Abstract-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.

This paper considers dynamic (multi-stage) signaling games involving an encoder and a decoder who have subjective models on the cost functions. We consider both Nash (simultaneous-move) and Stackelberg (leaderfollower) equilibria of dynamic signaling games under quadratic criteria. For the multi-stage scalar cheap talk, we show that the final stage equilibrium is always quantized and under further conditions the equilibria for all time stages must be quantized. In contrast, the Stackelberg equilibria are always fully revealing. In the multistage 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 Nash equilibria; whereas the Stackelberg equilibria always admit linear policies for scalar sources but such policies may be nonlinear for multi-dimensional sources. We obtain an explicit recursion for optimal linear encoding policies for multi-dimensional sources, and derive conditions under which Stackelberg equilibria are informative. * (decoder), who generates his M-valued optimal decision U upon receiving X. The policies of the encoder and decoder are assumed to be deterministic; i.e., x = γ e (m) and u = γ d (x) = γ d (γ e (m)). Let c e (m, u) and c d (m, u) denote the cost functions of the encoder and the decoder, respectively, when the action u is taken for the corresponding message m. Then, given the encoding and decoding policies, the encoder's induced expected cost is J e γ e , γ d = E [c e (m, u)], whereas, the decoder's induced expected cost is J d γ e , γ d = E c d (m, u) . If the transmitted signal x is also an explicit part of the cost functions c e and/or c d , then the communication between the players is not costless and the formulation turns into a signaling game problem. Such problems are studied under the tools and concepts provided by game theory since the goals are not aligned. Although the encoder and decoder act sequentially in the game as described above, how and when the decisions are made and the nature of the commitments to the announced policies significantly affect the analysis of the equilibrium structure. Here, two different types of equilibria are investigated: the Nash equilibrium, in which the encoder and the decoder make simultaneous decisions, and the Stackelberg equilibrium, in which the encoder and the decoder make sequential decisions where the encoder is the leader and the decoder is the follower 1 . In this paper, the terms Nash game and the simultaneous-move game 2 will be used interchangeably, and similarly, the Stackelberg game and the leader-follower game will be used interchangeably.In the simultaneous-move game, the encoder and the decoder announce their policies at the same time, and a pair of policies (γ * ,e , γ * ,d ) is said to be a Nash equilibrium [4] if where Γ e and Γ d are the sets of all deterministic (and Borel measurable) functions from M to X and from X to M, respectively. As observed from the definition (1), under the Na...

We investigate the equilibrium behavior for the decentralized cheap talk problem for real random variables and quadratic cost criteria in which an encoder and a decoder have misaligned objective functions. In prior work, it has been shown that the number of bins in any equilibrium has to be countable, generalizing a classical result due to Crawford and Sobel who considered sources with density supported on [0, 1]. In this paper, we first refine this result in the context of log-concave sources. For sources with two-sided unbounded support, we prove that, for any finite number of bins, there exists a unique equilibrium. In contrast, for sources with semi-unbounded support, there may be a finite upper bound on the number of bins in equilibrium depending on certain conditions stated explicitly. Moreover, we prove that for log-concave sources, the expected costs of the encoder and the decoder in equilibrium decrease as the number of bins increases. Furthermore, for strictly log-concave sources with two-sided unbounded support, we prove convergence to the unique equilibrium under best response dynamics which starts with a given number of bins, making a connection with the classical theory of optimal quantization and convergence results of Lloyd's method. In addition, we consider more general sources which satisfy certain assumptions on the tail(s) of the distribution and we show that there exist equilibria with infinitely many bins for sources with two-sided unbounded support. Further explicit characterizations are provided for sources with exponential, Gaussian, and compactly-supported probability distributions.

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In this study, dynamic and repeated quadratic cheap talk and signaling game problems are investigated. These involve encoder and decoders with mismatched performance objectives, where the encoder has a bias term in the quadratic cost functional. We consider both Nash equilibria and Stackelberg equilibria as our solution concepts, under a perfect Bayesian formulation. These two lead to drastically different characteristics for the equilibria. For the cheap talk problem under Nash equilibria, we show that fully revealing equilibria cannot exist and the final state equilibria have to be quantized for a large class of source models; whereas, for the Stackelberg case, the equilibria must be fully revealing regardless of the source model. In the dynamic signaling game where the transmission of a Gaussian source over a Gaussian channel is considered, the equilibrium policies are always linear for scalar sources under Stackelberg equilibria, and affine policies constitute an invariant subspace under best response maps for Nash equilibria.