How long can an information cascade last? Well it can last forever. Once you've got an odd and an even numbered user showing the same action, say one, one, this would initiate a cascade of ones. But you can also stop and cascade with a little bit release of private information. For example, suppose at some point this person says, instead of just writing down the public action which will be one no matter what, I will also shout out my private signal which turns out to be, say zero And then the next person coming in, suppose she also receives the prize, signal zero Then on the one hand, she knows there is definitely a private signal of zero and a zero. On the other hand, it could have been just a private signal of one and zero. Back when where the cascade started, and this user just flip a coin and write down one. Or it could have been private signal sequence eleven. Right. Either way we could have started this cascade. So this user may decide to actually go with zero and write down a public action of zero. And that would break the entire chain of information cascade. And this is what we call the Emperor's new clothes effect." A little boy shouting now there's no cloth, can initiate a reversal of the cascade because even though people know there might be a lot of people doing the same thing they could have been following just a small tiny turning point of private signals many steps up before the current time and this explains quite a bit of social phenomena when cascades stop ranging from certain trends in the fashion industry to totalitarian regimes collapsing. There are quite a few other variations. One is the piece, not being the same. Different user would have a different P, the probability of knowing the correct number through the private signal. And suppose you, as a designer of the experiment knows all these P's then in order to start a cascade earlier to get an earlier onset of cascade. You would put those P's larger peas users, earlier in the crowd. You make this one, the one with the largest, P, and this would trigger an earlier onset of cascade through these rational ovation agents thinking. However, if you would like to have a checkpoint later in the crowd to possibly stop a wrong cascade, then you'll put the large piece people down there. They can also be multiple levels of numbers. We have been looking at a binary number guessing game. So whichever is more likely than the other will be written down as the public action, as the guess. But sometimes, for example, on a street corner, If one person decide to tilt the head towards the sky because of a nosebleed probably other people would just continue to pass by. But suppose there are ten people all tilting their heads, looking at the sky, then the next passage, passers-by, would say, well there might be something wrong with the sky. Wrong enough that I can see tend public action and that will trigger me to stop walking and also tilt my head to the sky. And once that starts, then other people will follow suit, and the crowd will get bigger and bigger until the new emperor's effect kicks in. Somebody says, hey, the, the first guy just had a nosebleed and that's why he tilted to look at the sky, and the crowd may disperse exactly at that point. So, depending on your curiosity level and how much you need to do, keep walking, a different person would have a different threshold of a crowd needed in order for her to stop walking and join the crowd as well. There are many implications to this study of dependence of information, or of action, on other people's action. And this partially explains social phenomena, such as crowd making the decision, ignoring each individual's private information. It partially explains chain reaction that are fragile and easy to revert. For example, one implication is why U.S. Presidential election in the primary season, had was called the Super Tuesday precisely to hold many states' primarily election in each party, so that there's no sequential, to avoid sequential decisioning. Now, we have looked at this social influence model, information cascade, through a simple rational Bayesian agent thinking along basically a linear topology. And in the rest of this lecture and the next lecture, we look at quite a few different extensions. But before then let's walk through one more extended numerical example, especially to look at the probability of correct versus wrong cascade. So first underlined number is one, this is different from the previous derivation where we assumed that underlying number to be either zero or one Now we say it is fixed at one and therefore an up cascade of 1's is correct cascade and a down cascade of 0's is an incorrect cascade. And we can write down the possible evolution through a tree. The first user, looking at private signal, it could be zero or one. It branches out different path in, along the tree. And then the second user could get zero or one, zero and one. If it actually get private signal zero first, and then one, then you flip a coin, that's what f denotes, to decide with 50% chance of write down one, 50% chance you write down zero. And then you'll get this possible outcome just out of two the first two users, of the two public actions y1, y2 mean 0,1 or 0,0. Now you can follow through the other branches of this tree in the same way. So now we can quickly write down the following. Just look at the first two people. We know that the probability of no cascade just like what we discussed before is that if x1 is zero, x2 is one and you flip a coin to maintain one. Or if x1 is one, x2 is zero and you flip a coin to maintain zero, this will be the probability 1-P times P times half. This is P times 1-P times half and you add up the two together you get P times 1-P same as before. Now the probability of an up cascade, however, is different because we now explicitly assume the correct number is fixed at one. So the up cascade is probability that x1 is one, x2 is one or x1 is one, x2 is zero but you flip a coin and decide to write down one. The probabilities are respectively P P + P times one - P times half, and that equals to P times one + P over half. And a probability of a down cascade after two user. You can write a similar expression. It is one - P^2 + one - P times P / two, That equals one - P times 1,2 - P / two. So that's the expressions for these three possible events.for P times one - P for no cascade, P times one + P / two for up cascade and P, one - P times two for a down cascade. Now this is also the probability of a correct cascade. This underline number is, is soon to be fixes, one and this is an incorrect. Cascade. Now this is just for the first two users. We can now write down the expression for a general two end users after even number of users. What would happen? The derivation is a slightly involved but you can either do that as home exercise or look at a textbook derivation. Basically follows the same principle as before with Cnet. First of all, no cascade that probability is simply P1-p, one - P the whole thing n times. Now up or correct cascade turns out to be Pp+1. P + one one - P - P^2 n times / two one - P + P^2 And the probablity of down and incorrect cascade turns out to be one - P two - P one - P - P^2 n times / two minus, two one - P + P^2. This is just basically, a denominator that we use to add up the terms. This factor of n is really a factor expressing, the probability, no cascade. Okay? But it's, the other part of the numerators are different because we already assumed the underlying number is one. Therefore, that breaks the symmetry. And therefore, up and down probabilities are not the same anymore, after one or n pairs of users. Now we can plot this on a graph, or actually four different, cases, okay? One pair, so two user. Two pair, four user. Five pair, ten user. 100 pairs, that's 200 users. Now, in fact, you can only see, visually, three lines for one to five pairs. Because beyond five pairs, it pretty much stays as the same. Visually, you cannot use naked eye to tell the difference anymore. And I am plotting here the probability of the correct cascade. As a function of P, the common probability of getting the correct part of signal across all the users, ranging from half to one, has to be bigger than half and one is the largest you can get. Now of course, one it is exactly one and the correct cascade is always a 100% probability. But as you can see it starts out actually much less than that as P moves very close to half-half, with half of the chance you actually going to get the correct underline number half of chance you get incorrect one. So, it's not a surprise that probability of correct cascade increases as a function of each individual getting the correct private signal. Now we also observe that. As n increases, The chance of no cascade actually drops dramatically. And also, the probability of correct cascade is still quite small even of, when n is large. And this is the key observation we wanna highlight in this graph, is that the probability of correct cascade is still small even when n is large. You think that bigger n should be able to give you wisdom of crowd and therefore the probability of getting a correct cascade should increase with n but that does not happen precisely because wisdom of crowd. Dependents on the independence of users' opinion whereas in, information cascade is exactly the opposite where the dependence in fact complete dependence of previous users' actions, and that destroys this independent assumption.