On multi-dimensional and noisy quadratic signaling games and affine equilibria

This study investigates extensions of the quadratic cheap talk and signaling game problem, which has been introduced in the economics literature. Two main contributions of this study are the extension of Crawford and Sobel´s cheap talk formulation to multi-dimensional sources, and the extension to noisy channel setups as a signaling game problem. We show that, in the presence of misalignment, the quantized nature of all equilibrium policies holds for any scalar random source. It is shown that for multi-dimensional setups, unlike the scalar case, 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 power term scaled by a multiplier. Conditions for the existence of affine equilibrium policies as well as general informative equilibria are presented for both the scalar and multi-dimensional setups.