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Verbalized Bayesian Persuasion

Wenhao Li, Yue Lin, Xiangfeng Wang, Bo Jin, Hongyuan Zha, Baoxiang Wang

TL;DR

This work extends Bayesian persuasion to real-world dialogue by proposing Verbalized Bayesian Persuasion (VBP), a framework that maps BP problems to verbalized mediator-augmented games and solves them with a Prompt-PSRO-based solver. It introduces a verbalized formulation of BP, including commitment, obedience, and information-obfuscation mechanisms, and proves convergence properties for static and multistage BP. Empirical results show VBP reproduces classic BP equilibria and discovers effective, language-based persuasion strategies in complex, multi-stage dialogues and real-world scenarios. The approach offers a scalable, language-centric interface for analyzing strategic communication in economics and societal applications, while outlining pathways to address limitations and broaden applicability.

Abstract

Information design (ID) explores how a sender influence the optimal behavior of receivers to achieve specific objectives. While ID originates from everyday human communication, existing game-theoretic and machine learning methods often model information structures as numbers, which limits many applications to toy games. This work leverages LLMs and proposes a verbalized framework in Bayesian persuasion (BP), which extends classic BP to real-world games involving human dialogues for the first time. Specifically, we map the BP to a verbalized mediator-augmented extensive-form game, where LLMs instantiate the sender and receiver. To efficiently solve the verbalized game, we propose a generalized equilibrium-finding algorithm combining LLM and game solver. The algorithm is reinforced with techniques including verbalized commitment assumptions, verbalized obedience constraints, and information obfuscation. Numerical experiments in dialogue scenarios, such as recommendation letters, courtroom interactions, and law enforcement, validate that our framework can both reproduce theoretical results in classic BP and discover effective persuasion strategies in more complex natural language and multi-stage scenarios.

Verbalized Bayesian Persuasion

TL;DR

This work extends Bayesian persuasion to real-world dialogue by proposing Verbalized Bayesian Persuasion (VBP), a framework that maps BP problems to verbalized mediator-augmented games and solves them with a Prompt-PSRO-based solver. It introduces a verbalized formulation of BP, including commitment, obedience, and information-obfuscation mechanisms, and proves convergence properties for static and multistage BP. Empirical results show VBP reproduces classic BP equilibria and discovers effective, language-based persuasion strategies in complex, multi-stage dialogues and real-world scenarios. The approach offers a scalable, language-centric interface for analyzing strategic communication in economics and societal applications, while outlining pathways to address limitations and broaden applicability.

Abstract

Information design (ID) explores how a sender influence the optimal behavior of receivers to achieve specific objectives. While ID originates from everyday human communication, existing game-theoretic and machine learning methods often model information structures as numbers, which limits many applications to toy games. This work leverages LLMs and proposes a verbalized framework in Bayesian persuasion (BP), which extends classic BP to real-world games involving human dialogues for the first time. Specifically, we map the BP to a verbalized mediator-augmented extensive-form game, where LLMs instantiate the sender and receiver. To efficiently solve the verbalized game, we propose a generalized equilibrium-finding algorithm combining LLM and game solver. The algorithm is reinforced with techniques including verbalized commitment assumptions, verbalized obedience constraints, and information obfuscation. Numerical experiments in dialogue scenarios, such as recommendation letters, courtroom interactions, and law enforcement, validate that our framework can both reproduce theoretical results in classic BP and discover effective persuasion strategies in more complex natural language and multi-stage scenarios.

Paper Structure

This paper contains 68 sections, 1 theorem, 3 equations, 21 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Bayesian Persuasion
  4. Learning Methods for BP
  5. Policy- and Prompt-Space Response Oracle
  6. Related Works
  7. Game-Theoretic Solvers with LLMs
  8. Deception in Multi-Agent Learning
  9. Conversational Persuasiveness of LLMs
  10. LLMs in Strategic Interactions
  11. Remark 1
  12. The Mediator-Augmented Game Formulation for BP
  13. VBP Solver
  14. Verbalized Formulation for BP
  15. Verbalized Committment Assumption
  16. ...and 53 more sections

Key Result

Proposition 5.1

VBP returns an $\varepsilon$-approximate Bayes correlated equilibrium in static BP and an $\varepsilon$-approximate Bayes-Nash equilibrium in multistage BP.

Figures (21)

  • Figure 1: Extending classic BP examples to verbalized mediator-augmented, extensive-form games.
  • Figure 2: Left: BP timing in the EFG; Right: Illustration of the Prompt-PSRO.
  • Figure 3: Verbalize Bayesian persuasion framework.
  • Figure 4: Performance comparison on classic static BP problems. Averaged over $20$ seeds. In the $3$ BP problems, the probability of lying refers to describing a weak student as strong, an innocent defendant as guilty, or an unpatrolled segment as patrolled. Conversely, the probability of honesty refers to accurately describing a strong student, a guilty defendant, or a patrolled segment.
  • Figure 5: Performance comparison on general static BP problems. Averaged over $20$ seeds. The physical meaning of the probabilities of lying and honesty is consistent with Figure \ref{['fig:classic-results']}.
  • ...and 16 more figures

Theorems & Definitions (3)

  • Definition 4.1
  • Proposition 5.1
  • proof