  We consider communications between an encoder and a decoder, viewed as two
decision makers, which have subjective beliefs on the probabilistic model of
the source distribution. Even though the decision makers employ the same cost
function, induced expected costs are different from the perspective of the
encoder and decoder due to their subjective probabilistic beliefs, which
requires a game theoretic treatment. Depending on the commitment nature of the
encoder to its policies, we analyze this signaling game problem under Nash and
Stackelberg equilibrium concepts. In particular, we consider a communication
scenario through a Gaussian noise channel with a power constrained encoder. We
show that the Stackelberg equilibrium cost of the encoder is upper semi
continuous, under the Wasserstein metric, as encoder's prior approaches to
decoder's prior and in the particular case of Gaussian subjective priors it is
also lower semi continuous, which proves the robustness of the equilibrium
around the team setup. We further prove that the optimality of affine policies
for Gaussian signaling under Stackelberg equilibria breaks down due to the
presence of prior mismatch. We also investigate the informativeness of
Stackelberg equilibria under affine policy restriction when there is prior
mismatch and show that under certain conditions the equilibria become
non-informative, that is information transmission ceases to exist. For the Nash
setup, we provide necessary and sufficient conditions under which there exist
informative affine Nash equilibria. Furthermore, we show that there exist fully
informative Nash and Stackelberg equilibria for the cheap talk problem (i.e.,
no additive noise term and no power constraint at the encoder) as in the team
theoretic setup under an absolute continuity condition.
