Maximizing Social Welfare and Agreement via Information Design in Linear-Quadratic-Gaussian Games

Information design in an incomplete information game involves a designer that aims to influence players' actions through signals generated from a designed probability distribution to optimize its objective function. For quadratic design objective functions, if the players have quadratic payoffs that depend on the players' actions and an unknown payoff-relevant state, and signals on the state that follow a Gaussian distribution conditional on the state realization, the information design problem is a semidefinite program (SDP) (Ui, 2020). In this note, we seek to characterize the optimal information design analytically by leveraging the SDP formulation, when the design objective is to maximize social welfare or the agreement among players' action. We show that full information disclosure maximizes social welfare when there is a common payoff state, the payoff dependencies among players' actions are homogeneous, or when the signals are public. When the objective is to maximize the agreement among players' actions, not revealing any information is optimal. When the objective is a weighted combination of social welfare and agreement terms, we establish a threshold weight below which full information disclosure is optimal under public signals for games with homogeneous payoffs. Numerical results corroborate the analytical results, and identify partial information disclosure structures that are optimal.