Optimal disclosure of information to privately informed agents

We study information design with multiple privately informed agents who interact in a game. Each agent's utility is linear in a real-valued state. We show that there always exists an optimal mechanism which is laminar partitional and bound its ``complexity''. For each type profile, such a mechanism partitions the state space and recommends the same action profile within a partition element. Furthermore, the convex hulls of any two partition elements are such that either one contains the other or they have an empty intersection. We highlight the value of screening: the ratio of the optimal and the best payoff without screening can be equal to the number of types. Along the way, we shed light on the solutions to optimization problems over distributions subject to a mean-preserving contraction constraint and additional side constraints, which might be of independent interest.