So, if you're the first person to go up, what should you do? Well, currently on the blackboard, there is nothing written. So you can only go by your private signal PRV I. But you know that this private signal's more likely to be at the true value than it is to be the false value. So therefore, if it's zero, you should guess zero, and if it's one, you should guess one. It's in your best interest to just guess whatever PRV I is. So, the first person's going to guess PRV I and write that on the board. It could be a zero, could be a one. And now the next person goes. So, if you're the second person to go up, what should you do? Well, I see the public action of the first person, whatever that was whatever PUB I was. And I see my own private signal, PRV II. But you could inver, in this case you can infer what PRV I was just by knowing what PUB I is. So, if PUB I is a one, then I could actually tell if the first person had a private signal of one. And if it's zero, I can tell the first person had a private signal of zero because I know exactly what the first person was facing. So, I know that when the first person was up at the board, that he faced his decision, and he only had his private signal to go by. And we're assuming that everyone acts rationally. So, therefore, I can exactly predict that he would have gone by whatever his private signal was. So, I effectively have two private signals. I have PRV I, and I have PRV II. PRV I and PRV II. So again, just to make sure that that's clear, that I know what the first person's private signal was because I know that his public action must have been the same as this private signal because of the logic we already went through with the first person. So, if both of these are zero, obviously, I'm going to guess zero. So, the second person should write down a zero if they're both zero. So, this is the second person going up. He should write a zero then. And if both of them are one, then you should guess a one. So, in the case of both of them are one, he should just guess one. And the reason for this is that it's just a stronger condition than what it was for the first person. So the first person knows that he was more likely to be shown the correct answer than the incorrect answer. Therefore, he writes that down. Now, if the second person sees two private signals which are the same value, then you know that that's gotta be even more likely now to be the correct value because it's more, it's much less likely that you're going to have two instances where you're shown the incorrect answer. But now, if both of them are different, you really have no information. So, if PRV I, or per, the, whatever person wrote up on the board was a one, and now I received a zero. Now I'm not really going to know I'm not really going to have any, any information from that because either one of them could be the correct answer. And obviously, one of them is, and one of them isn't. So, you don't really know anything, and the same thing if PRV I was zero, and PRV II was one. Really you have no information. So, what you do is just flip a coin and guess randomly. And then you know that you have a 0.5 chance of getting correct, and 0.5 chance of not getting it correct. So, now comes the first chance of the information cascade starting. If you're the third-person, the question is, what should you do? Well, you have PUB I and PUB II, which are the first two public actions of the first two users. And you have your own private signal, PRV III. And then you want to determine what PUB III is going to be. So, we need to analyze two different cases. The first case is that PUB I and PUB II are different, which is like what's drawn up on the board here. Like, there's a zero, one or a one, zero that you see as being the first two public actions. Well, in this case you can infer what person two's dilemma must have been because if the two, if the first two private signals were the same, then you know that he would have shown the same public actions. So, you would've had two matches. So, you know that the first two private signals must have been different and that the person two must have flipped a coin and chosen randomly. And you also know that, that when a person flipped a coin, that he was he got the opposite of what the first person's private signal was. So, you can infer quite a lot about what happened knowing that the two private signals were different. What it basically means is that there's no information previously, 'kay? It doesn't help at all because you know that the two private signals must have been different. And you also know that the person flipped a coin, and he chose whichever value he did. So, you're in the same shoes as person one. Therefore, you should just choose your own private signal as being the value that you're going to show. So, if it was a one, you just put a one down. And you don't, you don't rely on anything that happened before. And then, similarly, you're in the same shoes as person one. And, and similarly, you're in the same shoes as person one. And then, person four is going to be in the same shoes as person two. So the cycle now is repeating. And the case, but that's only in this case that PUB I and PUB II are different. The other case, suppose PUB I and PUB II are the same, right. So, they're either both one, or they're both zero. So, if private signal three is the same, then obviously, you're going to pick that number because now you just have even more even more of a chance of it being correct. You've more evidence that it's the right thing. And the reason for that, intuitively, is that you can still know which situation the first person was in. He must have just written down PRV I. So, now you know PRV I and PRV III are the same. So, regardless of what person two did, whether he had the same private signal, or if he just flipped a coin, then you have even more of a chance of it being the same than the second person did even if the private signals were the same for his case. So, in this case, you're going to pick this number. So, if PUB I and PUB II are the same as your private signal PRV III, then you're just going to pick the number and write down PRV III. But it turns out that even if PRV I is different, you're still going to write what they did. The math is going to work out that way. So, even if you have the case where you see two ones on the board, and then your private signal that you're shown was a zero, you're still going to write down a one. And you're going to follow what they did because that's enough information. Regardless of what, whether person two flipped a coin or not, that's enough information to show you that it's more likely that this is the correct answer than your own private signal. So, and the other case, too, if they're both zeros written down for both the public actions, and you got a one, you would still write down a zero here. It's a very strong result. The math works out that way. We're not going to go through it because it's a little complicated to prove that, but it just happens. So, in this case, even if your private signal is different, all that you need to start the cascade is for the first two public actions to have been the same. This starts the, what we talked at before, is information cascade where people start to ignore their own private signal because it's not logical anymore for you to follow your own private signal. It's in your best interest to just keep with the crowd.
