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PaperclipPerfector  3mo ago (text post)   3119 thread views

Small-Scale Question Sunday for August 11, 2024

Do you have a dumb question that you're kind of embarrassed to ask in the main thread? Is there something you're just not sure about?

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I'm very interested in the idea of common knowledge. It's been talked a lot about by the Scotts here and here

The crucial concept here is common knowledge. We call a fact “common knowledge” if, not only does everyone know it, but everyone knows everyone knows it, and everyone knows everyone knows everyone knows it, and so on.

I sort of understand this but I want to understand it better. Can someone explain this to me? Why is something not common knowledge if everyone knows that everyone knows it? What is the difference between that and the next level (everyone knows that everyone knows that everyone knows it)? I want to get a more intuitive grasp of that.

The concept often comes up in the context of dictatorships, so that's the context I'll use to illustrate it.

Alice, Bob and Carol live in the Soviet Union during Stalin's regime. Alice hates Stalin and wishes him dead. But Alice has never read a column or editorial which was even mildly critical of Stalin (Stalin controls Pravda), and also knows that everyone who criticises Stalin in any capacity immediately vanishes to the gulag, never to be heard from again. For fear of this happening to her, Alice never criticises Stalin in front of Bob and Carol. Unbeknownst to Alice, Bob and Carol also hate Stalin, but have performed exactly the same risk calculus and decided never to publicly criticise Stalin. Hence conversations between Alice, Bob and Carol consist of three people loudly, conspicuously praising Stalin and successfully deceiving the others that they sincerely admire Stalin and think he's the bee's knees - but all three of them hate him and erroneously believe that they're the only one of the three to think so.

There's a sophomoric theory of how dictatorships come to an end: "people admired Stalin, but then a critical mass of people turned against him and the public rose up to overthrow him". The toy example above illustrates why this theory is wrong: a critical mass of people hating the dictator is necessary but not sufficient to effect an overthrow of the dictatorship. It's perfectly possible for a simple or supermajority of the population to hate the dictator, but for the dictator to remain in power if enough of the people who hate him erroneously believe that their opinion is a minority or fringe opinion. It's not enough for Alice, Bob etc. to hate Stalin: Alice must also know that Bob hates Stalin, and Bob likewise - Stalin must be widely despised, and it must be common knowledge that Stalin is widely despised.

Alice, Bob and Carol live in the Soviet Union during Stalin's regime. Alice hates Stalin and wishes him dead. But Alice has never read a column or editorial which was even mildly critical of Stalin (Stalin controls Pravda), and also knows that everyone who criticises Stalin in any capacity immediately vanishes to the gulag, never to be heard from again. For fear of this happening to her, Alice never criticises Stalin in front of Bob and Carol. Unbeknownst to Alice, Bob and Carol also hate Stalin, but have performed exactly the same risk calculus and decided never to publicly criticise Stalin. Hence conversations between Alice, Bob and Carol consist of three people loudly, conspicuously praising Stalin and successfully deceiving the others that they sincerely admire Stalin and think he's the bee's knees - but all three of them hate him and erroneously believe that they're the only one of the three to think so.

This much makes sense to me, but beyond this it gets tough for me. This sounds like "everyone knows Stalin sucks, but everyone doesn't know that everyone knows Stalin sucks". But let's say everyone did know that everyone knows Stalin sucks. Why is that not common knowledge already? Why is it important that everyone knows that everyone knows that everyone knows that Stalin sucks?

I think the answer is roughly the same as that to my favorite riddle: https://xkcd.com/blue_eyes.html

So I'm a few days late, but I'm struggling with the riddle. I can't get it to work out in my head.

I know the answer is that all n blue-eyed people leave on the nth ferry. I understand why that works for low values of n, and how it builds on itself, as each blue-eyed person expects the (n-1) scenario to play out, and then when it doesn't they realize they have blue eyes.

But at some point, everyone can see multiple blue-eyed people. And everyone can see that everyone else can see multiple blue-eyed people. At that point I can't see how the Guru has provided anyone with any new information or any new common knowledge. Nor has the empty outbound ferry on Day 1. So I don't understand how all the blue-eyed people know to leave on Day 100.

The note provides the inductive base case.

(blue, brown, note falls from the sky saying someone has blue eyes)

(1, 0, False): No information on their eyes. They never leave.

