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A definition of near-common knowledge yields quantitative versions of agreement theorems for any probability space.

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2026-06-27 07:42 UTC pith:ZVGIIVH6

load-bearing objection The paper gives a (δ,ε)-common knowledge definition for arbitrary probability spaces and states quantitative versions of Aumann and Nielsen, but the key step of carrying the classical proofs over to the approximate operator is not obviously automatic in uncountable settings. the 2 major comments →

arxiv 2606.11902 v1 pith:ZVGIIVH6 submitted 2026-06-10 econ.TH math.PR

Delta-Epsilon-Common Knowledge and Quantitative Agreement Theorems

Christina Pawlowitsch , Stefan Schrott , Daniel Toneian This is my paper
classification econ.TH math.PR
keywords common knowledgeAumann agreement theoremquantitative agreementprobability spacesnoisy communicationposterior beliefsepistemic game theoryapproximate knowledge
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper defines a measure called (δ,ε)-common knowledge that tells how close agents come to sharing common knowledge of an event. This measure applies to arbitrary probability spaces rather than only countable ones. The authors then prove that when an event meets this approximate common-knowledge condition, the agents' posterior probabilities for the event cannot differ by more than an explicit bound depending on δ and ε. The same style of bound is obtained for random variables and for repeated exchange of posteriors. These results directly address settings with noisy or imperfect communication.

Core claim

The authors define (δ,ε)-common knowledge for general probability spaces and prove that if an event is (δ,ε)-common knowledge, then the agents' posterior probabilities differ by at most a bound depending on δ and ε. They extend this to Aumann's agreement theorem for events, Nielsen's version for random variables, and iterative communication of posteriors.

What carries the argument

(δ,ε)-common knowledge, a quantitative relaxation of the standard common knowledge operator that bounds the probability that the event is not common knowledge.

Load-bearing premise

The quantitative bounds hold when the strict common knowledge operator is replaced by the paper's approximate version in the original proofs.

What would settle it

A counterexample probability space where two agents are (δ,ε)-commonly knowledgeable of an event but their posteriors differ more than the predicted bound.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • If two agents are (δ,ε)-commonly knowledgeable of an event E, their conditional probabilities P(E|I1) and P(E|I2) differ by at most a bound depending on δ and ε.
  • The same quantitative bound applies when posteriors are exchanged iteratively.
  • The results cover non-countable spaces, allowing continuous probability distributions.
  • In noisy communication, the disagreement shrinks as communication improves toward common knowledge.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The bounds could be used to model belief formation in markets where information is shared only approximately.
  • One could test the explicit bounds by constructing finite examples with controlled noise levels and checking the resulting posterior gaps.
  • The construction may allow quantitative analysis of higher-order beliefs in settings beyond the three theorems treated here.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 2 minor

Summary. The paper introduces a definition of (δ,ε)-common knowledge applicable to arbitrary (including uncountable) probability spaces and derives quantitative analogues of Aumann's Agreement Theorem, Nielsen's extension to random variables, and iterative posterior communication results, with explicit applicability to noisy communication settings.

Significance. If the quantitative bounds are shown to hold without hidden measurability or continuity restrictions, the work would supply a usable approximate-common-knowledge framework for general spaces, directly extending the classical partition-based arguments to settings with noise or bounded rationality; this is a substantive technical contribution to epistemic game theory.

