0:37

And it will be very powerful in terms of what we can work, how we can work with it

Â and what kinds of things we can deduce. it also turns out that, that people, even

Â operating in this very somewhat naive updating way, will be very accurate and

Â can do good things and, in terms of convergence.

Â [COUGH] and so the, this can have nice convergent properties even though people

Â don't act in a very sophisticated way. the limitations are going to be that you

Â don't have a lot of strategy in this model.

Â So it's going to be more a mechanical model in terms of the way that it

Â updates. But it can still be a very useful model

Â for understanding some, behaviour. So what we'll do here is we're going to

Â spend a little more time with this model. We'll go through basic definitions and

Â then after we've done those we'll talk about things like when is there

Â convergence? When is there a consensus?

Â this model is going to be very nice in terms of allowing us to figure out who

Â has influence. And who have influenced this model's

Â going to relate back to some of the things that we talked about earlier in

Â their centrality measures, like eigenvector centrality.

Â So it'll give a nice foundation for some of those measure that we saw earlier in

Â the course. And we can also ask, you know, when is it

Â that the estimates that people are, are making turn out to be accurate?

Â So there's a lot that we can do with this model.

Â It's a very useful in, in many ways. Okay, so the st, the structure now is

Â going to be a little different than what we saw before.

Â here, the information's going to come in only once at the beginning.

Â So, people are going to start with some initial beliefs, and then you're talking

Â to your neighbors, you're talking to your friends, so there'll be repeated

Â communication. And we'll see how the, information

Â disseminates, who has influence. what's the convergence speed?

Â how does network structure impact all of this?

Â So there's a lot of things that we can analyze here.

Â fairly accurately. Okay bounded rationality here so what's

Â going to happen is individuals are going to repeatedly average the beliefs that

Â they have. So they'll get information from their

Â neighbors. They, so for instance we might say what,

Â what's the probability that I think that there's global warming?

Â Okay, well initially I don't know much. I, I start with a probability of 0.7

Â based on what I've seen and read. I talked to some other people that have

Â information. And maybe somebody's a 0.9, somebody else

Â is a 0.6. I'll take those, I'll, I'll do a weighted

Â average of the 0.9, the 0.6 and my own 0.7.

Â Come up with a new belief. And now we talk again.

Â And, and so you know, I, those people will have talked to other individuals,

Â and they'll come up with new beliefs. And then I'll update their, their new

Â beliefs with my new belief and, and so forth.

Â 3:57

But the second that we talked, and now my friends have learned things from other

Â friends the, as we talked about before in our, in the previous video those, those

Â things can be actually quite complicated, right?

Â So now, there's a lot to take into account and so forth and in this model,

Â what I'm going to do is I'm just going to average them again.

Â I'm not going to, to change the weights on different individual, because that

Â might be that somebody I know is talking to, to other people who might have better

Â information than this other person. Maybe I should start weighting them more

Â on the second time than the first time, and so forth.

Â so these aren't going to adjust over time.

Â And it can also be that maybe I put too much weight on my own beliefs.

Â And there's some experimental evidence that, individuals underweight, neighbors,

Â there's some experiments by Choi Deo and Kariv which find these kinds of things.

Â there's a number of experiments that, that have some evidence that people will

Â underweight, neighbors. And so this kind of model will, will

Â capture that in some cases. Okay, so this model is, we'll talk about

Â the version, which comes out of DeGroot in 1974.

Â it was also, there's also earlier versions of it due to French and Herrari,

Â it's a model that has appeared in a number of different literatures and You

Â know, I guess the fact that it's been rediscovered, and,used in different

Â literatures, attests to the fact that it's a very natural, model to write down

Â in terms of specifying communication and learning.

Â 5:33

So, there's an Individuals and what we're going to work with in terms of a network

Â here Is actually going to be weighted and directed.

Â And, it's going to be a stochastic matrix.

Â So we'll, we'll work with the notation T as sort of a, a trust matrix.

Â So here what's, what individuals do is they have some, these are going to be

Â weights that I put on different individuals when updating my belief.

