Recently someone pointed out to me that there was no good canonical post that explained the use of common knowledge in society. Since I wanted to be able to link to such a post, I decided to try to write it.

The epistemic status of this post is that I hoped to provide an explanation for a standard, mainstream idea, in a concrete way that could be broadly understood rather than in a mathematical/logical fashion, and so the definitions should all be correct, though the examples in the latter half are more speculative and likely contain some inaccuracies.

Let's start with a puzzle. What do these three things have in common?

Dictatorships all through history have attempted to suppress freedom of the press and freedom of speech. Why is this? Are they just very sensitive? On the other side, the leaders of the Enlightenment fought for freedom of speech, and would not budge an inch against this principle.
When two people are on a date and want to sleep with each other, the conversation will often move towards but never explicitly discuss having sex. The two may discuss going back to the place of one of theirs, with a different explicit reason discussed (e.g. "to have a drink"), even if both want to have sex.
Throughout history, communities have had religious rituals that look very similar. Everyone in the village has to join in. There are repetitive songs, repetitive lectures on the same holy books, chanting together. Why, of all the possible community events (e.g. dinner, parties, etc) is this the most common type?

What these three things have in common, is common knowledge - or at least, the attempt to create it.

Before I spell that out, we’ll take a brief look into game theory so that we have the language to describe clearly what’s going on. Then we’ll be able to see concretely in a bunch of examples, how common knowledge is necessary to understand and build institutions.

Prisoner's Dilemmas vs Coordination Problems

To understand why common knowledge is useful, I want to contrast two types of situations in game theory: Prisoner’s Dilemmas and Coordination Problems. They look similar at first glance, but their payoff matrices have important differences.

The Prisoner's Dilemma (PD)

You’ve probably heard of it - two players have the opportunity to cooperate, or defect against each other, based on a story about two prisoners being offered a deal if they testify against the othe...