Aumann's Agreement Theorem

One of the fascinating insights from rational decision theory and Bayesian epistemology is Aumann's Agreement Theorem. Proposed by Robert Aumann in 1976, the theorem states:

"Two rational agents who have common knowledge of each other's beliefs and priors cannot agree to disagree. If their beliefs differ, rational Bayesian updating based on shared information must lead them to converge on the same credence."

Understanding Aumann's Agreement

At its core, the theorem hinges on:

1. Perfect Bayesian Rationality: Both agents update their beliefs strictly according to Bayes' theorem.

2. Common Knowledge: Each knows the other's beliefs, priors, and updating methodology completely, including knowing that the other knows this, and so on ad infinitum.

Given these conditions, persistent disagreement is impossible. If two rational agents find themselves disagreeing, it must mean:

• They hold different priors.

• They have asymmetrical information.

• They are misunderstanding each other's reasoning processes.

Practical Limitations

Despite its theoretical elegance, Aumann’s theorem is challenging to apply directly in real-world scenarios:

• Differences in Priors: Real agents rarely have identical prior beliefs, making full convergence unlikely.

• Imperfect Information: Common knowledge assumptions rarely hold in practice. Information asymmetry or incomplete transparency in reasoning prevents convergence.

• Cognitive Biases: Humans exhibit systematic biases that deviate from perfect Bayesian rationality.

Credence Divergence in Practice

Consider philosophical or theological disagreements, such as differing credences regarding metaphysical claims like the logical necessity of the Christian Trinity. Such divergence arises naturally due to differences in foundational philosophical assumptions (priors), evaluation methods, and epistemic thresholds for acceptance.

For example, my own credence in a particular theological claim might hover around 1-2%, reflecting skepticism but openness, while another rigorously empirical, conditionalist thinker might place their credence closer to 0.01-0.1%. Both positions can be entirely rational given their respective priors and epistemic standards.

Philosophical Implication

The profound insight from Aumann’s theorem is not that disagreements cannot exist, but rather that rational agents should always investigate why disagreements persist. Persistent divergence in rational credence signals underlying informational asymmetries or differences in priors worth exploring explicitly.

Thus, rational disagreement—far from being merely tolerated—is a diagnostic tool. It prompts rigorous epistemic inquiry and a clearer articulation of our hidden assumptions.

Conclusion

Aumann’s theorem reminds us that while perfect rational convergence may be unattainable practically, striving for it pushes us toward deeper epistemic clarity. Rational disagreement, used thoughtfully, sharpens our understanding rather than diminishing it.