Just to clarify, Hallq uses "mutual knowledge" as if it's synonymous to "common knowledge", but game theorists use the two terms as a contrast: mutual knowledge of A is when everyone knows A, common knowledge of A is when everyone knows that everyone knows that everyone knows (...) A. So this is about raising to common knowledge things that were merely mutual knowledge.
Followup to: Belief in Belief
One of those insights that made me sit upright and say "Aha!" From The Uncredible Hallq:
The power of real deception - outright lies - is easy for even us nerds to understand.
The notion of a lie that the other person knows is a lie, seems very odd at first. Up until I read the Hallq's explanation of Pinker, I had thought in terms of people suppressing uncomfortable thoughts: "If it isn't said out loud, I don't have to deal with it."
Like the friends of a terminal patient, whose disease has progressed to a stage that - if you look it up online - turns out to be nearly universally fatal. So the friends gather around, and wish the patient best hopes for their medical treatment. No one says, "Well, we all know you're going to die; and now it's too late for you to get life insurance and sign up for cryonics. I hope it isn't too painful; let me know if you want me to smuggle you a heroin overdose."
So even that is possible for a nerd to understand - in terms of, as Vassar puts it, thinking of non-nerds as defective nerds...
But the notion of a lie that the other person knows is a lie, but they aren't sure that you know they know it's a lie, and so the social situation occupies a different state from common knowledge...
I think that's the closest I've ever seen life get to imitating a Raymond Smullyan logic puzzle.
Added: Richard quotes Nagel on a further purpose of mutual hypocrisy: preventing an issue from rising to the level where it must be publicly acknowledged and dealt with, because common ground on that issue is not easily available.