  In this study, Nash and Stackelberg equilibria of single-stage and
multi-stage quadratic signaling games between an encoder and a decoder are
investigated. In the considered setup, the objective functions of the encoder
and the decoder are misaligned, there is a noisy channel between the encoder
and the decoder, the encoder has a soft power constraint, and the decoder has
also noisy observation of the source to be estimated. We show that there exist
only linear encoding and decoding strategies at the Stackelberg equilibrium,
and derive the equilibrium strategies and costs. Regarding the Nash
equilibrium, we explicitly characterize affine equilibria for the single-stage
setup and show that the optimal encoder (resp. decoder) is affine for an affine
decoder (resp. encoder) for the multi-stage setup. For the decoder side,
between the information coming from the encoder and noisy observation of the
source, our results describe what should be the combining ratio of these two
channels. Regarding the encoder, we derive the conditions under which it is
meaningful to transmit a message.