major comments (2)
  1. [Definition of (δ,ε)-common knowledge and the statements of the quantitative Aumann/Nielsen theorems] The central claim (abstract and §1) that quantitative agreement bounds follow once the classical common-knowledge operator is replaced by the paper's (δ,ε) version rests on the unexamined assertion that per-level discrepancies of size δ and ε remain controlled after the infinite intersection that defines common knowledge. In uncountable spaces this requires an explicit argument that the approximate fixed-point or partition intersection still forces posteriors (or random-variable expectations) to agree within a function of δ and ε; the manuscript must supply this derivation or a counter-example check, as direct substitution does not automatically bound error propagation through the knowledge hierarchy.
  2. [Section containing the communication theorem and its proof] Theorem on iterative posterior communication (the noisy-communication result): the proof must verify that the (δ,ε) operator preserves the martingale property or convergence used in the classical argument when the underlying space is not countable; otherwise the claimed quantitative bound on disagreement after finite rounds of communication may fail to hold uniformly.
minor comments (2)
  1. [Introduction and definitions] Notation for the (δ,ε) operator should be introduced with an explicit comparison table to the classical common-knowledge operator to clarify which properties are preserved and which are relaxed.
  2. [Definition section] The abstract states results hold for 'any (and not just countable) probability spaces'; the manuscript should add a short remark on whether the construction requires the probability space to be complete or to satisfy any other standard measure-theoretic regularity condition.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and constructive report. The two major comments correctly identify places where the manuscript would benefit from more explicit arguments to substantiate the quantitative bounds in general probability spaces. We address each point below and will make the indicated revisions.

read point-by-point responses

  1. Referee: [Definition of (δ,ε)-common knowledge and the statements of the quantitative Aumann/Nielsen theorems] The central claim (abstract and §1) that quantitative agreement bounds follow once the classical common-knowledge operator is replaced by the paper's (δ,ε) version rests on the unexamined assertion that per-level discrepancies of size δ and ε remain controlled after the infinite intersection that defines common knowledge. In uncountable spaces this requires an explicit argument that the approximate fixed-point or partition intersection still forces posteriors (or random-variable expectations) to agree within a function of δ and ε; the manuscript must supply this derivation or a counter-example check, as direct substitution does not automatically bound error propagation through the knowledge hierarchy.

    Authors: We agree that the manuscript requires an explicit derivation showing control of the (δ,ε) discrepancies after the infinite intersection that defines common knowledge. Although the definition is constructed level-by-level to bound per-level errors, the passage to the intersection in arbitrary spaces needs a separate argument. We will add a new lemma (or proposition) in §2 that derives the required bound on the intersection using the definition and the properties of the probability measure, thereby justifying the quantitative statements of the Aumann and Nielsen theorems. revision: yes

  2. Referee: [Section containing the communication theorem and its proof] Theorem on iterative posterior communication (the noisy-communication result): the proof must verify that the (δ,ε) operator preserves the martingale property or convergence used in the classical argument when the underlying space is not countable; otherwise the claimed quantitative bound on disagreement after finite rounds of communication may fail to hold uniformly.

    Authors: The referee correctly notes that the proof of the noisy-communication theorem must confirm preservation of the relevant convergence (or an analogous quantitative bound) under the (δ,ε) operator when the space is uncountable. The current argument relies on the classical martingale property without an explicit check for the approximate operator. We will revise the proof in the relevant section to supply this verification, either by direct estimation of the disagreement after each round or by establishing that the (δ,ε) operator inherits the necessary convergence properties from the underlying filtration. revision: yes

Circularity Check

0 steps flagged

No significant circularity; extension of independent classical theorems via new definition

full rationale

The paper introduces a novel (δ,ε)-common knowledge operator defined directly on arbitrary probability spaces and claims quantitative agreement results by adapting Aumann's and Nielsen's proofs. No equations, fitted parameters, or self-referential reductions appear in the abstract or reader's summary. The central claims rest on extending externally established theorems rather than any self-definition, fitted-input prediction, or self-citation chain that collapses the result to the paper's inputs. This is the standard case of a self-contained mathematical extension with no detectable circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only abstract available; no free parameters, axioms, or invented entities are described.

pith-pipeline@v0.9.1-grok · 5613 in / 1078 out tokens · 14669 ms · 2026-06-27T07:42:56.059794+00:00 · methodology

0 comments
read the original 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, $(\delta,\epsilon)$-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.

discussion (0)

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Reference graph

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