Why two rational people cannot agree to disagree on the facts
In short: If two rational people share the same starting assumptions and have access to all relevant information, they cannot logically maintain different beliefs. This mathematical certainty suggests that disagreement isn't just a matter of opinion—it is a signal of hidden information or fundamentally different starting points.
Robert Aumann demonstrated a core game theory rule in his 1976 work "Agreeing to Disagree": if two Bayesian actors share initial beliefs and their personal views are public, they are unable to "agree to disagree" regarding an event's likelihood. This means that if it is common knowledge what each agent believes, and both agents are rational actors who update their beliefs using Bayes' rule, their posterior beliefs must be identical.
The mechanism relies on the concept of common knowledge, which is a recursive hierarchy of awareness. For a belief to be common knowledge, Agent A must know that Agent B knows the belief, and Agent B must know that Agent A knows that Agent B knows it, and so on. In this model, if agents were to hold different posterior probabilities while knowing each other's opinions, it would create a logical contradiction. If they truly knew each other's thoughts, the information would have already been integrated into their own updated beliefs.
The theorem highlights that persistent disagreement between rational actors implies one of two things: either the agents do not share a common prior (they started with different fundamental assumptions), or their individual beliefs are not common knowledge (there is some information one side has that the other does not). While variations exist—such as Monderer and Samet's work on common-belief or Ziv Hellman's exploration of "close" priors—Aumann's core finding remains a cornerstone for understanding how information and rationality constrain human and artificial intelligence.
Additionally, the presence of a common prior requires this theorem to hold true. Essentially, if you can't agree on the odds, you can't agree on the facts.
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Sources
- Summary by Intellect · source: Aumann's agreement theorem
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