(Idea from Robin Hanson and Tyler Cowen’s 2004 paper Are Disagreements Honest?)
One common argument relating to common priors is that two rational agents with all the same information (including no information at all) could have no possible grounds on which to disagree. Priors by definition refer to the state of knowledge before either agent had any evidence relevant to a given proposition. So there is no information that either agent could have that would allow a difference in priors.
A response to this is that some information that we have is inherently private and unique to us. For instance, you and I might have differences in intelligence, in ways of conceptualizing the world, or in the things we innately find intuitively plausible. All of these differences may count as important information in shaping our priors on a given subject, before we ever encounter a single piece of evidence relevant to the subject.
Here’s a really weird argument for why even these differences should not count. If we use anthropic reasoning, and treat our own existence and the details of our brain and body as just another thing to be conditioned on, then even these private intimate details are simply contingent facts about the world that are to be treated as evidence. Before you’ve conditioned on your own existence, you should be agnostic as to which set of brain/body/mind out of all the possible sets of observers “you” will end up being. You must imagine yourself behind Rawls’ veil of ignorance, a disembodied reasoner that is identical to all other such reasoners. So there is no conceivable reason why your prior should differ from anybody else’s – you must treat yourself as literally the same entity as them pre anthropic conditioning.
In less out-there terms, if you encounter somebody with an apparently different prior from you, then you should consider “Hmm, what if I were born as this person, instead of myself?” The answer to which is, of course, you would have had the same priors as them. Which means that your difference in “priors” is actually a difference of posteriors resulting from conditioning on the arbitrary choice of body/brain/experiences you ended up with.
In addition, by Aumann’s agreement theorem, any apparent differences in priors that become common knowledge should quickly go away, once they are realized to be merely differences in posteriors. Essentially, any differences in priors that last between two rational individuals are signs that they are arbitrarily favoring their own existence in considerations of what prior they should use.