So Prime Minister Narendra Modi has set off this little storm on Twitter by talking about the relationship between India and Canada being similar to the “$latex 2ab$ term” in the expansion of $latex (a+b)^2 $.

India and Canada are like that “2ab” that comes from the formula of (a+b)square: PM #ModiinCanada pic.twitter.com/1IpCJ0aQK8

— ANI (@ANI_news) April 16, 2015

Essentially, Modi was trying to communicate that the whole of the relationship between India and Canada is greater than the sum of parts, and it can be argued that the lack of a “$latex cos \theta$” term there implies that he thinks India and Canada’s interests are perfectly aligned (assuming a vector sum).

But that is for another day, for this post is about net neutrality. So how does 2ab explain net neutrality? The fundamental principle of the utility of the Internet is Metcalfe’s law which states that the value of a telecommunications network is proportional to the square of the number of entities in the network. In other words, if a network has n entities, the value of these n entities being connected is given by the formula $latex k n^2 $. We can choose the unit in which we express utility such that we can set $latex k = 1$, which means that the value of the network is $latex n^2$.

Now, the problem with not having net neutrality is that it can divide the internet into a set of “walled gardens”. If your internet service provider charges you differentially to access different sites, then you are likely to use more of the sites that are cheaper and less of the more expensive sites. Now, if different internet service providers will charge different websites and apps differently, then it is reasonable assume that the sites that customers of different internet services access are going to be different?

Let us take this to an extreme, and to the hypothetical case where there are two internet service providers, and they are not compatible with each other, in that the network that you can access through one of these providers is completely disjoint from the network that you can access through the other provider (this is a thought experiment and an extreme hypothetical case). Effectively, we can think of them as being two “separate internets” (since they don’t “talk to” each other at all).

Now, let us assume that there are $latex a$ users on the first internet, and $latex b$ users on the second (this is bad nomenclature according to mathematical convention, where a and b are not used for integer variables, but there is a specific purpose here, as we can see). What is the total value of the internet(s)?

Based on the formula described earlier in the post, given that these two internets are independent, the total value is $latex a^2 + b^2$. Now, if we were to tear down the walls, and combine the two internets into one, what will be the total value? Now that we have one network of $latex (a+b)$ users, the value of the network is $latex (a+b)^2$ or $latex a^2 + 2 ab + b^2$ . So what is the additional benefit that we can get by imposing net neutrality, which means that we will have one internet? $latex 2 ab$, of course!

In other words, while allowing internet service providers to charge users based on specific services might lead to additional private benefits to both the providers (higher fees) and users (higher quality of service), it results in turning the internet into some kind of a walled garden, where the aggregate value of the internet itself is diminished, as explained above. Hence, while differential pricing (based on service) might be locally optimal (at the level of the individual user or internet service provider), it is suboptimal at the aggregate level, and has significant negative externalities.

#thatswhy we need net neutrality.