Hi folks. So, now we're still talking about learning, and indeed now we're going to move to talking about the DeGroot model. And in terms of our outline, we talked a little bit about Bayesian learning. And now we'll talk much more about and there's a lot more to say about Bayesian learning, but we're not going to have time for that here. But we can look now at the DeGroot model, which is a model over repeated communication an a more naive updating. So, it has advantages an disadvantages. [COUGH] In terms of advantages the model is going to be, quite simple an mathematically elegant, and it'll bring network structure in a very easy way. And it will be very powerful in terms of what we can work, how we can work with it and what kinds of things we can deduce. it also turns out that, that people, even operating in this very somewhat naive updating way, will be very accurate and can do good things and, in terms of convergence. [COUGH] and so the, this can have nice convergent properties even though people don't act in a very sophisticated way. the limitations are going to be that you don't have a lot of strategy in this model. So it's going to be more a mechanical model in terms of the way that it updates. But it can still be a very useful model for understanding some, behaviour. So what we'll do here is we're going to spend a little more time with this model. We'll go through basic definitions and then after we've done those we'll talk about things like when is there convergence? When is there a consensus? this model is going to be very nice in terms of allowing us to figure out who has influence. And who have influenced this model's going to relate back to some of the things that we talked about earlier in their centrality measures, like eigenvector centrality. So it'll give a nice foundation for some of those measure that we saw earlier in the course. And we can also ask, you know, when is it that the estimates that people are, are making turn out to be accurate? So there's a lot that we can do with this model. It's a very useful in, in many ways. Okay, so the st, the structure now is going to be a little different than what we saw before. here, the information's going to come in only once at the beginning. So, people are going to start with some initial beliefs, and then you're talking to your neighbors, you're talking to your friends, so there'll be repeated communication. And we'll see how the, information disseminates, who has influence. what's the convergence speed? how does network structure impact all of this? So there's a lot of things that we can analyze here. fairly accurately. Okay bounded rationality here so what's going to happen is individuals are going to repeatedly average the beliefs that they have. So they'll get information from their neighbors. They, so for instance we might say what, what's the probability that I think that there's global warming? Okay, well initially I don't know much. I, I start with a probability of 0.7 based on what I've seen and read. I talked to some other people that have information. And maybe somebody's a 0.9, somebody else is a 0.6. I'll take those, I'll, I'll do a weighted average of the 0.9, the 0.6 and my own 0.7. Come up with a new belief. And now we talk again. And, and so you know, I, those people will have talked to other individuals, and they'll come up with new beliefs. And then I'll update their, their new beliefs with my new belief and, and so forth. And the, the idea of it being Non- Bayesian is the fact that the weights aren't going to be adjusting over time. So, if I was a Bayesian and I started out with my belief of 0.7 and somebody else's belief of 0.9 that there's global warming and somebody else that's 0.6. Then a Bayesian would have a way of weighting those, and depending on the variances and the precision of different information, you would end up averaging those and putting different weights on those things. And so it would be, indeed, what you would do is take an average of those and that would come your new belief. Or an appropriate weighted, incorporation of those. But the second that we talked, and now my friends have learned things from other friends the, as we talked about before in our, in the previous video those, those things can be actually quite complicated, right? So now, there's a lot to take into account and so forth and in this model, what I'm going to do is I'm just going to average them again. I'm not going to, to change the weights on different individual, because that might be that somebody I know is talking to, to other people who might have better information than this other person. Maybe I should start weighting them more on the second time than the first time, and so forth. so these aren't going to adjust over time. And it can also be that maybe I put too much weight on my own beliefs. And there's some experimental evidence that, individuals underweight, neighbors, there's some experiments by Choi Deo and Kariv which find these kinds of things. there's a number of experiments that, that have some evidence that people will underweight, neighbors. And so this kind of model will, will capture that in some cases. Okay, so this model is, we'll talk about the version, which comes out of DeGroot in 1974. it was also, there's also earlier versions of it due to French and Herrari, it's a model that has appeared in a number of different literatures and You know, I guess the fact that it's been rediscovered, and,used in different literatures, attests to the fact that it's a very natural, model to write down in terms of specifying communication and learning. So, there's an Individuals and what we're going to work with in terms of a network here Is actually going to be weighted and directed. And, it's going to be a stochastic matrix. So we'll, we'll work with the notation T as sort of a, a trust matrix. So here what's, what individuals do is they have some, these are going to be weights that I put on different individuals when updating my belief. Okay. And so what I do is person I, everybody starts at time 0 with some initial belief. Right? So we've all got whatever information we had from the past, whatever experiences we have. So we all start with some prior, big question, will there be a recession next year. Or, is there global warming or you know is this politician a good politician? so we're all start with our beliefs and we're going to put these in zero one. You could have these be vectors, you know the model actually extends quite naturally to have multiple dimensional versions of beliefs and Beliefs on many things and, and so forth. We'll just work with a simple case where you've got a belief and we'll keep it in 0, 1. So this is my belief of what's the probability that there is global warming. Okay, now the belief at time t that I have is just going to be a weighted average of the beliefs of my neighbors and my neighbors, this is captured through the Tij, right, so I put some weight, person I puts some weight on j's belief, which is captured by Tij. So stochastic here, is telling us that the sum of Tij's when we sum across J, so this thing is each one of the Tij is not negative so I can't put negative weight on equal, I put some positive weight and out of the people I listen to I decided I'm going assign a total weight of one. So it could be that in this model, it could be that Tii is positive right, so I put weight on my own past belief. somebody that never listens to anybody would have Tii equal to 1. Right. That would be that I just listen to myself, I never pay attention, my belief just stays what it is and you can't convince me of anything