Â Okay. And so what I do is person I, everybody

Â starts at time 0 with some initial belief.

Â Right? So we've all got whatever information we

Â had from the past, whatever experiences we have.

Â So we all start with some prior, big question, will there be a recession next

Â year. Or, is there global warming or you know

Â is this politician a good politician? so we're all start with our beliefs and

Â we're going to put these in zero one. You could have these be vectors, you know

Â the model actually extends quite naturally to have multiple dimensional

Â versions of beliefs and Beliefs on many things and, and so forth.

Â We'll just work with a simple case where you've got a belief and we'll keep it in

Â 0, 1. So this is my belief of what's the

Â probability that there is global warming. Okay, now the belief at time t that I

Â have is just going to be a weighted average of the beliefs of my neighbors

Â and my neighbors, this is captured through the Tij, right, so I put some

Â weight, person I puts some weight on j's belief, which is captured by Tij.

Â So stochastic here, is telling us that the sum of Tij's when we sum across J, so

Â this thing is each one of the Tij is not negative so I can't put negative weight

Â on equal, I put some positive weight and out of the people I listen to I decided

Â I'm going assign a total weight of one. So it could be that in this model, it

Â could be that Tii is positive right, so I put weight on my own past belief.

Â somebody that never listens to anybody would have Tii equal to 1.

Â Right. That would be that I just listen to

Â myself, I never pay attention, my belief just stays what it is and you can't

Â convince me of anything and I'm extremely stubborn.

Â but if I listen to anybody else then I'm going to put some weight on their belief,

Â some weight on my own belief. And what people were doing is forming a

Â new belief by repeatedly talking to others.

Â And incorporating that and, and updating their beliefs.

Â Okay. So, a very simple, natural updating

Â process. So, let's take a quick look at an

Â example. So let's suppose that person one listens

Â to everybody equally, right so a third on everybody.

Â So, here we see person one putting weight one third on one, one third on two, one

Â third on three. Person two puts weight a half on one and

Â a half on two. So weights one and two equally, but

Â doesn't listen to three at all 3 listens to 1, and herself so we get half on on, 1

Â half on 3. Okay.

Â So that would be the t matrix associated with this and sometimes it's going to be

Â useful to keep track of the diagrams in terms of the cycles and so forth and here

Â we can that you know, now we're going to have self links, self loops.

Â So some people are listening to themselves, and we've got waits.

Â So now we have a weighted directed matrix, and uh,different individuals can

Â pay different attention. This is one where each of the individuals

Â happens to put equal weight on each of their friends.

Â You don't have to have that, we'll look at different examples afterwards, but

Â this is a, a simple starting point. Okay?

Â So now what we can do is begin to see how updating works under this.

Â So suppose that, for instance, the initial beliefs of the three individuals

Â were person 1 started with a belief of 0, person 3 started with a belief of 0, and

Â it was only person 2 that started with a positive belief.

Â [COUGH] Okay, so person one puts an equal weight on each of these.

Â So they are going to weight a third of zero, a third of zero and third of one.

Â Their next period of belief, so at time zero, they have this, belief of person

Â one. At time

Â One, is now a third. So after one iteration of updating,

Â they've switched to a belief of a third. Okay.

Â So you can just do this for each individual.

Â This person is waiting a half of a one and a zero.

Â They're going to go to a half. This person just saw two zeros, right?

Â So they're still stuck at zero. So they didn't update at all.

Â And we can see that the amount that the people updated depended on which weights

Â they had and who they were listening to. And, but this person's belief now, in

Â the, in the second period Now becomes positive because at this point now, you

Â know, it took one period for this person to start having a positive belief.

Â Now, once this person has a positive belief, this person's belief starts going

Â up and what we can see is, through this averaging process people's beliefs are

Â going to be tend to be pulled towards each other.

Â 10:51

The people with the low beliefs are being pulled up to positive beliefs by talking

Â through two and then eventually the information that passes from two to one

Â passes on to three and the the one in twos.

