“”“We study information design in games with a continuum of actions such that the players’ payoffs are concave in their own actions. A designer chooses an information structure—-a joint distribution of a state and a private signal of each player—-and evaluates it according to the designer’s expected payoff under the equilibrium play in the induced Bayesian game. We show an information structure is designer optimal whenever it induces the equilibrium play that can be implemented by an incentive contract in an auxiliary principal-agent problem with a single agent who observes the state and controls all actions.

We use this result to characterize optimal information structures in a variety of settings, including price competition, first-order Bayesian persuasion, and venture capital fundraising. If the state is normally distributed and the payoffs are quadratic, then in many cases Gaussian information structures are optimal. Fully informing a subset of players can also be optimal and robustly so, for all state distributions.”“”