The Convergence of Belief Change
I've changed my mind about countless matters, but most if not all such changes have been vaguely akin to switching from brand A to brand B. In some deep sense, however, I feel that I've never really changed how I think about or evaluate things. This may sound like a severe case of cerebral stenosis, but I think the condition is universal. Although people change their minds, they do so in an invariant, convergent sort of way, and I find this to be of more interest than the brand switches, important as they sometimes are.
I take heart that this stance can be viewed as something other than stubborn rigidity from the so-called Agreement Theorem of Nobel Prize-winning game-theorist Robert Aumann. His theorem can be roughly paraphrased as follows: Two individuals cannot forever agree to disagree.
An important definition allows for a slightly fuller statement. Information is termed "common knowledge" among a group of people if all parties know it, know that the others know it, know that the others know they know it, and so on. It is much more than "mutual knowledge," which requires only that the parties know the particular bit of information, not that they be aware of the others' knowledge.
Aumann showed that as agents' beliefs, formed in rational response to different bits of private information, gradually become common knowledge, their beliefs change and eventually coincide.
Thus in whatever rational ways each of us comes to change his or her mind, in the long run the rest of us will follow suit. Of course, as Keynes observed, in the long run, we're all dead.
Another problem is Aumann's result doesn't say anything about the convergence of irrational agents.
