0:00

'Kay, so to finish up some of our discussion the DeGroot model.

Â Let's ask a question now of, of sort of applying this and understanding when it

Â is that a society of people who are updating in this sort of just weighted

Â average method. Are actually going to come to a consensus

Â which is accurate, right? So when is it that their beliefs will

Â actually be reasonably accurate beliefs? so information aggregation in this

Â setting. So is the consensus that they're going to

Â come to accurate. so we have to put some structure on this

Â and actually this question the reason I originally become interested in DeGroot

Â model was out of conversations with a student, a former student of mine Ben

Â Golub /g. And we started asking the question of

Â when was it that people's beliefs would actually converge to the right sort of

Â thing even if they were acting in a fairly simple form.

Â And so how does this depend on network structure?

Â How does it depend on the influence? How does it relate to speed of

Â convergence? There's a whole series of questions that

Â we can ask here. And we'll just take a quick look at how

Â this looks in the context of the DeGroot model.

Â And let's suppose that there is a true state out there which we'll call mu.

Â So this nature, for example there's really going to be some probability that

Â there's going to be global warming. And that's something that that true and

Â it's out there and each person's belief at time 0 is, is is different from the

Â true belief. So there's some true number out there,

Â and everybody has some error. And the error what we're going to make

Â sure of is that the errors of different individuals have 0 mean and finite

Â variance. Okay, and if you want to keep all these

Â things in 0, 1, you can do that if you like they don't have to be in 0, 1.

Â You can keep them in 0, 1 by, by making sure that the variance is actually

Â bounded so that you, you can't have your belief differ from mu by.

Â So that beliefs don't go outside 0, 1, but that's actually not necessary for

Â this analysis. Okay, so we've got the beliefs, everybody

Â has some error, and now we run into group model.

Â And so what we'd like to have is, if people keep talking to themselves,

Â talking to different individuals it would be nice if the, the situation eventually

Â converged to a true mu. So that by talking to people we would

Â actually really learn what the true mu was.

Â And here we can allow these, these epsilon i's to have different variances

Â across individuals but let's make sure that they're independent conditional on

Â mu. So each, the, the errors that one person

Â makes, so one person might be have a high belief, another person has a low belief.

Â So people are making errors, but those errors aren't correlated.

Â Okay. So, let's consider lie, large societies,

Â so we want to ask when is the crowd going to be wise.

Â So when is it when they get together and talk to a network, that they're actually

Â going to come to a reasonably accurate mu.

Â So what we want is, we want a situation where the probability that the limit as

Â we get look at the limiting beliefs over time compared to mu.

Â The probability that that's differing by more than some delta vanishes in large

Â societies. Okay, so there's a bunch of quantifiers

Â here, so we're looking at the limit belief.

Â And we want to make sure that the limit belief being bigger than some delta, that

Â vanishes and we want to do that as a society becomes large.

Â Okay? So larger and larger societies, when is

Â it that they're going to be accurate. Okay?

Â So if obviously, if there's only two individuals, then we've only got two

Â signals. Then if we're each making an error, even

Â an average of those two errors isn't going to give us a very accurate number.

Â so to get accuracy when people are making errors, we're going to have to average

Â over a large number. But then, the question is when is it the,

Â that the overall society can average in a, in a useful manner?

Â Okay. So, one thing that's very useful is a

Â variation on the weak law of large numbers, you can prove this easy using

Â Chebychev's inequality. So let's consider a situation where all

Â these errors are independent, so there's a bunch of people making independent

Â errors. each person has a 0 mean in their error,

Â so they're either above or below. but in expectation, they have a 0, so

Â they're centered at 0. And they each have finite variance, so

Â that nobody's infinitely ignorant. then, when we look at, let's suppose that

Â we were doing some influences. Whatever those influences are, so society

Â n has a, a vector s1, through sn. So we look at those.

Â We'll call those the si n's. All right.

Â So in society n, you've got an s1 through sn, then we have a different one for when

Â we add this extra person. So each one of these societies has a

Â vector. the Weak Law of Large Numbers tells us

Â that if we look at averaging those error. So what's going to happen is the

Â influence is going to capture how much of each persons error enters into the

Â overall societal error. The sum of the weights on those errors is

Â going to be 0. If and only if the largest influence that

Â anybody has goes to 0, okay. So if anybody retains influence, then

Â what we're going to do is end up retaining weight on that person's error.

Â Their, their believe is going to be a non-trivial part of the overall society's

Â belief. And so, it's going to be necessary that

Â everybody have a, a, a negligible weight in the limit.

Â So as society aggregates, they have to disperse so that we're putting weight on

Â different individuals. If we keep all listening to the same

Â person, we're going to have inaccuracy. And, actually that's going to be enough,

Â so as long as society spreads it out >> Given everybody has finite variance,

Â averaging a bunch of these variables will give us a good answer.

Â So wise crowds occur if and only if the maximum influence vanishes.

Â Okay, so thats a nice simple result that than tells us that we are going to get

Â convergence if and only if. To, to, the right belief if and only if

Â when we look at this in larger and larger societies, each one of these things is

Â tending to 0. Right, so we have all the beliefs tend to

Â 0. and the max tending to 0.

Â 6:43

Okay. So what's a sufficient condition for

Â this. suppose that we look at a situation where

Â there's, where you actually have reciprocal attention.

Â So, let's make T not only row stochastic but column stochastic.

Â So, everybody gives some weight out, but we also get the same weight in.

Â Everybody is getting, somebody is paying attention to them.

Â So, that the weight that they have out is, is equal.

Â Then you would get s equals 1 over n, right.

Â So that's a situation where T would be wise.

Â So, if everybody got weighted as much as weight in as they were giving out, we'd

Â be in good shape. so reciprocal trust would be something

Â that would imply wisdom. So, making sure that the trust that comes

Â to in any individual is the same as what goes out, that's a very strong condition,

Â though. So, generally in society, we're going to

Â have some heterogeneity in terms of overall, how much somebody gets paid

Â attention to. And so, what's important is that when

Â we're looking at this, there's no single individual that's getting too much of the

Â weight from other individuals who matter. Right?

Â So there's, if there was some i that had for instance, everything body putting

Â weight at least to a on them, then their overall influence would be at least a.

Â So there can't be anybody who gets, you know, too much detention.

Â So you can't have too strong an opinion leader.

Â That's going to be an, an obvious condition.

Â If anybody's getting too much weight in, their eventual belief is going to

Â influence society. And so the network's going to have to be

Â that, as it becomes larger and larger it can be that, you know, each individual is

Â only listening to a few people. So people are getting a lot of weight

Â from a few neighbors. But it can't be that overall, they're

Â getting weight from the whole society at a rate of at least a.

Â So, if there's anybody that's getting too much weight in, then that's going to be

Â detrimental and you won't end up getting convergence.

Â Now, you can, you can generalize these kinds of conditions.

Â So, in the paper with Ben Golub, we give more explicit characterizations of the

Â conditions. You can't have any group that's too

Â influential. You have to have some balance across

Â groups, and as long as things work out to be reasonably balanced, then you end up

Â with convergence. And if not, then you can end up with the

Â wrong kinds of beliefs in the limit. Okay, so that takes us through a little

Â bit of understanding the DeGroot model and learning convergence and so forth.

Â we'll, we'll wrap up some of our discussion of learning next and then we

Â can start turning to games on networks.

Â