[Paper Review] Information Design with Elicitation and Strategic Coordination

Alessandro Bonatti, Munther A. Dahleh+1 more|arXiv (Cornell University)|Feb 23, 2023

Auction Theory and ApplicationsDecision Sciences3 citations

TL;DR

This paper studies optimal information design and mechanism design in symmetric n-player games with quadratic payoffs and Gaussian uncertainty, where a data provider sells signals to privately informed players to induce efficient coordination. The key result is that welfare-optimal mechanisms induce maximal positive correlation in actions for strategic complements and maximal negative correlation for strategic substitutes, with deterministic linear recommendations that depend on the state and types without full revelation.

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.

Motivation & Objective

• To study joint information and mechanism design in games with incomplete information, where players have private types and common uncertainty over a state of nature.
• To characterize implementable mechanisms that truthfully elicit types and recommend actions forming a Bayes Correlated Equilibrium.
• To derive welfare- and revenue-optimal mechanisms within a class of Gaussian mechanisms for symmetric games with quadratic payoffs.
• To understand how type elicitation and action recommendation interact in incentive-compatible designs.
• To analyze the structure of optimal signals and action recommendations in games of strategic complements and substitutes.

Proposed method

• The authors use the Revelation Principle to reduce the problem to designing a mechanism that truthfully elicits types and recommends actions forming a Bayes Correlated Equilibrium (BCE).
• They characterize all implementable joint distributions of state, types, and actions under Gaussian assumptions, focusing on multivariate normal distributions.
• The optimal mechanisms are derived by solving a constrained optimization problem over covariance parameters of the action and signal distributions.
• The solution involves deriving necessary and sufficient conditions for global incentive compatibility, accounting for double deviations in reporting and action choices.
• The analysis distinguishes between games of strategic complements (s > 0) and substitutes (s < 0), showing divergent optimal correlation patterns.
• The optimal mechanisms are shown to be deterministic conditional on (state, type reports), with action recommendations as linear functions of the state and types.

Experimental results

Research questions

• RQ1 What is the structure of the optimal mechanism that truthfully elicits types and recommends actions to achieve welfare maximization?
• RQ2 How does the optimal mechanism differ between games of strategic complements and strategic substitutes?
• RQ3 Can the type-elicitation and action-recommendation stages be separated in mechanism design, or do they interact in a way that invalidates standard BCE approaches?
• RQ4 What is the role of private types versus the common state in shaping the optimal signal structure?
• RQ5 How does the optimal mechanism compare to the complete-information Nash equilibrium in terms of action correlations and welfare outcomes?

Key findings

• For games of strategic complements, the welfare-optimal mechanism maximally correlates players’ actions by aligning recommendations with the common state and individual types.
• For games of strategic substitutes, the welfare-optimal mechanism maximally anticorrelates actions, reducing coordination among players despite shared state information.
• When the prior variance of types is sufficiently large, action recommendations become deterministic and linear in the state and types, without revealing full information.
• The optimal mechanism overweights players’ own types and underweights the common state relative to the complete-information Nash equilibrium, especially in games of complements and substitutes.
• The welfare-optimal mechanism is globally incentive compatible, with covariance signs between actions and signals matching the strategic structure (positive for complements, negative for substitutes).
• The optimal mechanism differs systematically from the complete-information outcome: |σa_iω^W| < |σa_iω^N| and |σa_iθ_i^W| > |σa_iθ_i^N|, indicating reduced reliance on the state and increased reliance on private types.