Common Knowledge and Aumann’s Agreement Theorem
Two people of indeterminate gender—let’s not make any assumptions here—go on a date. Afterward, one of them says to the other: “Would you like to come up to my apartment to see my etchings?” The other says, “Sure, I’d love to see them.”
This is such a cliché that we might not even notice the deep paradox here. It’s like with life itself: people knew for thousands of years that every bird has the right kind of beak for its environment, but not until Darwin and Wallace could anyone articulate why (and only a few people before them even recognized there was a question there that called for a non-circular answer).
In our case, the puzzle is this: both people on the date know perfectly well that the reason they’re going up to the apartment has nothing to do with etchings. They probably even both know the other knows that. But if that’s the case, then why don’t they just blurt it out: “would you like to come up for some intercourse?” (Or “fluid transfer,” as the John Nash character put it in theBeautiful Mind movie?)
.. The crucial concept here is common knowledge. We call a fact “common knowledge” if, not only does everyone know it, but everyone knows everyone knows it, and everyone knows everyone knows everyone knows it, and so on.
.. The moral is that the mere act of saying something publicly can change the world—even if everything you said was already obvious to every last one of your listeners. For it’s possible that, until your announcement, not everyone knew that everyone knew the thing, or knew everyone knew everyone knew it, etc., and that could have prevented them from acting.
.. But real humans tend to be unable to reason reliably past three or four levels in the knowledge hierarchy.
.. (My favorite Soviet joke involves a man standing in the Moscow train station, handing out leaflets to everyone who passes by. Eventually, of course, the KGB arrests him—but they discover to their surprise that the leaflets are just blank pieces of paper. “What’s the meaning of this?” they demand. “What is there to write?” replies the man. “It’s so obvious!” Note that this is precisely a situation where the man is trying to make common knowledge something he assumes his “readers” already know.)
.. The kicker is that, to prevent something from becoming common knowledge, all you need to do is censor the common-knowledge-producing mechanisms: the press, the Internet, public meetings. This nicely explains why despots throughout history have been so obsessed with controlling the press, and also explains how it’s possible for 10% of a population to murder and enslave the other 90% (as has happened again and again in our species’ sorry history), even though the 90% could easily overwhelm the 10% by acting in concert. Finally, it explains why believers in the Enlightenment project tend to be such fanatical absolutists about free speech—why they refuse to “balance” it against cultural sensitivity or social harmony or any other value, as so many well-meaning people urge these days.