The Power of Mutual Knowledge

There’s a puzzle I first encountered on Steven Landsburg’s blog “The Big Questions.” It involves an island of 100 blue-eyed and 100 brown-eyed natives being visited by a foreigner. There is a strictly observed religious tradition to never talk about anyone else’s eye-color, and to commit ritual suicide within a day if you ever discover your own eye color. (There are no reflective surfaces on the island.) But of course everyone can see everyone else’s eye color. Everyone with blue eyes knows there are at least 99 blue-eyed people and 100 brown-eyed people, just as everyone with brown eyes knows there are at least 99 brown-eyed people and 100 blue-eyed people. They just don’t know their own eye color. A foreigner (who happens to have blue eyes) arrives by boat, spends several months visiting and learning their ways, then sails away. Just as he leaves, he says, “Well, how interesting that there would be blue-eyed people in this part of the world!” And he sails off.

At first glance, he didn’t tell them anything. “Of course, everybody already knew that there are blue eyed people on the island! The foreigner’s statement adds no information.” But if you work through the puzzle, you discover the surprising result that everyone commits ritual suicide on the 100^th day. It's a subtle story about mutual knowledge slowly creeping in and eventually having horrendous consequences. (Note that Landsburg is making a very different point than I am.)

I have two dueling thoughts on this. My first thought is, “This is way too complicated for anyone to actually figure out. Nobody is smart enough to actual work this out and figure out their own eye-color. In the real world, everyone would be safe.”

My second thought is, “Social life is unimaginably more complex than a simple rule about eye-color and ritual suicide. Of course people are constantly working out complex implications of mutual knowledge. Of course blurting shit out makes people uncomfortable. It may only 'reveal' information that everybody knows. But it reveals that everybody knows that everybody knows that everybody knows, ad infinitum.”

Imagine saying something unflattering about a coworker. “Everyone in this room knows you’re not qualified.” Everyone, including the accused, might already know, and everyone might suspect that everyone else already thinks it. But plausible deniability has been taken away. Now every time this coworker looks someone in the eye, he’ll see shame staring back at him. The boss, who was willing to tolerate the under-performer out of pity, doesn't have plausible deniability when someone asks, "How can you keep him on your team?" The coworker who was picking up the under-performer's slack feels emasculated if he continues. Everyone could live with the uncomfortable truth before it became mutual knowledge. It doesn't have to be such an obvious accusation, either. More in line with the puzzle, it could be a snippy comment about someone not carrying his weight. It's obvious enough who the target was, so mutual knowledge seeps in. 

You could think of other examples. You're in a group of friends, two of whom have an obvious mutual crush, and perhaps another friend in the group is jealous. Maybe everyone knows this dynamic exists, and maybe everyone suspects that everyone else knows. But blurting it out would be really uncomfortable. Even someone who indirectly hinted at it (perhaps with a light joke or teasing) might be scolded or shamed for creating an awkward moment. If you don't viscerally feel the discomfort of this scenario, think about how the group might split into factions. The jealous rival might feel compelled by shame to avoid the flirting couple. Other friends might feel compelled to choose between factions. Even when everybody knows and everybody suspects that everybody knows, everyone still has plausible deniability. 

In the same vein, merely stating that "some people" have cynical attitudes and do illicit activities may implicate you. In an alternative version of the puzzle given above, there is a society of 100 couples. Every husband cheats on his wife, and every wife knows about every infidelity of every other woman's husband (just not her own). In this version, she must murder her husband within 24 hours if she figures out he is a cheater. By the same logic as the blue-eyed and brown-eyed islander story, if some incautious outsider blurts out what everyone already knows, something awful happens. On day 100, all the cheating husbands die. 

This isn't about trivial matters of social faux pas and embarrassment. Dictators don't like crowds, because crowds tend to turn into angry protests. And these reveal to the world that, yes, everyone else is dissatisfied with the status quo. It's hard to maintain the fiction of a "100% approval rating" or a "bountiful harvest" in the light of this kind of public demonstration. Nicolae Ceausescu was brought down when people started chanting at a public speech and he lost control of the audience. The Arab Spring seems like another example of this. Why wouldn't Hosni Mubarak just sit in office and hold power? Why not just ignore the protesters and wait it out, like American presidents do all the time? I think this "mutual knowledge" dynamic is at play and cracks the armor of a dictatorship much more than it does in a democracy.

I am stealing some of these ideas about mutual knowledge from a Steven Pinker book, though at this point I couldn't even tell you which one. The Blank Slate? Or maybe it was How the Mind Works.

Are there other good examples of this dynamic at work?
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Think about an island with one blue-eyed and one brown-eyed person. On this island, the foreigner’s statement would cause the blue-eyed person to discover his eye-color. The blue-eyed person knows that the other person’s eyes are brown. Knowing he must be the blue-eyed person, he commits ritual suicide. The brown eyed person, seeing this, realizes that he must have brown eyes, or the blue-eyed person wouldn’t have discovered his eye-color and killed himself. “If I had blue eyes, he would have waited a day..”

Now think about an island with two blue-eyed and two brown-eyed people. The blue-eyed people know there’s at least one blue-eyed person. The brown-eyed people know there are at least two blue-eyed people. The foreigner’s statement might first cause each blue-eyed person to think, “Oh, he’s talking about that blue-eyed person. If that blue-eyed person sees three brown-eyed people he’ll commit ritual suicide within 24 hours. If not…” So when the blue-eyed person doesn’t commit ritual suicide within 24 hours, the other blue eyed person says, “Uh, oh. He was talking about both of us!” This is symmetric. They both commit ritual suicide. The brown-eyed people have worked this out, too, and so they know there were 2 blue-eyed people, not three. This allows them to work out that they must both have brown eyes.

Now think about an island with 3 blue-eyed and 3 brown-eyed people…work this one out yourself. By induction, this process keeps going. “On the Xth day, all X blue eyed and all X brown-eyed people commit ritual suicide.” And all because one loud-mouth visitor blurted something out.  

Or think about it this way. Obviously if there's only one blue eyed person on the island, the foreigner's statement that there's a blue-eyed person reveals that person's eye color to him. Ritual suicide.
Given this, if there are two blue-eyed people, the foreigner's statement will reveal that there's at least one blue-eyed person. On day 2, each blue eyed person works out that the other sees a blue-eyed instead of a brown-eyed person and commits ritual suicide.
Given this, if there are three blue-eyed people, after day 2 each blue-eyed person works out that there must be three blue-eyed people.
And so on. There's no magic number where this induction stops working.