and I'm extremely stubborn. but if I listen to anybody else then I'm going to put some weight on their belief, some weight on my own belief. And what people were doing is forming a new belief by repeatedly talking to others. And incorporating that and, and updating their beliefs. Okay. So, a very simple, natural updating process. So, let's take a quick look at an example. So let's suppose that person one listens to everybody equally, right so a third on everybody. So, here we see person one putting weight one third on one, one third on two, one third on three. Person two puts weight a half on one and a half on two. So weights one and two equally, but doesn't listen to three at all 3 listens to 1, and herself so we get half on on, 1 half on 3. Okay. So that would be the t matrix associated with this and sometimes it's going to be useful to keep track of the diagrams in terms of the cycles and so forth and here we can that you know, now we're going to have self links, self loops. So some people are listening to themselves, and we've got waits. So now we have a weighted directed matrix, and uh,different individuals can pay different attention. This is one where each of the individuals happens to put equal weight on each of their friends. You don't have to have that, we'll look at different examples afterwards, but this is a, a simple starting point. Okay? So now what we can do is begin to see how updating works under this. So suppose that, for instance, the initial beliefs of the three individuals were person 1 started with a belief of 0, person 3 started with a belief of 0, and it was only person 2 that started with a positive belief. [COUGH] Okay, so person one puts an equal weight on each of these. So they are going to weight a third of zero, a third of zero and third of one. Their next period of belief, so at time zero, they have this, belief of person one. At time One, is now a third. So after one iteration of updating, they've switched to a belief of a third. Okay. So you can just do this for each individual. This person is waiting a half of a one and a zero. They're going to go to a half. This person just saw two zeros, right? So they're still stuck at zero. So they didn't update at all. And we can see that the amount that the people updated depended on which weights they had and who they were listening to. And, but this person's belief now, in the, in the second period Now becomes positive because at this point now, you know, it took one period for this person to start having a positive belief. Now, once this person has a positive belief, this person's belief starts going up and what we can see is, through this averaging process people's beliefs are going to be tend to be pulled towards each other. The people with the low beliefs are being pulled up to positive beliefs by talking through two and then eventually the information that passes from two to one passes on to three and the the one in twos. These people the one in threes beliefs are bringing two down and so overall overtime these things will tend to converge into something Right, and in this case if you go through it actually converges to 2 7ths, 2 7ths, and 2 7ths. Okay, so if you just keep running this process, keep running it, it goes to 2 7ths, 2 7ths, 2 7ths. Okay, now you, we'll learn exactly how you could have guessed the 2 7ths, 2 7ths, 2 7ths in a little bit. but, it basically, this is going to be some measure of sort of how influential this particular person to is because their the only person to start it with a positive belief. And then we can see, you know what is the eventual belief. We'll everybody, came with, brought in by this averaging process towards the same belief and in this case it was 2 7th. Okay? Okay and let's work with a slightly different example. So now one where everybody doesn't put equal weight on each other. So now we'll do one where we have person 1, person 2, person 3. 1 and 2 act as before but person 3 now puts 3 4th away on herself and 1 4th on person 2. Okay. And we switch the beliefs around so now person one starts with a belief of 1. So we can go through this process right. So, another example if you do this over time. You know, person one goes to a third. Here person three still lags for a little while. Beliefs go to a half. and what would happen in this world. You can go through. Now it's going to converge to 3 11ths. Okay? It converges to something different than it did before. It depends on what the network looks like. And it also just depends on what those initial beliefs were. So both the shape of the network and the initial beliefs are going to tell us what things converge to, okay? Now, what's nice about this model is that it's essentially [INAUDIBLE] weighted averaging over time. So it's a nice linear system. And this linear system is something that we know a lot about, in terms of the mathematics. And these things are going to converge and we'll have a nice way of talking about what do these numbers turn out to be, how do they depend on the initial beliefs and how do they depend on the network. So there'll be an explicit solution that we can just calculate out directly in terms of what those numbers are and what those relations [INAUDIBLE] So that's going to be a nice part of the model, is we'll, we'll learn where, what's the limit point of this process. And we can actually also say a lot about the speed of convergence here and how that depends on the network and so forth. Okay, so what's happening here is, you know, when a person talks to their friends effectively, you know, after one period they've incorporated information from these people. After 2 periods, they've i, indirectly incorporated information from 2 distances because this information passed on to people that were at distance 1 after, you know, the first period. And, and so, what happens is, is we end up getting if e, if these people are listening. To other individuals that information goes on and, and gets incorporated by the individual over time. So, as time goes on we've got more reach in terms of where the information is coming in from and the fact that people are averaging things means that we're going to end up with a nice conversion here so, things it's going to have nice properties. Other interpretations. So this model actually has a lot of interpretations and we've been talking about averaging beliefs. We could also think about social influence on actions. So suppose that what I'm doing is I'm just choosing some behavior instead of a belief over time. So think of B I as behavior and what I'm doing is actually trying to match the behavior of my neighbor. And I paid certain amounts of attention to different people. And so what I lose, I try to match what people did over time, right. So, so that's going to be something where you've got, you know, social influence on actions. You can think of how those actions, work through. there's also some ties to Markov processes and, and probabilities. You can think of these as probabilities of things. I'll talk a little bit about that, As we go along there will also be some relationships to page ranks and other kinds of things. but we can also in terms of this, you can also just think of this as a kind of game, where you are trying to match the actions of your neighbors and, and, and you want to do that over time and you end up with the same kind of dynamic. So this is going to be a very simple tractable model and what we'll do next now is start talking about what it's convergences properties. When does it converge? What can it converge to? Can we say something about how that depends on the network and so forth?
