I know. I know that I know. For as long as the human race existed, knowledge embodied power. In the life of a society, however, what becomes even more important is the fact that we share certain knowledge with fellow human beings, and that we, moreover, are aware of each other’s knowledge. This special type of awareness is termed common knowledge: I know that you know that I know that you know – and so on to infinity.

In theory, common knowledge is surprisingly hard to achieve. One of the classic examples is the Coordinated Attack Problem, where two armies need to coordinate on the timing of the attack. If they attack separately, they fail. Sending a courier with a message to the other army’s camp seems like a very simple solution to the problem. But, there is a very small probability that a courier does not reach the other camp. How would Camp One know that the friendly Camp Two in fact received the message? Of course, Camp Two can send a courier back confirming the receipt. But again, there is a small probability that the courier does not reach the destination. If he doesn’t then Camp One would not know whether Camp Two knows the timing of the attack, and therefore will not attack at the specified time. If, however, Camp One successfully receives the message, they would need to once again confirm by sending a courier back to Camp Two. This exchange can go on forever, and paradoxically the armies will not be able to coordinate, no matter how many couriers cross the valley between the camps. Thus, in a world, where achieving common knowledge is a virtually impossible task, we seem to be destined to live with ambiguity.

In practice, of course, we do not need to send and receive infinite confirmations to coordinate on a certain equilibrium. Dalkiran and Hoffman (2011) for example, argue that all we need to do is to believe with sufficiently high probability that the other person knows that we know and so on. A public announcement about an event, or a publicly visible symbol, would facilitate this kind of common belief. Without the announcement, however, it is possible that we will end up having only mutual knowledge of the event (I know that I know and you know that you know, but we are not aware, or not sure, that the other knows as well).

It is surprising how the ambiguity about what others know can and has been exploited in public life throughout centuries – sometimes with very significant social consequences. The disconnect between mutual knowledge and common knowledge has been used to perpetuate the power of the ruling elite, in campaigns of disinformation, to make the public believe in something that was contrary to what people privately knew or observed. In the Soviet Union, for example, public announcements have been used very skillfully to perpetuate the myths of efficiency and prosperity, contrary to what people witnessed in their everyday life.

Like any weapon, however, the ambiguity can be used to topple as well as perpetuate. Consider for example that in order for me to rise against the regime, I need to know that sufficiently many others share my discontent. Can I do it without the risk of publicly speaking my mind? I can certainly show up on a public square carrying a rose, a carnation, wearing an orange scarf, or a tri-color cockade. It is quite possible, that this is precisely what I felt like wearing on that day regardless of my political beliefs - no regime could fault me for that. But when all of a sudden I see another person carrying the same symbol, once our eyes meet, the ambiguity is removed. The civic society, the common knowledge, the revolution is born: I know that you know, that I know, that you know that I know. All the way to infinity.