Information Design with Elicitation and Strategic Coordination

We study linear-quadratic games of incomplete information with Gaussian uncertainty, where each player's payoff depends on a privately observed type and a common state. The designer observes the state, elicits types, and sells action recommendations. We characterize all implementable mechanisms with Gaussian joint distributions of actions and fundamentals, and identify the players-optimal, consumer-optimal, and revenue-maximizing designs. In games of strategic complements (substitutes), these optimal mechanisms maximally correlate (anticorrelate) players' actions. When type uncertainty is large, recommendations become deterministic linear functions of the state and reports, but remain only partially revealing.