  This paper addresses information design with linear best responses of agents,
quadratic objective functions of an information designer, and a payoff state
distributed according to a Gaussian distribution. We formulate the problem as
semidefinite programming (SDP) and use the duality principle to characterize an
optimal information structure. There exists a Gaussian information structure
that is optimal among all information structures. A necessary and sufficient
condition for optimality is that the realizations of the induced action profile
and a state satisfy linear constraints derived from the primal and dual SDP. As
a result, an observed action profile typically reveals the true state even if
individual agents have only partial knowledge. In symmetric network games, an
optimal information structure inherits this symmetry, which facilitates the
computation of an optimal information structure.
