Generalising Aumann's Agreement Theorem

TL;DR

This paper extends Aumann's agreement theorem to quantum and generalized probability theories, showing that rational agents cannot agree to disagree under these frameworks, emphasizing the role of conditioning in agreement.

Contribution

It demonstrates that Aumann's agreement impossibility holds in quantum and generalized probability theories, broadening the theorem's applicability.

Findings

Agreement to disagree is impossible in quantum theory.

Agreement to disagree is forbidden in any generalized probability theory.

The probabilistic version depends on how information is conditioned.

Abstract

According to Aumann's celebrated theorem, rational agents cannot agree to disagree. In other words, agents who once shared a common prior probability distribution and who have common knowledge about their posteriors cannot assign different probability distributions to a given proposition. Common knowledge imposes strong restrictions on assigned probabilities. In fact, Aumann's agreement theorem was one of the first attempts to formalise and explore the role played by common knowledge in decision theory. Recently, the debate over possible (quantum) extensions of Aumann's results has resurfaced. This paper contributes to this discussion. First, we argue that agreeing to disagree is impossible in quantum theory. Secondly, by building on the quantum argument, we show that agreeing to disagree is also forbidden in any generalised probability theory. The upshot is that in its probabilistic version, the agreement theorem is a direct consequence of how we choose to condition upon acquiring new information.