Of course, a single fax machine is pointless (as you cannot even send faxes to yourself, since the fax protocol is “sync”). The 2nd fax machine is already more useful because it means the 2 users can now exchange messages with one another… The network’s value is directly related to its ability to “connect” people. Since everyone can be connected with everyone, Metcalfe’s law applies: the value of the fax network is the square of its number of participants.

Now, buying a fax machine when everybody else has one is not a very bold move, while buying the first one of your group of contacts is a pretty significant one, because the machine will only be useful if more members of the group get one too.

One could argue that economies of scale actually work against early adopters: they take more risks (what if no one else buys a fax machine?) and end up paying more than the people who join later: the more fax machines are produced, the cheaper they become because the producers’ investments are amortized over more machines. How can we reverse that and increase the reward for early adopters if the network eventually grows?

Enter discount tokens!

Let’s say each purchaser of a fax machine receives points to get discounts on more fax machines, at the time of the purchase. They can keep their points, and use or sell them to people who want to purchase fax machines at any time. The number of points they get is based on how early they purchased their fax machine. For example, let’s assume that the number of points someone gets is 100k/number of sold fax machines^2 . The first person gets 100k points, the 2nd gets 25k, the 3rd gets 11k… etc. Why do we use the ^2 ? Because of Metcalfe’s law: we know that the network’s value grows as the square of fax machine sold: so the value each participant receives should decrease similarly.

On the other end, the amount of discount each point yields is also based on the amount of sold fax machines: let’s take 1 point is 0.01ct * number of previously sold fax machines . If they used their points right away, the first user would then be able to get 100,000 * 1 / 100 * 1/100 = $10 in discount on their second fax machine. However, if the 1st buyer “waited” for 1,000 machines to have been sold, their 100,000 points would actually be worth $10,000 ( 100,000 * 1 / 100 * 1/100 * 1000 ) in discounts.

Similarly, the 1,000th person to join the network only receives 0.1 point, worth $0.01 if they used it right away… or $10 is they waited for the network to have 1M participants.

This model incentivizes people to become early participants in the fax machines network. If they do, and if the network eventually becomes large, they are rewarded for their initial “bet”. However, if somebody purchases a fax machine when tens of thousands of people have done so before, they do not see much of the benefits.

It is pretty clear now that this scheme is very (maybe too much) lucrative for early adopters. What is interesting is that the cost for the manufacturer is capped because the sum of 1/n² converges to π²/6 , so there will always be less than 164,493 points outstanding. Similarly, the “value” of a point is based on the number of machines previously sold. This means that the value of each point only increases if the network’s size increases. In practice, at any time, the the manufacturer will only ever spend at most 16.4% of their revenues in discounts.

We could of course improve this token design to make it slightly less rewarding for the very first users but also more valuable for the 1,000th person and further, based on a threshold after which the network effects are enough for people to become members of the network without any additional reward.