Common knowledge (logic)
Based on Wikipedia: Common knowledge (logic)
The Puzzle That Breaks Your Brain
Imagine an island where everyone has either blue eyes or green eyes. There are exactly one hundred blue-eyed people. Everyone can see everyone else's eye color, but nobody knows their own—there are no mirrors, no reflective surfaces, and discussing eye color is forbidden. There's also a terrible rule: if you ever figure out that you have blue eyes, you must leave the island at dawn the next morning.
For years, nothing happens. The blue-eyed people see ninety-nine other blue-eyed people. The green-eyed people see one hundred. Everyone coexists peacefully.
Then a stranger arrives. She gathers everyone together and announces: "At least one of you has blue eyes."
She hasn't told anyone anything they didn't already know. Every single person on the island can see at least one blue-eyed person. Yet one hundred days later, all one hundred blue-eyed people leave the island simultaneously.
How? Why? What changed?
The answer lies in a subtle but profound distinction that philosophers and mathematicians call the difference between "mutual knowledge" and "common knowledge." Understanding this distinction doesn't just solve a logic puzzle—it illuminates something fundamental about how groups of people coordinate, how markets function, and why public announcements can trigger cascading effects even when they reveal nothing that wasn't already known.
What You Know Versus What Everyone Knows You Know
Let's start with the basics. When we say someone "knows" something, we mean they believe it's true, and it actually is true. Simple enough.
Now imagine a group of people. We might say there's "mutual knowledge" of some fact when everyone in the group knows it. If I know the meeting is at three o'clock, and you know the meeting is at three o'clock, and Sarah knows the meeting is at three o'clock, then we have mutual knowledge of the meeting time.
But here's where it gets interesting. Mutual knowledge isn't the same as common knowledge.
Common knowledge requires something more. It's not enough that everyone knows the meeting is at three. Everyone must also know that everyone knows. And everyone must know that everyone knows that everyone knows. And so on, infinitely.
This sounds like philosophical hair-splitting. It isn't.
The Private Message Problem
Suppose I need to coordinate with you on a project. I send you an email saying "Let's meet at three." You now know the meeting time. But do I know that you know? Maybe your email was down. Maybe it went to spam.
So you send a reply: "Got it, see you at three." Now I know that you know.
But do you know that I know that you know? Maybe my email is down now. Maybe your reply bounced.
So I send another message: "Great, confirmed." Now you know that I know that you know.
But do I know that you know that I know that you know?
You can see where this is going. No finite exchange of private messages can ever establish common knowledge. There's always another level of uncertainty.
This isn't just a theoretical curiosity. It's the reason why, in computer science, the "coordinated attack problem" is provably unsolvable. Two generals trying to coordinate an attack using unreliable messengers can never achieve perfect certainty about coordination, no matter how many confirmations they exchange.
The Magic of Public Announcements
Here's what makes public announcements special: they create common knowledge in a single stroke.
When the stranger stands in the town square and announces "at least one of you has blue eyes" to everyone simultaneously, something remarkable happens. Everyone hears it. Everyone sees everyone else hearing it. Everyone knows that everyone knows that everyone saw everyone hear it. The entire infinite tower of knowledge-about-knowledge springs into existence instantaneously.
This is why the stranger's announcement changes everything on the island, even though it conveys no new first-order information. Before the announcement, everyone knew there were blue-eyed people. But the blue-eyed people didn't have common knowledge of this fact. After the announcement, they do.
Solving the Blue Eyes Puzzle
Let's work through the logic, starting with the simplest case.
Imagine there's only one blue-eyed person—call her Alice. Alice sees only green eyes around her. When the stranger announces "at least one of you has blue eyes," Alice immediately realizes she must be the one. She leaves at the first dawn.
Now imagine there are two blue-eyed people: Alice and Bob. Alice sees Bob's blue eyes. Bob sees Alice's blue eyes. When the stranger makes her announcement, both Alice and Bob already know there's at least one blue-eyed person—they can each see one.
But Alice doesn't know that Bob knows. From Alice's perspective, it's possible that Bob sees only green eyes (since Alice doesn't know her own eye color). If that were true, Bob would learn something new from the announcement and leave on the first dawn.
The first dawn comes. Nobody leaves.
This is new information. Alice now knows that Bob didn't think he was the only blue-eyed person. That means Bob must see at least one pair of blue eyes. Since Alice sees only Bob with blue eyes, those blue eyes must be her own.
Bob reasons identically. On the second dawn, both leave.
The Infinite Regress
The pattern continues. With three blue-eyed people, the first two dawns pass with no departures. On the third dawn, all three leave. With one hundred blue-eyed people, ninety-nine dawns pass before everyone figures it out on the hundredth day.
What's happening is a cascade of reasoning about reasoning about reasoning. Each passing dawn where nobody leaves eliminates one layer of uncertainty. The announcement doesn't tell anyone anything directly—it provides a synchronization point, a shared starting gun that allows everyone to begin counting together.
Before the announcement, consider what a blue-eyed person—let's call her Carol—knows when there are exactly three blue-eyed people (Carol, David, and Eve).
Carol knows there are blue-eyed people. She can see David and Eve.
Carol knows that David knows there are blue-eyed people. David can see Eve.
But Carol doesn't know that David knows that Eve knows there are blue-eyed people. From Carol's perspective, she might not have blue eyes. If she doesn't, then David only sees Eve with blue eyes. And if David only sees Eve, then David might think Eve sees no blue eyes at all.
The announcement breaks this chain of uncertainty. Now Carol knows that David knows that Eve knows there's at least one blue-eyed person—because everyone heard the announcement, everyone saw everyone hear it, and so on forever.
