What Do We Really Know About What Other People Know?

A waiter, some barbecue, and a very strange chain of knowledge

Here’s a simple situation. A waiter spills gravy on a guest’s dress. The guest is angry. The waiter says, “I’m sorry. It was my fault.”

Both of them already knew the waiter was at fault. The guest knew it, the waiter knew it, and they both knew the other knew it. So why did the waiter say it out loud?

Because he wanted the guest to know that he knew he was at fault. Not just that he was at fault, but that he knew it. And by saying it publicly, he made sure of something more: the guest now knows that he knows she knows that he knows it was his fault. That’s a lot of nested “knows.” Philosophers call this kind of thing higher-order knowledge — knowing about what someone knows about what someone knows, and so on.

This seems like an elaborate puzzle about a simple apology. But it leads to a strange and powerful idea: common knowledge, which is different from just everyone knowing something. Common knowledge is what happens when everyone knows something, and everyone knows that everyone knows it, and everyone knows that everyone knows that everyone knows it, and so on, forever.

The barbecue problem

Imagine a group of friends having a picnic. They’re eating barbecue ribs. After the meal, some of them have barbecue sauce on their faces. Nobody can see their own face. There’s a cook who can see everyone.

The cook comes out with ice cream, looks around, and announces: “At least one of you has barbecue sauce on your face. I’m going to ring this bell over and over. When you figure out you’re messy, wipe your face. Then we’ll have dessert.”

Now, here’s the thing. If only one person is messy, that person looks around, sees everyone else is clean, and realizes immediately: “It must be me.” They wipe their face after the first ring.

But if two people are messy, something interesting happens. Let’s call them Alex and Bailey. When the cook rings the bell the first time, Alex looks at Bailey. Alex sees that Bailey is messy. But Alex doesn’t know about himself. And Bailey looks at Alex and sees Alex is messy. So neither wipes their face after the first ring.

Then the bell rings a second time. Alex thinks: “If I were clean, then Bailey would have seen only clean faces and would have known she was the only messy one after the first ring. But she didn’t wipe her face. So I must be messy too.” And Bailey reasons the same way about Alex. So after the second ring, both wipe their faces.

If three people are messy, it takes three rings. If four, four rings. And so on.

Here’s the puzzle: the cook’s announcement told everyone something they already knew. If there are two messy people, each already knew that at least one person was messy. They could see that with their own eyes. Yet the announcement changed everything. Without it, nobody would ever figure out they were messy. With it, they eventually do.

The announcement made the fact common knowledge instead of just something everyone happened to know. Before the announcement, each person knew there was at least one messy person. But they didn’t know that everyone else knew it too. The public announcement changed that. It turned a fact that everyone knew into a fact that everyone knew everyone knew, and so on up the chain.

So what exactly is common knowledge?

Let’s say some fact — call it A — is true. Philosophers distinguish between several levels of knowing:

First-order knowledge: I know A.
Mutual knowledge: Everyone in a group knows A.
Common knowledge: Everyone knows A, and everyone knows that everyone knows A, and everyone knows that everyone knows that everyone knows A, and so on, forever.

Common knowledge is like an infinite ladder of “knows.” If you try to climb all the way up, you never reach the top. But sometimes a public event — like the cook’s announcement — plants the ladder on solid ground.

Why does this matter?

Common knowledge turns out to be important for all sorts of things that people do together. Think about these situations:

Coordination problems. Suppose you and your friend get separated in a big department store. You didn’t plan a meeting spot. Where do you look? You’ll probably go somewhere that seems “obvious.” But what’s obvious depends on what you think the other person thinks is obvious — which depends on what you think they think you think is obvious… This infinite loop is exactly the problem of common knowledge. You need to know that the other person will be in the same place, and they need to know that you know, and so on. That’s why established meeting spots work so well — they’re common knowledge.

Agreements and promises. Imagine two farmers whose crops ripen at different times. Each needs the other’s help to harvest, or the crop will rot. Both know this. But will they help each other?

The farmer whose crop ripens first thinks: “If I help my neighbor now, then when my crop ripens later, he’ll already have what he needs from me. He has no reason to help me then. So I shouldn’t help him now.” The other farmer thinks the same way. So neither helps, and the crops rot. Both would be better off if they cooperated. But without common knowledge that they’ll both cooperate, each does what seems safe in the moment.

This is a famous puzzle in philosophy. It shows that even when everyone knows they’d all be better off working together, that knowledge isn’t enough. They need something more — common knowledge of their intentions, or some way to create it.

Conventions. Think about which side of the road to drive on. It doesn’t really matter whether everyone drives on the left or the right, as long as everyone does the same thing. But for that to work, everyone needs to know what everyone else will do. Not just know it — know that everyone knows it, and so on. That’s why driving rules are laws. They create common knowledge.

But is common knowledge even possible?