Â These people the one in threes beliefs are bringing two down and so overall

Â overtime these things will tend to converge into something Right, and in

Â this case if you go through it actually converges to 2 7ths, 2 7ths, and 2 7ths.

Â Okay, so if you just keep running this process, keep running it, it goes to 2

Â 7ths, 2 7ths, 2 7ths. Okay, now you, we'll learn exactly how

Â you could have guessed the 2 7ths, 2 7ths, 2 7ths in a little bit.

Â but, it basically, this is going to be some measure of sort of how influential

Â this particular person to is because their the only person to start it with a

Â positive belief. And then we can see, you know what is the

Â eventual belief. We'll everybody, came with, brought in by

Â this averaging process towards the same belief and in this case it was 2 7th.

Â Okay? Okay and let's work with a slightly

Â different example. So now one where everybody doesn't put

Â equal weight on each other. So now we'll do one where we have person

Â 1, person 2, person 3. 1 and 2 act as before but person 3 now

Â puts 3 4th away on herself and 1 4th on person 2.

Â Okay. And we switch the beliefs around so now

Â person one starts with a belief of 1. So we can go through this process right.

Â So, another example if you do this over time.

Â You know, person one goes to a third. Here person three still lags for a little

Â while. Beliefs go to a half.

Â and what would happen in this world. You can go through.

Â Now it's going to converge to 3 11ths. Okay?

Â It converges to something different than it did before.

Â It depends on what the network looks like.

Â And it also just depends on what those initial beliefs were.

Â So both the shape of the network and the initial beliefs are going to tell us what

Â things converge to, okay? Now, what's nice about this model is that

Â it's essentially [INAUDIBLE] weighted averaging over time.

Â So it's a nice linear system. And this linear system is something that

Â we know a lot about, in terms of the mathematics.

Â And these things are going to converge and we'll have a nice way of talking

Â about what do these numbers turn out to be, how do they depend on the initial

Â beliefs and how do they depend on the network.

Â So there'll be an explicit solution that we can just calculate out directly in

Â terms of what those numbers are and what those relations [INAUDIBLE] So that's

Â going to be a nice part of the model, is we'll, we'll learn where, what's the

Â limit point of this process. And we can actually also say a lot about

Â the speed of convergence here and how that depends on the network and so forth.

Â Okay, so what's happening here is, you know, when a person talks to their

Â friends effectively, you know, after one period they've incorporated information

Â from these people. After 2 periods, they've i, indirectly

Â incorporated information from 2 distances because this information passed on to

Â people that were at distance 1 after, you know, the first period.

Â And, and so, what happens is, is we end up getting if e, if these people are

Â listening. To other individuals that information

Â goes on and, and gets incorporated by the individual over time.

Â So, as time goes on we've got more reach in terms of where the information is

Â coming in from and the fact that people are averaging things means that we're

Â going to end up with a nice conversion here so, things it's going to have nice

Â properties. Other interpretations.

Â So this model actually has a lot of interpretations and we've been talking

Â about averaging beliefs. We could also think about social

Â influence on actions. So suppose that what I'm doing is I'm

Â just choosing some behavior instead of a belief over time.

Â So think of B I as behavior and what I'm doing is actually trying to match the

Â behavior of my neighbor. And I paid certain amounts of attention

Â to different people. And so what I lose, I try to match what

Â people did over time, right. So, so that's going to be something where

Â you've got, you know, social influence on actions.

Â You can think of how those actions, work through.

Â there's also some ties to Markov processes and, and probabilities.

Â You can think of these as probabilities of things.

Â I'll talk a little bit about that, As we go along there will also be some

Â relationships to page ranks and other kinds of things.

Â but we can also in terms of this, you can also just think of this as a kind of

Â game, where you are trying to match the actions of your neighbors and, and, and

Â you want to do that over time and you end up with the same kind of dynamic.

Â So this is going to be a very simple tractable model and what we'll do next

Â now is start talking about what it's convergences properties.

Â When does it converge? What can it converge to?

Â Can we say something about how that depends on the network and so forth?

Â