0:15

But you can also stop and cascade with a little bit release of private information.

Â For example, suppose at some point this person says, instead of just writing down

Â the public action which will be one no matter what, I will also shout out my

Â private signal which turns out to be, say zero And then the next person coming in,

Â suppose she also receives the prize, signal zero Then on the one hand, she

Â knows there is definitely a private signal of zero and a zero.

Â On the other hand, it could have been just a private signal of one and zero.

Â Back when where the cascade started, and this user just flip a coin and write down

Â one. Or it could have been private signal

Â sequence eleven. Right.

Â Either way we could have started this cascade.

Â 1:18

So this user may decide to actually go with zero and write down a public action

Â of zero. And that would break the entire chain of

Â information cascade. And this is what we call the Emperor's new

Â clothes effect." A little boy shouting now there's no cloth, can initiate a reversal

Â of the cascade because even though people know there might be a lot of people doing

Â the same thing they could have been following just a small tiny turning point

Â of private signals many steps up before the current time and this explains quite a

Â bit of social phenomena when cascades stop ranging from certain trends in the fashion

Â industry to totalitarian regimes collapsing.

Â 2:13

There are quite a few other variations. One is the piece, not being the same.

Â Different user would have a different P, the probability of knowing the correct

Â number through the private signal. And suppose you, as a designer of the

Â experiment knows all these P's then in order to start a cascade earlier to get an

Â earlier onset of cascade. You would put those P's larger peas users,

Â earlier in the crowd. You make this one, the one with the

Â largest, P, and this would trigger an earlier onset of cascade through these

Â rational ovation agents thinking. However, if you would like to have a

Â checkpoint later in the crowd to possibly stop a wrong cascade, then you'll put the

Â large piece people down there. They can also be multiple levels of

Â numbers. We have been looking at a binary number

Â guessing game. So whichever is more likely than the other

Â will be written down as the public action, as the guess.

Â But sometimes, for example, on a street corner,

Â If one person decide to tilt the head towards the sky because of a nosebleed

Â probably other people would just continue to pass by.

Â But suppose there are ten people all tilting their heads, looking at the sky,

Â then the next passage, passers-by, would say, well there might be something wrong

Â with the sky. Wrong enough that I can see tend public

Â action and that will trigger me to stop walking and also tilt my head to the sky.

Â And once that starts, then other people will follow suit, and the crowd will get

Â bigger and bigger until the new emperor's effect kicks in.

Â Somebody says, hey, the, the first guy just had a nosebleed and that's why he

Â tilted to look at the sky, and the crowd may disperse exactly at that point.

Â 5:51

Now, we have looked at this social influence model, information cascade,

Â through a simple rational Bayesian agent thinking along basically a linear

Â topology. And in the rest of this lecture and the

Â next lecture, we look at quite a few different extensions.

Â But before then let's walk through one more extended numerical example,

Â especially to look at the probability of correct versus wrong cascade.

Â So first underlined number is one, this is different from the previous derivation

Â where we assumed that underlying number to be either zero or one Now we say it is

Â fixed at one and therefore an up cascade of 1's is correct cascade and a down

Â cascade of 0's is an incorrect cascade. And we can write down the possible

Â evolution through a tree. The first user, looking at private signal,

Â it could be zero or one. It branches out different path in, along

Â the tree. And then the second user could get zero or

Â one, zero and one. If it actually get private signal zero

Â first, and then one, then you flip a coin, that's what f denotes, to decide with 50%

Â chance of write down one, 50% chance you write down zero.

Â And then you'll get this possible outcome just out of two the first two users, of

Â the two public actions y1, y2 mean 0,1 or 0,0.

Â Now you can follow through the other branches of this tree in the same way.

Â 7:40

So now we can quickly write down the following.

Â Just look at the first two people. We know that the probability of no cascade

Â just like what we discussed before is that if x1 is zero, x2 is one and you flip a

Â coin to maintain one. Or if x1 is one, x2 is zero and you flip a

Â coin to maintain zero, this will be the probability 1-P times P times half.

Â This is P times 1-P times half and you add up the two together you get P times 1-P

Â same as before. Now the probability of an up cascade,

Â however, is different because we now explicitly assume the correct number is

Â fixed at one. So the up cascade is probability that x1

Â is one, x2 is one or x1 is one, x2 is zero but you flip a coin and decide to write

Â down one. The probabilities are respectively P P +

Â P times one - P times half, and that equals to P times one + P over half.

Â And a probability of a down cascade after two user.

Â You can write a similar expression. It is one - P^2 + one - P times P / two,

Â That equals one - P times 1,2 - P / two.

Â So that's the expressions for these three possible events.for P times one - P for no

Â cascade, P times one + P / two for up cascade and P, one - P times two for a

Â down cascade. Now this is also the probability of a

Â correct cascade. This underline number is, is soon to be

Â fixes, one and this is an incorrect. Cascade.

Â Now this is just for the first two users. We can now write down the expression for a

Â general two end users after even number of users.

Â What would happen? The derivation is a slightly involved but

Â you can either do that as home exercise or look at a textbook derivation.

Â Basically follows the same principle as before with Cnet.

Â First of all, no cascade that probability is simply P1-p, one - P the whole thing

Â n times. Now up or correct cascade turns out to be Pp+1.

Â P + one one - P - P^2 n times / two one - P + P^2 And the probablity of down

Â and incorrect cascade turns out to be one - P two - P one - P - P^2 n times /

Â two minus, two one - P + P^2. This is just basically, a denominator that

Â we use to add up the terms. This factor of n is really a factor

Â expressing, the probability, no cascade. Okay?

Â But it's, the other part of the numerators are different because we already assumed

Â the underlying number is one. Therefore, that breaks the symmetry.

Â And therefore, up and down probabilities are not the same anymore, after one or n

Â pairs of users. Now we can plot this on a graph, or

Â actually four different, cases, okay? One pair, so two user.

Â Two pair, four user. Five pair, ten user.

Â 100 pairs, that's 200 users. Now, in fact, you can only see, visually,

Â three lines for one to five pairs. Because beyond five pairs, it pretty much

Â stays as the same. Visually, you cannot use naked eye to tell

Â the difference anymore. And I am plotting here the probability of

Â the correct cascade. As a function of P, the common probability

Â of getting the correct part of signal across all the users, ranging from half to

Â one, has to be bigger than half and one is the largest you can get.

Â 12:31

Now of course, one it is exactly one and the correct cascade is always a 100%

Â probability. But as you can see it starts out actually

Â much less than that as P moves very close to half-half, with half of the chance you

Â actually going to get the correct underline number half of chance you get

Â incorrect one. So, it's not a surprise that probability

Â of correct cascade increases as a function of each individual getting the correct

Â private signal. Now we also observe that.

Â As n increases, The chance of no cascade actually drops

Â dramatically. And also, the probability of correct

Â cascade is still quite small even of, when n is large.

Â And this is the key observation we wanna highlight in this graph, is that the

Â probability of correct cascade is still small even when n is large.

Â You think that bigger n should be able to give you wisdom of crowd and therefore the

Â probability of getting a correct cascade should increase with n but that does not

Â happen precisely because wisdom of crowd. Dependents on the independence of users'

Â opinion whereas in, information cascade is exactly the opposite where the dependence

Â in fact complete dependence of previous users' actions, and that destroys this

Â independent assumption.

Â