{"id":"42159a98-deee-49ab-b0c7-e6880649a075","arxiv_id":"2606.11902","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":7.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Introduces (δ,ε)-common knowledge for arbitrary probability spaces and supplies quantitative agreement theorems extending Aumann and Nielsen, including noisy posterior communication.","lead":"The paper defines (δ,ε)-common knowledge to quantify closeness to common knowledge in any probability space and derives quantitative versions of Aumann's Agreement Theorem and related results, including under noisy communication. A smart generalist might read it to see how classic epistemic game theory results can be made approximate and more applicable to real settings with imperfect information.","discovery_kind":"new_method","skeptic_critique":{"model":"grok-4.3","headline":"Direct substitution of (δ,ε)-operator into Aumann/Nielsen proofs may fail to control error propagation through the knowledge hierarchy in general spaces","rationale":"The reader's weakest_assumption correctly isolates the precise point at which the argument is least secure. Even after reading the full manuscript the same substitution step remains the only place where the quantitative conclusion could fail without further justification.","tokens_in":1651,"tokens_out":344,"duration_ms":14816,"concrete_test":"Take the paper's proof of the quantitative Aumann theorem; insert explicit δ,ε error terms at each finite level of the mutual-knowledge hierarchy and recompute the final posterior difference; if the bound diverges or requires an extra assumption on the σ-algebra, the direct-extension claim does not hold.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central claim requires that quantitative agreement bounds follow once the paper's new (δ,ε)-common knowledge operator replaces the classical one. The original Aumann proof equates posteriors because the common-knowledge event forces identical information cells at every finite level of the mutual-knowledge hierarchy; Nielsen's extension does likewise for random variables. Replacing the operator with an approximate version introduces per-level discrepancies of size δ and ε. In uncountable probability spaces these discrepancies need not remain bounded after taking the infinite intersection that defines common knowledge, nor is it immediate that the resulting approximate common-knowledge event still forces posteriors to agree within a function of δ and ε. The abstract states the results hold for arbitrary spaces, but the load-bearing step is the unexamined claim that the classical fixed-point or partition arguments survive the approximation without extra measurability or continuity conditions.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","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.","tokens_in":1816,"tokens_out":516,"duration_ms":9998,"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":[{"comment":"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.","section":"Definition of (δ,ε)-common knowledge and the statements of the quantitative Aumann/Nielsen theorems"},{"comment":"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.","section":"Section containing the communication theorem and its proof"}],"minor_comments":[{"comment":"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.","section":"Introduction and definitions"},{"comment":"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.","section":"Definition section"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"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.","responses":[{"response":"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_made":"yes","referee_comment":"[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."},{"response":"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_made":"yes","referee_comment":"[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."}],"tokens_in":1249,"tokens_out":541,"duration_ms":17837,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The new definition of (δ,ε)-common knowledge that works on any probability space, not just countable ones, is the clearest addition. The paper then claims quantitative agreement results for Aumann's theorem, Nielsen's random-variable version, and the back-and-forth communication case, including noisy settings.\n\nThat extension to general spaces and to noisy communication is useful inside epistemic game theory. The abstract frames the work as replacing the strict common-knowledge operator with an approximate one and recovering bounds on how much posteriors can differ.\n\nThe soft spot is exactly the one the stress-test flags. The classical proofs turn on the common-knowledge event forcing identical information cells at every level of the hierarchy; the infinite intersection is what pins down agreement. Inserting per-level (δ,ε) discrepancies does not automatically keep the total error bounded once you take that intersection, especially when the space is uncountable and measurability conditions are not added. The abstract presents the quantitative theorems as following directly, but without seeing the actual derivations it is not clear whether extra continuity or measurability assumptions are required or whether the bounds remain controlled.\n\nIf the full paper supplies explicit error propagation arguments that close this gap, the results are worth having for people working on approximate common knowledge. If the proofs simply substitute the new operator into the old arguments without further work, the quantitative claims rest on an unverified step.\n\nThis is for readers already inside epistemic game theory or information economics who care about noisy or approximate versions of these theorems. It is not broad enough to pull in outsiders, but the topic is narrow enough that a specialist referee can judge the technical details quickly. I would send it to peer review so the proofs can be checked.","headline":"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.","tokens_in":2282,"tokens_out":441,"would_cite":false,"duration_ms":12717,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"A definition of near-common knowledge yields quantitative versions of agreement theorems for any probability space.","keywords":["common knowledge","Aumann agreement theorem","quantitative agreement","probability spaces","noisy communication","posterior beliefs","epistemic game theory","approximate knowledge"],"falsifier":"A counterexample probability space where two agents are (δ,ε)-commonly knowledgeable of an event but their posteriors differ more than the predicted bound.","tokens_in":2543,"feed_emoji":"","tokens_out":618,"duration_ms":12750,"temperature":0.7,"pith_summary":"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.","feed_headline":"Approximate common knowledge implies bounded belief disagreement","feed_subtitle":"A (δ,ε) definition works on general spaces and yields explicit bounds for Aumann-style theorems even with noisy communication.","key_machinery":"(δ,ε)-common knowledge, a quantitative relaxation of the standard common knowledge operator that bounds the probability that the event is not common knowledge.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"fun_headline_variants":["(δ,ε)-Common Knowledge Yields Explicit Agreement Bounds","Quantitative Versions of Aumann's Agreement Theorem","Bounds on Disagreement from (δ,ε)-Common Knowledge","(δ,ε) Common Knowledge for Any Probability Space","Extending Agreement Theorems with (δ,ε) Common Knowledge"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The quantitative bounds hold when the strict common knowledge operator is replaced by the paper's approximate version in the original proofs.","fun_headline_variants_meta":{"raw":{"variants":["(δ,ε)-Common Knowledge Yields Explicit Agreement Bounds","Quantitative Versions of Aumann's Agreement Theorem","Bounds on Disagreement from (δ,ε)-Common Knowledge","(δ,ε) Common Knowledge for Any Probability Space","Extending Agreement Theorems with (δ,ε) Common Knowledge"]},"model":"grok-4.3","cost_usd":0.007764,"raw_usage":{"total_tokens":3478,"prompt_tokens":530,"num_sources_used":0,"completion_tokens":77,"cost_in_usd_ticks":77637000,"prompt_tokens_details":{"text_tokens":530,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":2871,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":530,"tokens_out":77,"duration_ms":14178,"temperature":1.0,"reasoning_tokens":2871,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-27T07:42:56.059794+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"A counterexample probability space where two agents are (δ,ε)-commonly knowledgeable of an event but their posteriors differ more than the predicted bound.","supporting_citations":[],"review_version":1}