Delta-Epsilon-Common Knowledge and Quantitative Agreement Theorems

2026-06-10

AI Summary

This study addresses the quantification of “approximate common knowledge” among agents in general probability spaces and establishes corresponding quantitative consensus theorems. To this end, the authors introduce the novel notion of (δ,ε)-common knowledge, which for the first time formalizes a measurable approximation of consensus in arbitrary—possibly uncountable—probability spaces. Within this framework, they derive quantitative versions of Aumann’s classic agreement theorem and its extension by Nielsen to random variables, encompassing broad scenarios such as noisy communication and iterative posterior updating. By transcending the traditional restriction to countable spaces inherent in classical common knowledge theory, this work provides a rigorous measure-theoretic foundation for distributed inference and multi-agent consensus, along with explicit convergence bounds.

Abstract

Aumann defined common knowledge mathematically and established his now famous Agreement Theorem. We present a novel approach to quantifying how close individuals are to commonly knowing events, $(δ,ε)$-common knowledge, which is defined for any (and not just countable) probability spaces, and provide quantitative versions of the key results in this field. Specifically, we do this for Aumann's Agreement Theorem and Nielsen's extension thereof to random variables, as well as for the setting in which posteriors are communicated back and forth between individuals. Our results apply in particular to noisy communication settings.

Problem

• common knowledge
• agreement theorem
• quantitative analysis
• noisy communication
• probability spaces

Innovation

• (δ,ε)-common knowledge
• quantitative agreement theorem
• noisy communication
• posterior updating
• common knowledge