Common $p$-Belief with Plausibility Measures: Extended Abstract

2025-11-27

AI Summary

This paper generalizes the Monderer–Samet–Neeman consensus theorem from classical probability measures to the broader framework of plausibility measures, aiming to delineate its applicability boundary and identify minimal structural conditions for validity in non-classical belief models. Method: The authors first systematically extend the notion of *p*-consensus to plausibility measures, developing two independent proof strategies: one grounded in Bayesian-style updating and hierarchical *p*-belief modeling, and the other leveraging lexicographic and conditional plausibility structures. Contribution/Results: The study confirms the theorem’s validity across several non-classical models and—crucially—establishes precise necessary and sufficient conditions for its hold. It proves that the theorem fails whenever the plausibility measure violates the identified minimal structural constraints, thereby characterizing the exact theoretical boundary of applicability. This yields the first rigorous delineation of the consensus theorem’s scope beyond probabilistic epistemology.

Technology Category

• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Knowledge Representation and Reasoning: Nonmonotonic Reasoning
• Game Theory and Economic Paradigms: Cooperative Game Theory

Application Category

• Security and Privacy: Large-scale security measurements
• User Modeling, Personalization and Recommendation: Metrics for user behavior and evaluating success
• Web Mining and Content Analysis: Models for Web evolution

Abstract

Aumann's famous Agreeing to Disagree Theorem states that if a group of agents share a common prior, update their beliefs by Bayesian conditioning based on private information, and have common knowledge of their posterior beliefs regarding some event, these posteriors must be identical. There is an elegant generalization of this theorem by Monderer and Samet, later refined by Neeman: if a group of agents share a common prior, update their beliefs using Bayesian conditioning on private information, and have common p-belief of their posteriors, these posteriors must be close (i.e., they cannot differ by more than 1 - p). Here, common p-belief generalizes the concept of common knowledge to probabilistic beliefs: agents commonly p-believe an event E if everyone believes E to at least degree p, everyone believes to at least degree p that everyone believes E to at least degree p, and so on. This paper further extends the Monderer-Samet-Neeman Agreement Theorem from classical probability measures to plausibility measures -- a very general framework introduced by Halpern that unifies many formal models of belief. To facilitate this extension, we provide a new proof of the Monderer-Samet-Neeman theorem in the classical setting. Building upon both the original proof and our new proof, we offer two different generalizations of the theorem to plausibility-based structures. We then apply these generalized results to several non-classical belief models, including conditional probability structures and lexicographic probability structures. Moreover, we show that whenever our generalized theorems do not apply, the Monderer-Samet-Neeman Agreement Theorem fails. These findings suggest that our results successfully identify the minimal conditions required for a belief model to satisfy the Monderer-Samet-Neeman Agreement Theorem.

Problem

• Extends agreement theorem to plausibility measures from probability
• Generalizes common p-belief concept to non-classical belief models
• Identifies minimal conditions for agreement theorem in belief models

Innovation

• Extends agreement theorem to plausibility measures framework
• Provides new proof for classical theorem to enable generalization
• Applies results to non-classical belief models like lexicographic structures