# Tail Indexing and Bitcoin

In February 2010, Satoshi Nakamoto contextualises the finite limit of bitcoins by saying as the block subsidy reduces, transaction fees will provide compensation for nodes. He also qualifies this by saying in 20 years time, they’ll either be high bitcoin transaction volume or no volume at all.

If the latter prediction is true, it means in approximately 5 or 6 years bitcoins will have gone out of fashion. In this article, we look at this possibility by utilising tail-indexation of a probability distribution.

# Tail Distributions

In probability theory, tail distributions are probability distributions whose tails are not exponentially bounded. In many applications, it is the right side of the tail which is of interest. We can see this in the bitcoin supply growth rate which fits classically with long tailed distributions of a high-frequency or high-amplitude population (such as the initial bitcoin block rewards) followed by a low-frequency or low-amplitude population which gradually “tails off” asymptotically (i.e. because of future and subsequent halvings).

Long tail indexation is standard analysis for businesses to understand rank-size and rank-frequency distributions and is associated with the Pareto 80:20 Power Law, but the way in which Bitcoin used this in its protocol to predetermine monetary supply represented something new when the first coins were mined in 2009. We can see the close relationship in the cumulative and supply density of bitcoins following a Pareto distribution:

# Satoshi’s Familiarity with Tail Indexation

Satoshi’s familiarity with probability is first laid out in the original Bitcoin Whitepaper (most notably Section 11, *Calculations*), but then goes onto explain how coin generation works to the 80:20 rule, and then also explains the application of long tail theory in relation to bitcoin mining where the greater combined value lays in the longer part of the tail. In the whitepaper too, Satoshi designs the code to avoid an infinite tail.

There is however another insight which indicates Satoshi’s idea as to how his coins will obtain value: he admits to not knowing how software can reflect the real world value of things, other than to keep the supply of his bitcoins predetermined, so as the value of coins increase so do new users to create a positive feedback loop — this fits with the power law idea that a functional relationship between two quantities (in this case, supply and demand for bitcoins), vary as a power of another.

We can see how this has played out in the right side of the tail of the American dollar against the value of bitcoin:

# Future Bitcoin Volumes

This then leads to some concluding reflections on the future of bitcoin transaction volumes. If it comes to pass that there will be no bitcoin transactions in five years, then how will that be the case?

The idea here is that if tail indexing in the form of probability distribution is a contributing axiom for bitcoins to appreciate against sovereign fiat issuances, then it’s possible for some sort of external and official currency board or coalition to come along and index on bitcoins in this same way — because if bitcoins are following a long tail distribution, and according to the theory, the future coins have a gradually lowering statistical probability of being mined.

I’ve written previously on how this might be worked from an axiomatic specification to be run at governmental level (either unilaterally or multilaterally), but as yet — to the best of my knowledge — remains an experimental conjecture.