Here’s a problem. The definition of common knowledge involves an infinite chain: “A knows that B knows that A knows that B knows…” No actual human being can think through an infinite number of steps. So how could anyone ever have common knowledge of anything?

Some philosophers say we don’t need to think through all the steps. Common knowledge can be generated by a single public event, if everyone recognizes it as such. When the cook makes his announcement, everyone hears it at the same time, and everyone knows everyone else heard it, and everyone knows everyone knows that, and so on. The public event does the work for us. We don’t need to mentally climb the infinite ladder — we just need to know we’re at the bottom of it.

Other philosophers are more skeptical. They point out that in real life, we can never be completely sure that someone else heard the announcement, or that they understood it the same way, or that they drew the right conclusions. There’s always a tiny bit of uncertainty. And if there’s any uncertainty at all, common knowledge might break down.

Here’s a famous example. Two people want to meet for dinner. One knows which restaurant the chef is working at tonight. The other doesn’t. They agree that the one who knows will send an email to the other, and the other will automatically reply. But sometimes emails get lost. So if you send a message, you don’t know if the other person got it unless you get a reply. And if you get a reply, you don’t know if they know you got their reply unless you send another reply. And so on.

If the email system worked perfectly and sent infinite replies instantly, they’d eventually have common knowledge. But in reality, the chain of replies stops somewhere. And when it stops, they don’t have true common knowledge — they just have a lot of layers of mutual knowledge.

Strangely, in this situation, having almost common knowledge doesn’t help. People who have exchanged many messages behave as if they had exchanged none at all. They act as if they’re still in the dark, even though they’ve confirmed each other’s knowledge many times over. This is deeply weird, and philosophers still argue about why this happens and what it means.

What philosophers still disagree about

Nobody has fully settled what common knowledge is or whether real people ever have it. Here are some questions that are still open:

How much do we need? Some situations seem to require full common knowledge for people to coordinate. Others seem to work fine with just a few layers of mutual knowledge. Nobody has a complete theory of when each is needed.

Is common knowledge a thing that happens, or a way of describing situations? Some philosophers think common knowledge is a real psychological state that groups can be in. Others think it’s just a useful way to model certain situations, even if real people never actually have it.

Can we create common knowledge intentionally? The cook’s announcement worked. But do public announcements always create common knowledge? What about private conversations, or signals that might be missed?

What about beliefs instead of knowledge? Some philosophers argue that we don’t need full knowledge — just common belief with high enough probability. If everyone is 99.9% sure that everyone else knows something, maybe that’s enough for most practical purposes. But others think this misses the point, that common knowledge has special properties that no amount of high-probability belief can replace.

Key Terms

Term	What it does in the debate
Common knowledge	A fact that everyone knows, and everyone knows that everyone knows, and so on, forever
Mutual knowledge	A fact that everyone in a group knows, but without the nested “knows that they know” chain
Higher-order knowledge	Knowledge about what someone knows about what someone knows — any chain of two or more levels
Coordination problem	A situation where people need to match each other’s actions to succeed, like meeting at the same place
Public announcement	An event that everyone in a group witnesses together, often used to create common knowledge

Key People

David Lewis — A philosopher who first clearly defined common knowledge as an infinite chain of “knows” and showed how it relates to social conventions like driving on the right side of the road.
Robert Aumann — An economist and mathematician who gave a different, more technical definition of common knowledge using partitions of possible worlds, and proved that people who share common knowledge cannot “agree to disagree” about their beliefs.
Thomas Schelling — An economist who studied how people solve coordination problems by finding “obvious” meeting points, which often rely on common knowledge.
Ariel Rubinstein — An economist who created the email game example, showing that almost-common knowledge doesn’t work the same way as actual common knowledge.

Things to Think About

The cook’s announcement in the barbecue puzzle told people something they already knew, but it changed their behavior. Can you think of other situations where hearing something you already know still changes things? What about a teacher saying “Tomorrow’s test is on chapters 3-5” when everyone already read the syllabus?
When you and a friend have an inside joke, do you have common knowledge of the joke? Or just mutual knowledge? What would it take to turn one into the other?
The email game shows that almost-common knowledge can be useless — people act as if they know nothing. Does this match your experience? When has “I’m pretty sure they know” not been enough?
Some philosophers think common knowledge is rare in real life. But people coordinate successfully all the time. If we don’t have common knowledge, what do we have instead? Is it enough?

Where This Shows Up

Computer science: The “coordinated attack problem” in distributed computing is exactly this puzzle — two generals need to agree on when to attack, but messages can get lost. Programmers must design systems that work without common knowledge.
Game theory: Common knowledge assumptions are built into many models of how people make strategic decisions, from economics to international relations.
Social conventions: Why do people stand on one side of an escalator or the other? Why do we say “bless you” after sneezes? These are conventions sustained by common knowledge about what everyone expects.
Group behavior: Riots, protests, and other collective actions often depend on people knowing that enough others will also participate — and knowing that others know this too.