(1, 0, True): No one else could possibly have blue eyes. They leave on day 1.

(1, 1, False): Same as (1, 0, False). No one leaves.

(1, n, False): Same as (1, n-1, False). No on leaves.

(2, 0, True): On day 1, each reasons that if they are brown in (1, 1, True), the other person will leave. The other person doesn't leave. They each leave on day 2.

(n, 0, True): On day n-1, each reasons that if they are brown in (n-1, 0, True), the other n-1 people will leave on day n-1. This doesn't happen. All n people leave on day n.

(2, 0, False): On day 1, each reasons that whether they are blue or brown in (1, 1, False), the other person will never leave. The other person does not leave. This gives no information. No one ever leaves.

Etc

To throw another explanation into the arena: Alice, Bob and Carol currently hate Stalin but each thinks they are alone, so they don't rebel. If Alice found a credible note saying "Bob hates Stalin. Carol hates Stalin." Then she has learned a little, but her options don't increase all that much.

  • She can't try to convince Bob to form a rebellion. Since Bob hasn't gotten a note, he will just assume Alice is an agent trying to catch Bob doing something bad. (Haven't dictatorial regimes employed snitches? Sounds familiar right?)
  • She can't really tell Bob: "I know you hate Stalin." Like before, Bob will assume she is an agent. After all, Bob thinks he's the only one opposing Stalin! (Don't governments deploy sting operations to catch detractors? Glowies etc.)
  • She could try to be honest and vulnerable with Bob and say "I hate Stalin." This is actually risky. I can't search for it now and probably couldn't find it - a blog post about how these scenarios, and expectations, affect friendships. Since the social norm is to report your friends who hate Stalin, then Alice's admission is like saying: "Report me." Bob can maybe reason that Alice expects to be safe telling her friend Bob this. Or in other words, Alice has accused Bob of hating Stalin. This would freak Bob the fuck out. I know I would freak out if one of my friends said they were into ISIS or some kind of terrorist group.

Anyways, the above bullet points are just Alice's thought process. In reality, Bob also got a note saying "Alice hates Stalin. Carol hates Stalin." And Carol also got a similar note. The problem, hopefully you see, is that the notes are secret.

Why is it important that everyone knows that everyone knows that everyone knows that Stalin sucks?

Because without that, Alice, Bob and Carol are individually carrying out a risk calculus under the assumption that each of them is the only one who hates Stalin. If you hate Stalin and everybody else loves him, the expected payoff for publicly criticising him is that you are shipped off to the gulag. Thus everyone in the society is incentivised not to criticise Stalin, and he remains in power.

But when it becomes common knowledge, the payoff matrix changes: if I hate Stalin and I know that everybody else does too, the expected payoff for publicly criticising him is that it creates a chain reaction culminating in Stalin being forced out. Common knowledge that Stalin is widely despised is the only thing that can incentivise a self-interested agent in the system (an agent concerned with self-preservation) to act to change the system from within.

The scenario I'm describing above is a pretty quintessential Moloch trap. From outside the system, it's obvious that it's in Alice, Bob and Carol's best interest to rise up and overthrow Stalin. From within the system, none of them have any good reason to believe that attempting to overthrow Stalin would end in any other way than being immediately shipped off to the gulag or executed; hence, they all keep their heads down and the system endures.

L1: Everyone knows Stalin sucks. But they don't know that everyone knows this, so they don't think a rebellion could succeed.

L2: Everyone knows that everyone knows Stalin sucks. But they don't know that everyone knows this, so for all they know everyone else is on L1 and doesn't think a rebellion could succeed, so people expect other people to steer clear of a rebellion due to erroneously not thinking it could succeed, causing it to fail.

L3: Everyone knows that everyone knows that everyone knows Stalin sucks. But they don't know that everyone knows that, so for all they know everyone else is on L2 and expects rebellions to fail, so people expect other people to steer clear of a rebellion due to erroneously thinking that others will steer clear of it due to erroneously not thinking it could succeed, causing it to fail.

Etcetera.

On the other hand, L∞: A trusted newspaper says that everyone agrees Stalin sucks. Now there's no even hypothetical-within-a-hypothetical possibility that somebody might think a rebellion has no chance.