From Philosophy to Mathematics
The concept of common knowledge emerged almost simultaneously in multiple fields during the late 1960s. The philosopher David Lewis introduced it in his 1969 book "Convention," exploring how social conventions—like driving on the right side of the road—become self-reinforcing. The sociologist Morris Friedell defined it in a paper the same year.
But it was the economist and mathematician Robert Aumann who, in 1976, gave common knowledge its rigorous mathematical formulation. Aumann, who would later win the Nobel Prize in Economics, showed how to model common knowledge using set theory and probability. His work revealed why common knowledge matters so much for understanding coordination, bargaining, and strategic interaction.
Computer scientists became interested in the 1980s, recognizing that distributed systems—networks of computers trying to coordinate—faced exactly the same problems that philosophers had been puzzling over. The inability to achieve common knowledge through message-passing explained fundamental limitations in distributed computing.
Common Knowledge in the Real World
The blue eyes puzzle might seem artificial, but the dynamics it illustrates appear everywhere.
Consider financial markets. Everyone might privately believe a stock is overvalued. But as long as everyone thinks everyone else might still be a true believer, the bubble continues. It takes a public event—a damning news report, a regulatory announcement—to create common knowledge that "everyone knows this is overpriced." Then the crash happens.
Or consider political change. Under an authoritarian regime, many people might privately oppose the government. But as long as each person thinks they might be alone in their opposition, nothing happens. A public protest creates common knowledge of shared dissent. Suddenly, everyone knows that everyone knows, and coordination becomes possible.
The economist Timur Kuran has documented how this dynamic explains why authoritarian regimes can appear stable for decades and then collapse seemingly overnight. The underlying dissatisfaction was always there. What changed was its visibility—its transformation from mutual knowledge to common knowledge.
The Fixed-Point Puzzle
Mathematicians have grappled with a tricky problem in formalizing common knowledge: how do you capture an infinite tower of knowledge in a finite logical system?
Common knowledge of some proposition means: everyone knows it, everyone knows everyone knows it, everyone knows everyone knows everyone knows it, and so on without end. You can't write that as a normal logical formula—it would be infinitely long.
The solution uses a concept from mathematics called a fixed point. A fixed point is where applying some operation gives you back what you started with. Imagine a function that takes a number and doubles it, then subtracts one. The fixed point would be the number where doubling and subtracting one gives you the same number back (in this case, it's one: double one to get two, subtract one, you're back at one).
Common knowledge can be defined as the fixed point of a particular operation. Think of it this way: common knowledge of something is that thing, plus everyone knowing the common knowledge of it. This sounds circular, and it is—but it's a productive kind of circularity. It's like defining a recursive function that calls itself. The fixed-point definition captures the infinite regress in a finite formula.
Orders of Knowledge
Logicians distinguish different "orders" of shared knowledge. First-order mutual knowledge means everyone knows something. Second-order mutual knowledge means everyone knows that everyone knows. Third-order means everyone knows that everyone knows that everyone knows.
In the blue eyes puzzle with exactly three blue-eyed people, before the announcement there is second-order mutual knowledge that someone has blue eyes—everyone knows, and everyone knows that everyone knows. But there isn't third-order knowledge. That's exactly the layer the announcement provides.
Common knowledge is, in a sense, the limit of this sequence—infinite-order mutual knowledge. When you have it, you never run out of "everyone knows that everyone knows that..." levels.
Why This Matters Beyond Logic Puzzles
The distinction between mutual and common knowledge illuminates a deep truth about social reality: coordination requires not just shared beliefs, but shared awareness of those shared beliefs.
This is why rituals matter. When everyone stands for the national anthem, each person's standing signals something to everyone else, and everyone sees everyone seeing this. The ritual creates common knowledge of shared membership, shared values, shared identity.
This is why public ceremonies persist in an age of instant private communication. A wedding announcement posted on social media isn't quite the same as a ceremony where everyone gathers and witnesses. The simultaneous presence creates common knowledge in a way that sequential private notifications cannot.
This is why authoritarian regimes fear public gatherings. The content of what protesters say matters less than the fact that they're saying it together, in public, where everyone can see everyone seeing it. Private dissent is manageable. Common knowledge of dissent is revolutionary.
The Mathematician's Playground
The blue eyes puzzle and its variants have become a rich source of mathematical investigation. John Conway, the brilliant mathematician known for inventing the Game of Life, explored numerous variations. What if the announcement is made privately to each person rather than publicly? What if some people are known to be unreliable reasoners? What if there's uncertainty about who heard the announcement?
Each variation illuminates different aspects of how knowledge propagates through groups. The puzzles aren't just brain teasers—they're windows into the logical structure of social epistemology.
The Stranger's Gift
Return one last time to the island. The stranger's announcement seemed to contain no new information. Every islander could already see at least one pair of blue eyes. Yet the announcement changed everything.
What the stranger gave wasn't information in the usual sense. She gave them a shared reference point—a common origin from which synchronized reasoning could proceed. She transformed a fact that everyone knew privately into a fact that was publicly, commonly, infinitely known.
After a hundred dawns of silent nights, a hundred blue-eyed people pack their bags and depart together. They leave not because they learned what color their eyes were—in some sense, the visual information was always available to them. They leave because they finally reached the end of a chain of reasoning that required, at every step, the certain knowledge that everyone else was reasoning in lockstep.
Common knowledge made coordination possible. And coordination, in the end, is what transforms individual beliefs into collective action—whether that action is leaving an island at dawn, selling an overpriced stock, or standing up against a regime that everyone privately despises but publicly fears.
The stranger's simple statement—"at least one of you has blue eyes"—wasn't information. It was permission. Permission to reason together. Permission to act.