I stopped following Justin Wolfers on Twitter.  Not because I don’t want his tweets, they are great,  but because everyone I follow also follows Justin. They all retweet his best tweets and I see those so I am not losing anything.

Which made me wonder how increasing density of the social network affects how informed people are. Suppose you are on a desert island but a special desert island which receives postal deliveries.  You can get informed by subscribing to newspapers but you can’t talk to anybody.  As long as the value v of being informed exceeds the cost c you will subscribe.

Compare that to an individual in a dense social network who can either pay for a subscription or wait around for his friends to get informed and find out from them.  It won’t be an equilibrium for everybody to subscribe.  You would do better by saving the cost and learning from your friends.  Likewise it can’t be that nobody subscribes.

Instead in equilibrium everybody will subscribe with some probability between 0 and 1.  And there is a simple way to compute that probability.  In such an equilibrium you must be indifferent between subscribing and not subscribing.  So the total probability that at least one of your friends subscribes must be the q that satisfies vq = v – c.  The probability of any one individual subscribing must of course be lower than q since q is the total probability that at least one subscribes.  So if you have n friends, then they each subscribe with the probability p(n) satisfiying 1 – [1 - p(n)]^n = q.

(Let’s pause while the network theorists all rush out of the room to their whiteboards to solve the combinatorial problem of making these balance out when you have an arbitrary network with different nodes having a different number of neighbors.)

This has some interesting implications.  Suppose that the network is very dense so that everybody has many friends.  Then everyone is less likely to subscribe. We only need a few people to be Justin Wolfers’ followers and retweet all of his best tweets.  Formally, p(n) is decreasing in n.

That by itself is not such a bad thing. Even though each of your friends subscribes with a lower probability, on the positive side you have more friends from whom you can indirectly get informed.  The net effect could be that you are more likely to be informed.

But in fact the net effect is that a denser network means that people are on average less informed, not more. Because if the network density is such that everyone has (on average) n friends, then everybody subscribes with probability p(n) and then the probability that you learn the information is q + (1-q)p(n). (With probability q one of your friends subscribes and you learn from them, and if you don’t learn from a friend then you become informed only if you have subscribed yourself which you do with probability p(n).) Since p(n) gets smaller with n, so does the total probability that you are informed.

Another way of saying this is that, contrary to intuition, if you compare two otherwise similar people, those who are well connected within the network have a tendency to be less informed than those who are in a relatively isolated part of the network.

All of this is based on a symmetric equilibrium.  So one way to think about this is as a theory for why we see hierachies in information transmission, as represented by an asymmetric equilibrium in which some people subscribe for sure and others are certain not to.  At the top of the hierarchy there is Justin Wolfers.  Just below him we have a few people who follow him.  They have a strict incentive to follow him because so few others follow him that the only way to be sure to get his tweets is to follow him directly.  Below them is a mass of people who follow these “retailers.”

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