The simplest way to solve the jester’s puzzle is to make a table of the four cases (where the frog is, where the true inscription is), then determine for each case whether the inscriptions are in fact true or false as required for that case. (All the while making la-la-la-can’t-hear-you noises at any doubts one might have about whether self-reference can be formalised at all.) The conclusion is that the first box has the frog and the true inscription. That assumes that the jester was honest in stating his puzzle, but given his shock at the outcome of the king’s puzzle, that appears to be so.
But can self-reference be formalised? How, for example, do two perfect reasoners negotiate a deal? In general, how can two perfect reasoners in an adversarial situation ever interpret the other’s words as anything but noise?
“Are you the sort of man who would put the poison into his own goblet or his enemy’s? Now, a clever man would put the poison into his own goblet because he would know that only a great fool would reach for what he was given. I am not a great fool so I can clearly not choose the wine in front of you...But you must have known I was not a great fool; you would have counted on it, so I can clearly not choose the wine in front of me.” …etc.
Or consider a conversation between an FAI that wants to keep the world safe for humans, and a UFAI that wants to turn the world into paperclips.
In terms of individual rationality, I hope I would notice my own severe confusion and then assign >50% probability to the majority vote.
Noticing your own severe confusion should lead to investigating the reasons for the disagreement, not to immediately going along with the majority. Honest Bayesians cannot agree to agree either. They must go through the process of sharing their information, not just their conclusions.