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Bayesian Persuasion with Externalities: Exploiting Agent Types

Jonathan Shaki, Jiarui Gan, Sarit Kraus

TL;DR

We study Bayesian persuasion with externalities in a multi-agent setting where a principal signals the world state to agents and simultaneously correlates their actions. Agents are grouped into a small number of types to allow succinct representations, and three signaling channels—public, semi-private, and private—are analyzed. The paper introduces revelation-principle-style characterizations, blocking profiles, representative action vectors, and lottery policies, enabling polynomial-time optimization via linear programs when the maximum deviating set size $d$ is constant, while proving NP-hardness when $d$ is unbounded. It also develops semi-private signaling with Hall-type blocking-profile representations and extends the framework to private persuasion through lottery-based reductions. Overall, the work clarifies how externalities and coordination interact in Bayesian persuasion and provides practical, tractable methods for computing optimal policies under common structural constraints.

Abstract

We study a Bayesian persuasion problem with externalities. In this model, a principal sends signals to inform multiple agents about the state of the world. Simultaneously, due to the existence of externalities in the agents' utilities, the principal also acts as a correlation device to correlate the agents' actions. We consider the setting where the agents are categorized into a small number of types. Agents of the same type share identical utility functions and are treated equitably in the utility functions of both other agents and the principal. We study the problem of computing optimal signaling strategies for the principal, under three different types of signaling channels: public, private, and semi-private. Our results include revelation-principle-style characterizations of optimal signaling strategies, linear programming formulations, and analysis of in/tractability of the optimization problems. It is demonstrated that when the maximum number of deviating agents is bounded by a constant, our LP-based formulations compute optimal signaling strategies in polynomial time. Otherwise, the problems are NP-hard.

Bayesian Persuasion with Externalities: Exploiting Agent Types

TL;DR

We study Bayesian persuasion with externalities in a multi-agent setting where a principal signals the world state to agents and simultaneously correlates their actions. Agents are grouped into a small number of types to allow succinct representations, and three signaling channels—public, semi-private, and private—are analyzed. The paper introduces revelation-principle-style characterizations, blocking profiles, representative action vectors, and lottery policies, enabling polynomial-time optimization via linear programs when the maximum deviating set size is constant, while proving NP-hardness when is unbounded. It also develops semi-private signaling with Hall-type blocking-profile representations and extends the framework to private persuasion through lottery-based reductions. Overall, the work clarifies how externalities and coordination interact in Bayesian persuasion and provides practical, tractable methods for computing optimal policies under common structural constraints.

Abstract

We study a Bayesian persuasion problem with externalities. In this model, a principal sends signals to inform multiple agents about the state of the world. Simultaneously, due to the existence of externalities in the agents' utilities, the principal also acts as a correlation device to correlate the agents' actions. We consider the setting where the agents are categorized into a small number of types. Agents of the same type share identical utility functions and are treated equitably in the utility functions of both other agents and the principal. We study the problem of computing optimal signaling strategies for the principal, under three different types of signaling channels: public, private, and semi-private. Our results include revelation-principle-style characterizations of optimal signaling strategies, linear programming formulations, and analysis of in/tractability of the optimization problems. It is demonstrated that when the maximum number of deviating agents is bounded by a constant, our LP-based formulations compute optimal signaling strategies in polynomial time. Otherwise, the problems are NP-hard.

Paper Structure

This paper contains 32 sections, 26 theorems, 48 equations, 3 tables.

Table of Contents

  1. Introduction
  2. Contributions and Technical Novelties
  3. Motivating Examples
  4. Related Work
  5. The Problem
  6. The Principal's Policy
  7. Agents' Belief and Stability of the Policy
  8. The FDA Example
  9. Public Persuasion
  10. An Example
  11. Revelation Principle for Public Persuasion
  12. Computing an Optimal Public Policy
  13. Time Complexity
  14. Semi-private Persuasion
  15. Representing Semi-Private Blocking Profiles
  16. ...and 17 more sections

Key Result

Theorem 3.1

For any stable public policy $\sigma : \Omega \to \Delta({\mathcal{A}} \times G)$, there exists a stable public policy $\bar{\sigma} : \Omega \to \Delta(C)$ that yields as much utility for the principal as $\sigma$ does, where $C = \{ (\bar{{\mathbf{a}}}, \beta) : \bar{{\mathbf{a}}} \in {\bar{\mathc

Theorems & Definitions (50)

  • Definition 1: Stable policy
  • Definition 2: Representative
  • Definition 3: Blocking profile
  • Definition 4: Signature
  • Theorem 3.1
  • Theorem 3.2
  • Theorem 3.3
  • Lemma 4.0
  • Lemma 4.0
  • Definition 5: Semi-private blocking profile
  • ...and 40 more