I think Douglas has described the situation well. The heart of the Aumann reasoning as I understand it is the mirroring effect, where your disputant’s refusal to agree must be interpreted in the context of his understanding the full significance of that refusal, which must be understood to carry weight due to your steadfastness, which itself gains meaning because of his persistence, and so on indefinitely. It is this infinite regress which is captured in the phrase “common knowledge” and which is where I think our intuition tends to go awry and leads us to underestimate the true significance of agreeing to disagree.
Without this mirroring, the Bayesian impact of another person’s disagreement is less and we no longer obtain Aumann’s strong result. In that case, as Douglas described we should try to model the information possessed by the other person and compare it with our own information, doing our best to be objective. This may indeed benefit from a degree of modesty but it’s hard to draw a firm rule.
I think Douglas has described the situation well. The heart of the Aumann reasoning as I understand it is the mirroring effect, where your disputant’s refusal to agree must be interpreted in the context of his understanding the full significance of that refusal, which must be understood to carry weight due to your steadfastness, which itself gains meaning because of his persistence, and so on indefinitely. It is this infinite regress which is captured in the phrase “common knowledge” and which is where I think our intuition tends to go awry and leads us to underestimate the true significance of agreeing to disagree.
Without this mirroring, the Bayesian impact of another person’s disagreement is less and we no longer obtain Aumann’s strong result. In that case, as Douglas described we should try to model the information possessed by the other person and compare it with our own information, doing our best to be objective. This may indeed benefit from a degree of modesty but it’s hard to draw a firm rule.