I think there’s a slight misconception about Aumann’s agreement theorem here: “Common knowledge”, as Aumann defines it (and is leveraged in the proof), doesn’t just mean exchanging beliefs once: common knowledge means 1) knowing how they update after the first pass, then 2) knowing how they updated after knowing about the first pass, 3) knowing about how they updated after knowing about updating about knowing about the first pass, and so on
It’s only at the end of a potentially infinite chain of exchanging beliefs, that two rational agents are guaranteed to have the same beliefs. After a “first exchange”, two rationalists can very well still disagree, just less, and I’d honestly be kind of worried if they perfectly agreed with eachother after a first exchange
I think there’s a slight misconception about Aumann’s agreement theorem here: “Common knowledge”, as Aumann defines it (and is leveraged in the proof), doesn’t just mean exchanging beliefs once: common knowledge means 1) knowing how they update after the first pass, then 2) knowing how they updated after knowing about the first pass, 3) knowing about how they updated after knowing about updating about knowing about the first pass, and so on
It’s only at the end of a potentially infinite chain of exchanging beliefs, that two rational agents are guaranteed to have the same beliefs. After a “first exchange”, two rationalists can very well still disagree, just less, and I’d honestly be kind of worried if they perfectly agreed with eachother after a first exchange