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.

翻译：奥曼著名的“同意分歧定理”指出：若一组智能体共享一个共同的先验分布，基于私有信息通过贝叶斯条件化更新其信念，并且对某一事件的后验信念具有共同知识，则这些后验信念必须完全相同。蒙德雷尔和萨梅特提出了该定理的一个优美推广，后经内曼完善：若一组智能体共享一个共同的先验分布，基于私有信息使用贝叶斯条件化更新信念，并且对其后验信念具有共同的 $p$-信念，则这些后验信念必然接近（即差异不超过 $1 - p$）。此处，共同 $p$-信念将共同知识的概念推广至概率信念：智能体共同 $p$-信念事件 $E$，当且仅当每个智能体以至少 $p$ 的程度相信 $E$，每个智能体以至少 $p$ 的程度相信其他所有智能体也以至少 $p$ 的程度相信 $E$，依此类推。本文进一步将蒙德雷尔-萨梅特-内曼共识定理从经典概率测度推广至似然测度——这是由哈尔彭引入的一个非常通用的框架，统一了多种信念的形式化模型。为便于这一推广，我们在经典设定下为蒙德雷尔-萨梅特-内曼定理提供了一个新的证明。基于原始证明和我们的新证明，我们提出了该定理在基于似然的结构中的两种不同推广。随后，我们将这些推广结果应用于多种非经典信念模型，包括条件概率结构和字典序概率结构。此外，我们证明，每当我们的推广定理不适用时，蒙德雷尔-萨梅特-内曼共识定理便不成立。这些发现表明，我们的结果成功识别了信念模型满足蒙德雷尔-萨梅特-内曼共识定理所需的最小条件。