Metcalfe’s law does not state in any way that if you add users to a network then the value, and therefore the price, increases. This would be an absurd claim and the analogy that comes to mind is an increasing population full of infants using fax machines.
Metcalfe rather speaks to POTENTIAL value, if we are going think about the price and market cap of a network. That is to say that there is room for matchmaking connections for N² users, but there is nothing in such a law to suggest that any amount of users can efficiently use a certain network (later in this writing we will read Szabo alluding to such redundancies).
In other words it cannot be said that the addition of each user adds the same amount of value which is then multiplied across the network. In most cases it is probably easier to show such a claim cannot be true (how much value would be added from the last person in the world to have a fax network?).
Here we get Szabo’s extension of Metcalfe’s law in regard to emerging economics (through Adam Smith):
Metcalfe’s Law states that a value of a network is proportional to the square of the number of its nodes. In an area where good soils, mines, and forests are randomly distributed, the number of nodes valuable to an industrial economy is proportional to the area encompassed. The number of such nodes that can be economically accessed is an inverse square of the cost per mile of transportation. Combine this with Metcalfe’s Law and we reach a dramatic but solid mathematical conclusion: the potential value of a land transportation network is the inverse fourth power of the cost of that transportation.
Notice Szabo’s use of the word “potential”.
I had already linked to that Szabo quote before you mentioned it:
Szabo is incorrect. He fails to consider that we live in a relativistic universe. The maximum
potential of the network is not the reciprocal of the number of nodes to the fourth power. Szabo is computing the
potential as if the maximum is where every node communicates/trades to every other node and the cost of the transport being the limiting resource or cost. But as I showed in
my other comment in the blog which I linked you to, that value is meaningless unless it is considered relative to all other opportunities, i.e. opportunity cost is the limiting resource. Value is always relative, not absolute. Szabo is ascribing an absolute cost of communication and assuming that is the dominant opportunity cost from the individualized perspective of every node. But
I showed mathematically that rather it can be the grouping compatibilities that can be a limiting opportunity cost that can invert the assumption of greater relative value for the larger network. Networks increase the degrees-of-freedom of the node participants, thus the potential energy. To the extent that transport cost is a significant opportunity cost of the nodes, then Szabo's point applies, but as the cost and latency of communication decreases, there are a proliferation of opportunities which are significantly more valuable than those transport costs. As
Lima pointed out, the Inverse Commons was one of those huge value opportunities that was enabled by the Internet. The value from exchanging knowledge in the Inverse Commons over the Internet far exceeds any communication costs. So as you now see, money is not the only agent that can increase degrees-of-freedom in trade and increase surplus production. Communication networks can increase the access degrees-of-freedom for non-fungible knowledge, which becomes fungible collaborative within the Inverse Commons. So thus, we see fungible money becomes only a small component of the value creation, such as to pay for the communication infrastructure costs. This is why fungible money is diminishing in utility in the knowledge age. Fungible money is applicable to increasing the degrees-of-freedom for solving the coordination issues around physical resources. Yes atoms are heavy, but relative to knowledge production value, the atoms are asymptotically (an inexorable trend to) massless. So into the knowledge age we go, and fungible money will diminish in importance and
our insatiable quest will shift from power to knowledge.
Note this is second time
I caught Szabo with a fundamental error. Szabo does raise interesting historical examples and his anthropological research is sometimes interesting.
@traincarswreck, your four-color theorem theory of tripartite essential resources for human life is the most basic example of the fact that fungible money increases degrees-of-freedom over barter. The increase in efficiency of fungible money w.r.t. barter, increases as the diversity of physical resources (tangible goods) increases. With three bartered resources and no fungible money, one has a 1/3 chance of holding the resource that another may want to trade for, i.e. being bordered on one of the other two colors in your four color theory. The probability fraction decreases and complexity of risk mitigation increases as the diversity of goods to be bartered increases.