Metcalfe’s Law

Unless you work in the telecom industry, you may never have heard of “Metcalfe’s Law.”

And rightly so, as Metcalfe’s law is a simple and obscure mathematical observation relating to the impact/reach of a telecommunications network.

The law states that the potential of a network is simply the number of nodes squared, or mathematically:

n2

All this means, in its basic form, is that the network is more valuable the more people that use it.

For example, if you had a phone that could only call one other phone, then that network has a low value, as there are only two possible connections. Phone A can call Phone B, and Phone B can call Phone A.

If you add an extra phone to the network (for a total of 3 phones) then your total number of connections doesn’t just increase by one. Instead, if you have phone A, B and C you can now have connections:

A calls B.

B calls A.

A calls C.

C calls A.

B calls C.

C calls B.

So whereas a network of two phones had 2 connections, a network with 3 phones has 6 connections. This compounding growth curve is represented on a graph by the equation. The growth of value in a network scales exponentially, giving us what we call “the network effect.”

How does this work outside of telecom?

While it may make sense for a telephone network (or computers on the internet), how do we apply this to systems outside of telecoms?

My favorite example involves currency and gift cards. But before we dive into that example, we need to define a few terms. When we think about the world of gift cards, there are two types: “closed-loop” and “open-loop.” They get their name from broader concepts which can be applied to financial systems.

Closed-Loop Financial Systems

Closed-loop financial systems are ones in which the flow of money is controlled and can only be earned or spent in certain places.

For gift cards, this is like buying a gift card for a specific company such as McDonald’s.

You give $10 to McDonald’s, they give you a gift card valued at $10, and they know you can only spend it at McDonald’s.

Another example would be something like tokens in an arcade. You’ve purchased the tokens, which are the closed-loop currency, and can only use them in that arcade. Outside of the arcade they have very little value.

In fiat economic systems, we don’t really have any examples of a 100% closed-loop financial system. The closest we come is the Chinese Renminbi/Yuan which is a controlled currency. The Chinese government works hard to prevent Yuan from leaving the Chinese financial system, although it still does leak out through various black market channels.

It’s important to remember that closed-loop (and open-loop) aren’t categories. They are a sliding spectrum.

Open-Loop Financial Systems

On the other hand, open-loop financial systems are when money can be entered into a system that is not controlled. The most open example of this is any national currency, when you buy into it you are free to spend it anywhere in that country and can often find places around the world to easily accept and exchange your currency.

But, when we talk about gift cards, “open-loop” refers to a gift card that can be redeemed at multiple locations. One such example might be a gift card you can redeem at any store in your local shopping mall, or, if we want an even more open-loop card we’d use the example of pre-paid Visa gift cards, which can be redeemed just about everywhere.

Once again, as we can see, there is no example of a financial system that is 100% open, and there are multiple levels of “openness.”

Applying Network Value to Gift Cards

Now, let’s assume that I have four different financial instruments in my possession:

A $5 bill. A $5 gift card to Einstein Bro’s Bagel Co. A $5 gift card to Starbucks. A $5 Visa gift card.

At first glance, if we were asked which of these is the most valuable, it might be tempting to say “Trick question! They are all worth $5!” But, given what we now know about how networks are valued, we may take a different approach.

When we think about selling these items, we instantly know that the $5 is the most valuable, because we would never sell a $5 bill for anything less than $5. The underlying reason for this is a $5 bill can be used anywhere and so we don’t discount it at all.

As for the rest of the cards, let’s take a look at what they are worth on second hand networks. If we go over to GiftCardGranny.com, we can look up the value of the different cards.

Einstein Bro’s Bagel Co.

When we look up the gift card at Einstein Bro’s Bagel Co. we can see that the average giftcard for their store is selling at a 35% price discount:

This means if we attempted to sell our $5 Einstein Bro’s Bagel Co gift card, we’d probably only get $3.25 for it.

Why? By exchanging $5 of an open-loop currency for a closed-loop currency, we are restricting the number of places we can spend it, which makes it less valuable. So the supply and demand of “people who want Einstein Bro’s Bagel Co gift cards” and “people who have Einstein Bro’s Bagel Co gift cards” is out of balance, and thus, sellers must compete on price discounting to get their money back into the open-loop.

Starbucks

So, by this same logic then, we may expect to see that our $5 Starbucks gift card is worth about $3.25 as well, right?

Instead, the average Starbucks gift card is only 13.56% discounted, meaning our $5 card is worth about $4.32.

Why the difference? When we look at these cards as “networks” we have to remember how we apply Metcalfe’s law – the number of nodes matter.

For gift cards, these nodes are:

Starbucks locations. Number of people who want Starbucks.

Simply put, there are more Starbucks than Einstein Bro’s Bagel Co locations, and more people who prefer Starbucks to Einstein Bro’s Bagel Co. Therefore the network has more nodes and is more valuable.

Visa Gift Card

At this point, I think we all know what to expect:

The average Visa gift card trades at a discount of only 0.75%, making our $5 card worth roughly $4.96 – because it has a wider network with more nodes. More freedom for spending, more demand for buying.

The Value of Our Cards

So that makes our final value list:

The $5 bill (Worth $5) The Visa gift card (Worth $4.96) The Starbucks gift card (Worth $4.32) The Einstein Bro’s Bagel Co gift card (Worth $3.25)

Applying the Law to New Economic Systems and Crypto

Now that we have an understanding of Metcalfe’s Law, it would seem to suggest that we could simply count the number of nodes or transactions within an ecosystem and accurately get the price of a currency, right?

