It’s the distribution of money/tokens/coins that matters
And, then all bets are off
Normalization
In maths, there’s this thing called normalization. It’s not complicated, so don’t run. You have a bunch of buckets and each bucket contains some marbles. The marbles tally up to a total. In order to find the normalization, you divide the individual bucket marble count by the total and instead of a count, you get the proportion of the total that sits in each bucket. Now, that’s interesting and a lot more useful than a count. Here is why…
Normalization is useful because it can take something with a finite total, say, my lunch money plus yours, and the girl that sits to the left; and turn it into a probability distribution.
Let’s call the protagonists, Alice, Bob and Charlie and suppose that they have 20, 5, and 15 tokens respectively. Now, if this is normalized, then we divide by 40 and get the distribution as 1/2, 1/8 and 3/8 respectively. Let’s write it in order as (A, B, C) or (1/2, 1/8, 3/8). Basically, this means that if we found a random token on the ground, there is a 50% chance it’s Alice’s, a 12.5% chance it’s Bob’s and a 37.5% chance it’s Charlie’s.
Why does it matter?
Suppose that we have rules that increase the token supply. How will a rule change the distribution? Let’s look at a few examples and, for simplicity sake, assume that no spending/other rules apply for the duration of the example.
Example 1. The rule runs for 10 rounds, and each round A, B, and C each receive two tokens. What do the table and distribution look like after 10 rounds? It’s (2/5, 1/4, 7/20). The distribution is slightly more even in the sense that Alice and Charlie own proportionally fewer tokens, and Bob owns proportionally more.
What happens after 100 rounds? The distribution is (11/32, 41/128, 43/128). At this point, the distribution has almost converged, and they each own roughly the same proportion of the tokens; around one third.
Example 2. Suppose that we have a rule that depends on the initial number of tokens owned. For 10 rounds you gain 10% of your current token balance. At the end of 10 rounds, the table looks like this (1/2, 1/8, 3/8). After 100 rounds, the distribution is exactly the same: (1/2, 1/8, 3/8). For the record, the absolute amounts are Alice: 275612.25, Bob: 68903.06, and Charlie: 206709.19.
Example 3. Finally, suppose we impose the rule that gives you a token if you have more than 10 tokens and takes away a token if you have less than 10 tokens. How will that go? After 10 rounds, the distribution reaches a stable equilibrium point and doesn’t move from there. It just stays at (1/3, 1/3, 1/3) and they each have 10 tokens and nothing changes.
What does that have to do with the price of eggs?
These are contrived examples, and we can construct more complicated ones, but what do they have to do with reality?
Well, tokenomics and monetary systems are run by rules like these. And, when you evaluate a rule, the question that ought to matter to you is — how does the rule affect the distribution, and how does it affect my proportion of the whole in the longer term? In a world where tokens represent rights, privilege and economic power, this is the question that matters, when you consider the effect of a token or monetary policy. The distribution contains all this information, and when this information gets altered, then your rights are altered too…
PS. This article is my own opinion. I am a freelance mathematician for hire, developing algorithms, studying tokenomics, and thinking thoughts. Feel free to contact me via LinkedIn.
