Constrained Mediation: Bayesian Implementability of Joint Posteriors

2025-10-23

AI Summary

This paper studies how an information-constrained mediator can induce agents with private information to hold a desired joint posterior belief by designing public signals. Method: We propose a graph-theoretic modeling framework wherein states are vertices, information sets correspond to edges, and likelihood-ratio functions are embedded on edges to encode posterior structure. Building on this, we derive necessary and sufficient conditions for Bayesian implementability—namely, internal consistency and external consistency. Contribution/Results: Our characterization fully identifies the set of jointly attainable posterior distributions, revealing its fundamental equivalence to the set of Blackwell experiments. It precisely delineates the feasibility boundary for a single mediator to induce multiple posteriors and provides a computationally tractable, structured analytical tool for information design. The framework unifies conceptual insights with algorithmic utility, enabling rigorous analysis of signal design under informational constraints.

Technology Category

• Game Theory and Economic Paradigms: Mechanism Design
• Multiagent Systems: Mechanism Design
• Reasoning under Uncertainty: Graphical Models

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• User Modeling, Personalization and Recommendation: Psychology-informed user models and recommender systems

Abstract

We examine information structures in settings with privately informed agents and an informationally constrained mediator who supplies additional public signals. Our focus is on characterizing the set of posteriors that the mediator can induce. To this end, we employ a graph-theoretic framework: states are represented as vertices, information sets correspond to edges, and a likelihood ratio function on edges encodes the posterior beliefs. Within this framework, we derive necessary and sufficient conditions, internal and external consistency, for the rationalization of posteriors. Finally, we identify conditions under which a single mediator can implement multiple posteriors, effectively serving as a generator of Blackwell experiments.

Problem

• Characterizing implementable posteriors with constrained mediators
• Establishing graph-theoretic conditions for posterior rationalization
• Identifying when single mediators generate multiple Blackwell experiments

Innovation

• Graph-theoretic framework models states as vertices
• Information sets correspond to edges with likelihood ratios
• Mediator implements multiple posteriors via consistency conditions