if the optimal signaling strategy is computed for a discretized version obtained through a quantization scheme, and we provide an upper bound in terms of the quantization error.
In this article, we address Bayesian persuasion between a sender and a receiver with state-dependent quadratic cost measures for general classes of distributions. The receiver seeks to make mean-square-error estimate of a state based on a signal sent by the sender while the sender signals strategically in order to control the receiver’s estimate in a certain way. Such a scheme could model, e.g., deception and privacy, problems in multiagent systems. Existing solution concepts are not viable since here the receiver has continuous action space. We show that for finite state spaces, optimal signaling strategies can be computed through an equivalent linear optimization problem over the cone of completely positive matrices. We then establish its strong duality to a copositive program. To exemplify the effectiveness of this equivalence result, we adopt sequential polyhedral approximation of completely positive cones and analyze its performance numerically. We also quantify the approximation error for a quantized version of a continuous distribution and show that a semi-definite program relaxation of the equivalent problem could be a benchmark lower bound for the sender’s cost for large state spaces.
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.
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.
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 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.
This paper analyzes a finite horizon dynamic signaling game motivated by the well-known strategic information transmission problems in economics. The mathematical model involves information transmission between two agents, a sender who observes two Gaussian processes, state and bias, and a receiver who takes an action based on the received message from the sender. The players incur quadratic instantaneous costs as functions of the state, bias and action variables. Our particular focus is on the Stackelberg equilibrium, which corresponds to information disclosure and Bayesian persuasion problems in economics. Prior work solved the static game, and showed that the Stackelberg equilibrium is achieved by pure strategies that are linear functions of the state and the bias variables. The main focus of this work is on the dynamic (multi-stage) setting, where we show that the existence of a pure strategy Stackelberg equilibrium, within the set of linear strategies, depends on the problem parameters. Surprisingly, for most problem parameters, a pure linear strategy does not achieve the Stackelberg equilibrium which implies the existence of a trade-off between exploiting and revealing information, which was also encountered in several other asymmetric information games.