It includes a great test of whether a given discussion is Aumann-rational:

what rational disagreements should look like: they should follow unbiased random walks, until sooner or later they terminate in common knowledge of complete agreement.

as opposed to

suppose your friend tells you a liberal opinion, then you take it into account, but reply with a more conservative opinion. The friend takes your opinion into account, and replies with a revised opinion. Question: is your friend’s new opinion likelier to be more liberal than yours, or more conservative?

Obviously, more liberal! Yes, maybe your friend now sees some of your points and vice versa, maybe you’ve now drawn a bit closer (ideally!), but you’re not going to suddenly switch sides because of one conversation.

Yet, if you and your friend are Bayesians with common priors, one can prove that that’s not what should happen at all.