Many folks who are far better economists and mathematicians than I am, have tried (with varying levels of success) to apply this model to cryptocurrencies; and while many models fit backtests [Read: Issues with Backtesting], they fail to accurately predict the growth of a cryptocurrency based on either its number of nodes (users) or the number of transactions moving forward. (Although many of them are really awesome models).

Why is this?

There are two main reasons for this:

The original Metcalfe’s Law is designed to only measure the maximum potential value of a network. It does not measure the current or actual value. The original Metcalfe’s Law is designed to measure all nodes within a system at an equal value.

So while the general trend of “Network Transactions2” is historically true, this is more likely a matter of correlation, and not causation.

Different Transaction Values:

As we saw in our gift card example, not all nodes are of equal value or strength – and this is especially true of transactions in a financial network.

For example, here are four transactions:

I transfer $5 worth of Bitcoin between two wallets. I transfer $500 worth of Bitcoin between two wallets. I purchase something worth $5 using Bitcoin. I purchase something worth $500 using Bitcoin.

We can’t assume that these transactions have all added equal value to the network. In fact, we could debate if the first two added any real value at all.

The two remaining purchases were value within an ecosystem, but, at very different scales.

Different Node Values:

If we want to think of nodes in terms of network services/participants rather than transactions (which would be useful if you are applying the model to something like the Kin Ecosystem) then we can look at a different case.

Imagine two developers add Kin into their app:

“Developer A” has 1,000 daily active users (DAU) who love using their product, have an emotional connection to it and think it is an important part of their daily lives. “Developer B” has 500 DAU. They find the product useful, but in a solely functional manner.

In looking at Metcalfe’s Law we couldn’t equally weight these two systems, in fact because of the emotional component it wouldn’t even be fair to count the 500 DAU as 50% of the 1,000 DAU, as users with a strong emotional connection will pay more for something than those using it solely for function.

How do we solve for this?

That’s something I don’t have the answer to, and never will – at least not as a concrete formula. Metcalfe’s Law simply isn’t designed to predict the future price of a currency or network, primarily because to do so requires us to do complicated weighting and individual investigation for each node that makes it prohibitive.

How would we approach it? For us to evaluate the worth of a network I think we need to take into account:

The number of nodes.

The weight of that node compared to others within the system.

The value added by that node’s transactions.

The emotional weight users have to that node, measured by engagement KPIs.

The number of users on that node.

So, if Metcalfe’s classic law of n2 gives us the upper-bound value score of a network, then what we need to do is weight each node on some form of distribution and discount or increase the value of each node from there. In the end, we should end up with something that is a fraction of n2.

For a cryptocurrency like Kin, this might end up being something like:

((N1((f)(a)/2)+N2((f)(a)/2)+…+Nnth((f)(a)/2))2

Where:

Nn th is each individual node, represented by a count of N=1

is each individual node, represented by a count of N=1 f is the bell curve score of each node based on the value added to the network (solved as f(x) = y 1 + ((y 2 -y 1 )/(x 2 -x 1 )) (x-x 1 )).

+ ((y -y )/(x -x )) (x-x )). a is the bell curve score of each node based for user engagement, based on user engagement over number of users. (Same bell curve scoring as above).

Walking through the challenges

In this model, we take each node and give it a starting score of 1.

We then take their scores for “user engagement” and “value added” and grade them on a bell curve against all other nodes, giving us a weighted score for each. We then get the average of those scores. For example:

If N1 was the best node for “value-add” then it gets 100% or “1” and if it was a leading node for “user engagement” then it may get an 80% or “0.8” score for that.

We take the average of those two values and we get 0.9. We then discount the value of N by multiplying it by that weight. In this case, since the node started at “1” it is now a score of “0.9”.

We repeat that process for every node within the system and then we add up the final scores, after this we then square the sum of our result.

That will leave us with a number that is some sub-fraction of the classic n2 rule that is probably a more accurate predictor of the value of our network.

Does this really work? Is it accurate?

No, not at all. There are probably a number of issues with both my assumptions, and with the actual math equation. I’m not a mathematician. The point here is more to illustrate that in order to adapt Metcalfe’s Law, we would need to come up with complicated weighting mechanisms which makes it unrealistic to accurately predict the value of a currency.

The math gets very complicated very quickly, and the needed variables become almost impossible to measure.

Wait, so you are saying that we can’t predict the price of a crypto with Metcalfe’s Law?

Yes. Sadly, the goal of this article is to help you realize that Metcalfe’s Law is a mental model designed to help us think about measuring and growing value. It’s not something that we’ll likely ever be able to adapt and apply as a predictive tool.

At best, we’ll be able to adapt it in a way where we can accurately backtest/backfit data within acceptable bounds, but it likely won’t be a good indicator of raw price.

At the end of the day, Metcalfe’s Law won’t tell you the future price of your currency, but, understanding the principle of more nodes in a network equaling more value is important.

While many folks have created very complicated variants of Metcalfe’s Law to try and apply it to cryptocurrency and other financial systems, the only thing that has remained true is the simple n2.

Metcalfe’s Law is a theory. It’s a guideline to help you understand that networks grow value in a compounding and non-linear fashion. It wasn’t designed to predict the price of a network – the only thing you need to takeaway from Metcalfe’s Law is that if you want your cryptocurrency to be worth more, then adoption of earn and spend opportunities are the key.

TL;DR:

Metcalfe’s Law is about how “network effects” create compounding value. It will never be usable as a forward looking price predictor. But, it’s an important concept to understand in economics.