Basic Definitions

When is there convergence?

When is there a consensus?

Who has influence?

**==> When is the consensus accurate**

How does this depend on network structure?

How does it depend on influence?

How does it relate to speed of convergence

Suppose true state is $\mu$

Agent i sees $b_i(0) = \mu + \epsilon_i$

$\epsilon_i$ has 0 mean and finit variance, bounded below and above

signal distributions may differ acorss agents, but are independent conditional on $\mu$

Consider large societies

If they pooled their information, they would have an accurate estimate of $\mu$

For what sequences of societies does Prob: $$lim_t [|b^n(t) - \mu| > \delta] \rightarrow 0, \forall \delta$$

Prove this using Chebychev's inequality

Let $\epsilon_i's$ be independent, zero mean, and each have finite variance (bounded below). Then:

$plim \sum s_i^n \epsilon_i = 0$ iff $max_i s_i^n \rightarrow 0$

- let's suppose that we were doing some influences. Whatever those influences are, so society n has a vector s1, through sn. So we look at those. We'll call those the $s_i^n$'s. All right. So in society n, you've got an $s_1$ through $s_n$, then we have a different one for when we add this extra person. So each one of these societies has a vector. the Weak Law of Large Numbers tells us that if we look at averaging those error. So what's going to happen is the influence is going to capture how much of each persons error enters into the overall societal error. The sum of the weights on those errors is going to be 0. If and only if the largest influence that anybody has goes to 0. So if anybody retains influence, then what we're going to do is end up retaining weight on that person's error. Their believe is going to be a non-trivial part of the overall society's belief. And so, it's going to be necessary that everybody have a negligible weight in the limit. So as society aggregates, they have to disperse so that we're putting weight on different individuals. If we keep all listening to the same person, we're going to have inaccuracy. And, actually that's going to be enough, so as long as society spreads it out, given everybody has finite variance, averaging a bunch of these variables will give us a good answer.

**Wise crowds iff max influence vanishes**

Suppose that T is not only row stochastic(gives some weight out) **but also column stochastic**(so each agent get the same weight in, receives weight one). Then $s = (1/n,...1/n)$ is a unit lhs eigenvector, and so T is wise. So if everybody got weighted as much as weight in as they were giving out, we'd be in good shape.

So, **reciprocal trust** implies wisdom

But that is very strong condition, generally in society, we're going to have some heterogeneity in terms of overall, how mush somebody gets paid attention to.

What's important is that when we're looking at this, there's no single individual that's getting too much of the weight from other individuals who matter.

$s_i = \sum_i T_{ji}s_i$

If there is some i with $T_{ji} > a > 0$ for all j, then $s_i > a$, i.e. if everybody put a weight at least to "a" on them, then their overall influence would be at least "a", so there can't be anybody who gets too much detention.

So clearly cannot have too strong an "opinion leader"