So we've just seen how easy it is to start a cascade, of course, depending on the situation. And the question now is how long will a cascade last? Well, it'll last forever unless there's some kind of a disturbance, which really translates to a release of private signals. So people saying what their private signals are and telling the public what the private signals are, so that it's not just the public actions that are available when making, guesses. And how many such disturbances do we need? Well, even a few will often suffice, because they all know that they're playing follow the leader, so to speak, just to maximize the, the chance of them being correct, and they all know how the cascade started if they're rational. And, therefore, if they can recompute a probability based on a private signal, that it's now more probable that something else was the case, you can quickly reverse or end the cascade. So this is the counterpart of information cascade which is called the Emperor's New Clothes effect. And this name arises from the 19th-century short story in which a vain emperor, or this guy right here in the picture, is told that his new clothing is of the finest, best suit of fabric that is invisible only to those who are unfit for their positions. In reality, of course, he's actually not wearing any clothes at all, but, while everybody plays along, which are their public actions to just not say anything and just assume they can see his, clothing, including the Emperor himself who plays along. Because nobody wants to see him quote unquote, unfit, which is what their private signals are telling them because really they just see the guy walking around naked. Because he's not wearing any clothes but they don't want to seem like they're unfit for their positions. Then it only takes one kid to shout out, hey, he's not wearing anything at all. So, one person, maybe here, who doesn't understand what it means to be unfit for a position, just to say, hey, you know, you're not wearing any clothes until everybody becomes more confident that he is, in fact, not wearing any clothes. And then the cascade can break. So how do we break a cascade? Going back to our number-guessing thought experiments, suppose the cascade of ones has started. So we'll say the first two people have guessed and they started a cascade of ones, so there's some cascade of ones to three maybe. And, this is, the board, what people have written down. So, all ones so far. Now, then the next person comes up to the board. Suppose the person is shown the private signal of zero, right? So, the person sees that he gets the private signal of zero. And, he knows that some cascade has been triggered. He wants to see, maybe, if he can end the cascade. Not with himself because even though he sees a private signal of zero, remember it's still in his best guess to guess a one. And so, but then he shouts out that he got a zero, and he said my private signal was a zero. So now, when the next person comes along, so this is the next person in line, suppose the next person is now shown another private signal of zero. So now, this next person can say, okay, well I know that the person before me had a private signal of zero. And now I know that I have a private signal of zero. And then he thinks back to how the cascade was generated in the first place. The cascade started with the first two people, okay? In the best-case scenario, they both saw a private signal of a one, right? And then therefore wrote it and started the cascade. That is the only case that's comparable, now, to the chance that zero is the correct answer, because that would be seeing two private signals of one. Now these people have two private signals of zeros. But additionally, you know, obviously, that the first person had a private signal of one, or else he wouldn't have written one. The second person may not have had a private signal of one at all. So now you have a case where you can only know for sure that there was one private signal of one, right in the beginning, but now you have two private signals of zero here that you know. So, therefore, this person's now going to guess zero, because that's more likely now. And therefore is breaking the cascade. So there's a cascade here and now there's no cascade. And the reason that the cascade broke was that this person shouted the zero. So, remember, a cascade represents only what happened with a few people right around the time it started. That's the key point. If everybody knows that, which in this case, if they're all rational, they could figure that out, another block of a few people, may be able to break it, we show here.