Well typically in most dictatorships there is a cohort of between ten and seventy percent of the population that genuinely do support the regime for ideological or materialistic reasons. And those people are usually the ones holding the guns. Stalin is actually kind of an outlier in the ratio of sheer fear to genuine support. Also, you don’t have to be a genuine hardline supporter of the regime to rat somebody else out to save yourself.

My Grandpa used to chew tobacco in secret. Officially, this was not okay behavior. He wanted to do it, Grandma disapproved, so he "wasn't allowed to". Partly because it was bad for his health, partly because it's gross. So, not wanting to cause trouble, he pretended to quit, and then went off and did it in secret.

Obviously, Grandpa knows that grandpa chews tobacco. Let's call this level 1 knowledge. Knowledge of the object level fact itself.

He thought he was being sneaky about it, but he wasn't all that sneaky. Grandma found enough evidence to figure it out, so she has level 1 knowledge about him chewing. But since he obviously knows he chews, Grandma knows that he knows, so she has level 2 knowledge: knowledge that Grandpa has level 1 knowledge.

But she doesn't confront Grandpa. Because if she does they might have a fight, or he might decide that now the cat's out of the bag he might as well do it openly in front of people, which is gross. So she says nothing. Everyone knows that Grandpa chews, it is shared knowledge. But it's not common knowledge, because Grandpa doesn't know that she knows.

The lack of common knowledge is meaningful because it allows people to credibly pretend and act like they don't know even when they do.

Now, this didn't happen in real life (I think), but suppose that Grandpa is snooping around and happens to see a text (or more likely overhear a conversation, they're old after all), where Grandma was telling someone about this situation. He's like, "oh crap, I've been discovered. But Grandma chose not to confront me because she doesn't approve but also doesn't want to start a fight. If I want to I could just address this openly, admit what I've been doing, and then either stop or else have the fight and force my way to doing it openly. But I'm reasonably happy the way things have been going, and also I don't want to admit that I've been snooping, so I'll just keep chewing in secret and pretend I don't know that they know." Grandpa now has level 3 knowledge. Everyone knows that Grandpa chews, everyone knows that everyone knows, it's a bit of an "open secret", but not everyone knows that everyone knows that everyone knows, because nobody knows about Grandpa's snooping, so everyone else thinks that Grandpa is still oblivious to their knowledge, when he's not. This is meaningfully different from the previous scenario, where Grandpa actually thought he was being sneaky about the chewing, and is meaningfully different from common knowledge where it's all out in the open and nobody thinks they know more than someone else.

You can continue this pattern any finite number of steps upward, maybe Grandma planned for Grandpa to overhear her, and maybe Grandpa then discovers that it was a ruse, and so on and so on, but it becomes increasingly convoluted to understand or explain, and also there's little practical difference between level 19 knowledge and level 24 knowledge other than who happens to be on top at the moment, but they still differ from common knowledge, in that any large but finite number of steps creates an open secret that everyone knows but doesn't talk about, while the infinite recursion of common knowledge makes it just not a secret at all and it's harder for people to pretend it's a secret when it's obviously not.

The easier illustrative example of this is from poker. Poker pros talk about this all the time.

Level 1: I know what's in my hand.

Level 2: I know what's in my opponent's hand.

Level 3: I know what my opponent thinks is in my hand.

Level 4: I know what my opponent thinks I think is in his hand.

Level 5: I know what my opponent thinks I think he thinks is in my hand.

Etc.

Many poker players never advance beyond level 2. Even pros have trouble getting beyond 5.

Normally people cite this when it comes to these types of recursive logic.

Say you and I are the only two people on Earth.

Level 1: I know the sun rises in the East. You know the sun rises in the East. However, you might erroneously think I believe the sun rises in the West.

Level 2: I know the sun rises in the East. You know the sun rises in the East. I know you know the sun rises in the East. You know I know the sun rises in the East. However, you might erroneously think I believe you think the sun rises in the West.

I personally would consider it common knowledge at this point. But I can imagine a situation where, for example, you tell me to watch for the sunrise, and you have to clarify that you mean the East, because you think my understanding of you is that you believe the sun rises in the West. This wouldn’t happen at Level 3.

Level 3: I know the sun rises in the East. You know the sun rises in the East. I know you know the sun rises in the East. You know I know the sun rises in the East. I know you know that I know the sun rises in the East. You know that I know that you know the sun rises in the East.

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