Missionaries

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How do we create Common Knowledge?
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We start this week with a riddle. It’s a famous one, so no Googling…give yourself the chance to try and think it through. 🙂

Here it is as written in Terence Tao’s Blog:

“There is an island upon which a tribe resides. The tribe consists of 1000 people, with various eye colours.

Yet, their religion forbids them to know their own eye color, or even to discuss the topic; thus, each resident can (and does) see the eye colors of all other residents, but has no way of discovering his or her own (there are no reflective surfaces).

If a tribesperson does discover his or her own eye color, then their religion compels them to leave the island at noon the following day.

All the tribespeople are highly logical and devout, and they all know that each other is also highly logical and devout (and they all know that they all know that each other is highly logical and devout, and so forth).

Note: for the purposes of this logic puzzle, “highly logical” means that any conclusion that can logically deduced from the information and observations available to an islander, will automatically be known to that islander.

Of the 1000 islanders, it turns out that 100 of them have blue eyes and 900 of them have brown eyes, although the islanders are not initially aware of these statistics (each of them can of course only see 999 of the 1000 tribespeople).

One day, a blue-eyed foreign missionary visits to the island and wins the complete trust of the tribe.

One evening, he addresses the entire tribe to thank them for their hospitality.

However, not knowing the customs, the missionary makes the mistake of mentioning eye color in his address, remarking “how unusual it is to see another blue-eyed person like myself in this region of the world”.

What effect, if anything, does this faux pas have on the tribe?

With this (admittedly strange) riddle we return to the Common Knowledge game.

You’ll recall that we introduced the idea of a Common Knowledge game in last week’s email…

(And if you missed it, I’ve gone ahead and posted it on the blog in order to make sure we don’t leave our new subscribers behind.)

…as what happens when many people are making second, third, and fourth order decisions.

The crowd acts not based on how each individual person thinks, but on what they think about what other people think.

The question then becomes:

How do we know what other people think?

To be clear, I don’t think we ever know what other people think…not really.

For our purposes we are most interested in figuring out how other people decide what they think other people think.

In other words: “How do we know what’s common knowledge?”

First, let’s define common knowledge.

In one sense, common knowledge is just stuff that pretty much everyone knows.

The earth orbits the sun.

The Statue of Liberty is in New York City.

But common knowledge is more than just what we know…it’s what we know other people know.

From Wikipedia:

Common knowledge is a special kind of knowledge for a group of agents. There is common knowledge of p in a group of agents G when all the agents in G know p, they all know that they know p, they all know that they all know that they know p, and so on ad infinitum.”

It’s not just what you know – it’s what you know that I know you know I know.

How do we figure that out?

This brings us back to the island of the eye-color tribe.

Let’s start with the answer and work backwards (if you still want to take a stab at solving the riddle yourself, don’t read any further).

What effect does the missionary’s pronouncement have on the tribe?

All 100 tribe members with blue eyes leave the island at noon on 100 days after the missionary’s statement.

Why?

To help work this out, imagine that there was only one person with blue eyes on the island. What would happen then?

The missionary would pronounce that they see a person with blue eyes.

The blue-eyed island would immediately realize that the missionary must be referring to them; after all, they know that every other islander has brown eyes. Therefore, they would leave the island at Noon the next day.

Now, let’s make things slightly more complicated, and imagine that there are two blue-eyed islanders – let’s call them Marty and Beth.

The missionary states that they see a person with blue eyes.

Marty thinks: “Wow! He just called out Beth as having blue eyes. That means Beth will leave the island tomorrow.”

Beth thinks: “Yup – he’s talking about Marty. Poor Marty! He’ll have to leave the island tomorrow.

Tomorrow rolls around. Both Beth and Marty gather around with everyone else to watch the blue-eyed islander leave the island…

And no one leaves.

Beth and Marty stare at each other. The other islanders stand around awkwardly.

Beth thinks: “Wait a minute…Marty has blue eyes. He should know that he needs to leave, because he knows everyone else’s eye color, and can deduce that his eyes are blue.

But if he didn’t leave, that means that he thinks he doesn’t have to, because someone else should have deduced that their eyes are blue. And since I know that everyone else’s eyes are brown, that means….”

Marty thinks: “Wait a minute…Beth has blue eyes. She should know that she needs to leave, because she knows everyone else’s eye color, and can deduce that her eyes are blue.

But if she didn’t leave, that means that she thinks she doesn’t have to, because someone else should have deduced that their eyes are blue. And since I know that everyone else’s eyes are brown, that means….”

Beth and Marty together: “MY EYES MUST BE BLUE!”

And so Beth and Marty leave the island together at noon the next day.

This process continues for each new islander we add to the “blue eyes” group. And so the generalized rule for the riddle is:

If N is the number of blue-eyed islanders, nothing happens until the Nth day, whereupon all blue-eyed islanders leave simultaneously.

There’s a critical element to this riddle that many people (namely, me) miss:

Everyone already knows everyone else’s eye color.

It isn’t that the islanders are learning anything truly new about other people. They aren’t. It has nothing to do with what people know.

What changes the situation on the island is missionary’s public announcement.

It’s know that people suddenly know – it’s that they know that other people know.

Common Knowledge is created by public statement, even when all the information was already available privately.

It isn’t about eye color; it’s about the missionary.

Once we hear the missionary’s message, it takes time for everyone to process. It takes time for people to make strategic decisions.

The more ambiguity in the message, the more time it takes to have an effect (for example, “Beth has blue eyes” is an unambiguous message that would have had an immediate effect. “I see someone with blue eyes” is ambiguous, and takes N+1 days to have an effect.)

We ask the question at the beginning:

How do we know what other people think?

The answer: we create an understanding of what other people know by listening to the missionaries we believe everyone else listens to.

The message of the missionary has to be widely propagated and public. What are the channels that “everyone” watches? What are the sources that “everyone” follows?

Once the message is broadcast, there’s a time delay on the effect based on the amount of ambiguity in the message and the difficulty of the strategic decision-making that follows.

But.

Once everyone hears the message…

And everyone updates their beliefs about what other people believe…

Change can be both sweeping and immediate.

This is how massive changes can seemingly happen overnight. How “stable” regimes can be toppled, social movements can ignite, and stable equilibrium everywhere can be dramatically and irrevocably disrupted.

The Common Knowledge game teaches us some critical lessons:

1.) It isn’t the knowledge itself that matters, but what we believe other people know.

2.) You have better be aware of who the missionaries are, and what they’re saying.

Otherwise, it might soon be YOUR turn to leave the island.

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