Information design in concave games

We study information design in games with a continuum of actions such that the payoff of each player is concave in his action. A designer chooses an information structure--a joint distribution of a state and a private signal of each player. The information structure induces a Bayesian game and is evaluated according to the expected designer's payoff under the equilibrium play. We develop a method that allows to find an optimal information structure, one that cannot be outperformed by any other information structure, however complex. To do so, we exploit the property that each player's incentive is summarized by his marginal payoff. We show that an information structure is optimal whenever the induced strategies can be implemented by an incentive contract in a principal-agent problem that incorporates the players' marginal payoffs. We use this result to establish the optimality of Gaussian information structures in the settings with quadratic payoffs and a multivariate normally-distributed state. We analyze the details of optimal structures in a differentiated Bertrand competition and in a prediction game.