On the Structure of Equilibrium Strategies in Dynamic Gaussian Signaling Games

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.Comment: will appear in IEEE Multi-Conference on Systems and Control 201

Constrained Communication with Multiple Agents: Anonymity, Equal Treatment, and Public Good Provision

This paper studies information transmission subject to anonymity requirements and communication in public good provision without transfers. The structure of informative equilibria under anonymity or in public good provision can di¤er substantially from that of direct one-to-one communication, and in particular we distinguish i) informational distortion caused by the intrinsic divergence of preferences between the decision maker and each agent; and ii) informational distortion caused by the decision maker's weak response to each agent's message due to the equal treatment of all agents that results from anonymity or the nature of public goods. We examine the interaction between these two types of distortion and demonstrate that they may partly offset one another. Information transmission and welfare can be enhanced by introducing the second type of distortion through anonymity when the first type of distortion is severe. In public good provision where the intrinsic preference divergence between the utilitarian decision maker and each agent is absent, as the number of agents becomes larger the quality of communication diminishes and informative equilibria converge to the one that can be played by letting each agent report a binary message (e.g. "yes" or "no") even if their preferences and the decision are continuous.Cheap Talk, Anonymous Communication, Equal Treatment, Public Good Provision.

Strategic information transmission and stochastic orders

I develop new results on uniqueness and comparative statics of equilibria in the Crawford and Sobel (1982) strategic information transmission game. For a class of utility functions, I demonstrate that logconcavity of the density implies uniqueness of equilibria inducing a given number of Receiver actions. I provide comparative statics results with respect to the distribution of types for distributions that are comparable in the likelihood ratio order, implying, e.g., that advice from a better informed Sender induces the Receiver to choose actions that are more spread out

Quadratic Multi-Dimensional Signaling Games and Affine Equilibria

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.Comment: 15 pages, 4 figure

Noisy Talk

We examine the possibilities for communication between agents with divergent preferences in a noisy environment. Taking Crawford and Sobel’s [4] (noiseless) communication game as a reference point, we study a model in which there is a probability e ? (0, 1) that the received message is a random draw from the entire message space, independent of the actual message sent by the sender. Just as in the CS model, we find that all equilibria are interval partitional; but unlike in CS, coding (the proportion of the message space used by any given set of types) is of critical importance. Via the appropriate coding scheme, one can construct equilibria that induce finitely many, a countable infinity or even an uncountable infinity of actions. Furthermore, for a given number of actions, there is typically a continuum of equilibria that induce that many actions. Surprisingly, the possibility of error can improve the prospects for communication. We show that for small noise levels there is a simple class of equilibria that are almost always welfare superior to the best CS equilibrium. There exists an optimal noise level for which these equilibria achieve the efficiency bound for general communication devices. Furthermore, for a range of biases introducing any amount of noise can be beneficial.Communication, information transmission, cheap talk, noise.

Noisy talk

We investigate strategic information transmission with communication error, or noise. Our main finding is that adding noise can improve welfare. With quadratic preferences and a uniform type distribution, welfare can be raised for almost every bias level by introducing a sufficiently small amount of noise. Furthermore, there exists a level of noise that makes it possible to achieve the best payoff that can be obtained by means of any communication device. As in the model without noise, equilibria are interval partitional; with noise, however, coding (the measure of the message space used by each interval of the equilibrium partition of the type space) becomes critically important.Communication, information transmission, cheap talk, noise