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PaperclipPerfector  1yr ago (text post)   3383 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?

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.

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