I rewatched 12 Monkeys last week (because my wife was going through a Brad Pitt phase, although I think this movie cured her of that :), in which Bruce Willis plays a time traveler who accidentally got locked up in a mental hospital. The reason I mention it here is because It contained an amusing example of mutual belief updating: Bruce Willis's character became convinced that he really is insane and needs psychiatric care, while simultaneously his psychiatrist became convinced that he actually is a time traveler and she should help him save the world.

Perhaps the movie also illustrates a danger of majoritarianism: if someone really found a secret that could save the world, it would be tragic if he allowed himself to be convinced otherwise due to majoritarian considerations. Don't most (nearly all?) true beliefs start their existence as a minority?