Information Design with Elicitation and Strategic Coordination

TL;DR

This paper develops a tractable Gaussian framework for information design in linear-quadratic games with private types and a common state, analyzing how a platform can elicit private information, signal fundamentals, and extract payments. It provides a complete characterization of implementable Gaussian joint distributions and derives optimal obedient mechanisms under welfare, revenue, and consumer objectives, showing that optimal designs coordinate actions via maximal conditional correlations, with determinism or randomness depending on uncertainty. The results yield precise prescriptions for how to weight private types versus common state in action recommendations, and reveal nuanced trade-offs along the Pareto frontier between firms and downstream consumers, including the counterintuitive possibility that platform market power can sometimes improve consumer welfare. The framework also extends to variations like endogenous participation, delegation, and asymmetric settings, offering policy insights for platform-mediated coordination and information pricing.

Abstract

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.

Figures (8)

• Figure 1: Symmetric mechanisms satisfying obedience are parametrized by points inside the ellipse \ref{['eq:ellipse']} and by the noise correlation $\rho\in[-\frac{1}{n-1},1]$. The mechanism is deterministic conditioned on $(\theta,\omega)$ iff its lies on \ref{['eq:ellipse']}. The ellipse circumscribes a rectangle formed by four deterministic mechanisms corresponding to “extremal” information structures (see the text).
• Figure 2: Symmetric and incentive compatible mechanisms must satisfy $rt\sigma_{a_i\theta_j}\geq 0$. The hatched area contains the obedient mechanisms that are not incentive compatible: these admit profitable double deviations for the players. At the boundary $\sigma_{a_i\theta_j}=0$, the red line segment $[\varnothing,\text{SO}]$ contains the obedient mechanisms that are implementable without transfers.
• Figure 3: Obedience ellipse \ref{['eq:ellipse']} and firms-optimal mechanisms (F) in the coordinate system $x=\sigma_{a_i\theta_j}/rt\sigma_{\theta_i}^2$, $y=\sigma_{a_i\omega}/s\sigma_{\omega}^2$. We focus on the quadrant $x\geq x_0$, $y\geq y_0$ containing \ref{['eq:ellipse']}'s top-right quarter as well as the stationarity curve $(x(\lambda),y(\lambda))$ parametrized by $\lambda\geq \lambda_{\rm min}$ (in green). On the left, the curve originates outside \ref{['eq:ellipse']}, and the firms-optimal mechanism is attained at the intersection with \ref{['eq:ellipse']}. On the right, the curve originates inside \ref{['eq:ellipse']} and the optimal mechanism is obtained for $\lambda=\lambda_{\rm min}$.
• Figure 4: Dual effect of type-action covariance on information rents ($t=1$, $\sigma_{\theta_i}^2=1)$
• Figure 5: Comparison of obedient mechanisms

Theorems & Definitions (61)

• Example 1: Cournot Competition
• Example 2: Bertrand Competition
• Example 3: Beauty Contest
• Definition 1: Incentive Compatibility
• Proposition 1
• Remark 1
• Proposition 2
• Proposition 3
• Proposition 4
• Remark 2