Eliezer, “when two or more human beings have common knowledge that they disagree about a question of simple fact” is problematic because this can never happen with common-prior Bayesians. Such people can never have common knowledge of a disagreement, although they can disagree on the way to common knowledge of their opininons. Maybe you could say that they learn that they initially disagree about the question.
However I don’t think that addition fixes the problem. Suppose the first coin in my example is known by all to be biased and fall heads with 60% probability. Then the person who knows the first coin initially estimates the probability of HH as 50%, while the person who knows the second coin initially estimates it as 60%. So they initially disagree, and both know this. However they will both update their estimates upward to 100% after hearing each other, which means that they do not adjust their probability estimates towards the other.
I am having trouble relating the Modesty Argument to Aumann’s result. It’s not clear to me that anyone will defend the Modesty Argument as stated, at least not based on Aumann and the related literature.
Eliezer, “when two or more human beings have common knowledge that they disagree about a question of simple fact” is problematic because this can never happen with common-prior Bayesians. Such people can never have common knowledge of a disagreement, although they can disagree on the way to common knowledge of their opininons. Maybe you could say that they learn that they initially disagree about the question.
However I don’t think that addition fixes the problem. Suppose the first coin in my example is known by all to be biased and fall heads with 60% probability. Then the person who knows the first coin initially estimates the probability of HH as 50%, while the person who knows the second coin initially estimates it as 60%. So they initially disagree, and both know this. However they will both update their estimates upward to 100% after hearing each other, which means that they do not adjust their probability estimates towards the other.
I am having trouble relating the Modesty Argument to Aumann’s result. It’s not clear to me that anyone will defend the Modesty Argument as stated, at least not based on Aumann and the related literature.