I have read that Pareto Distributions come from modeling phenomena that have certain characteristics like: preferential attachment, positive feedback plus other considerations like a constant negative tax against that preferential attachment. The phenomena to model can be with infinite additions of items or the number of items can be finite and, conceptually, it can be thought as a (in)finite sequence of events in many cases.
For example a pool of paperclips for which in each round we follow these rules:
1) choose two paperclips randomly
2) if they are not linked, then link them and do 3) and if they are linked do 1)
3) add a new paperclip in the pool
4) do 1)
In the limit my intuition tells me that this phenomena will follow a Pareto distribution, where a certain x% of the paperclips-chains ordered by length hold a percentage y% of the paperclips, and where x% < y% and x%+y% = 100%. But I have no idea which will be the tail index (also called shape index or alpha) of this distribution or how can I derive it.
The most common tail index in natural phenomena modeled with Pareto distributions is α = log4(5) ≈ 1.16, which I believe corresponds to the famous x=20 and y=80. My question asks how can this tail index value be derived theoretically in general, from probability assumptions and "rules of a game".
And if indeed can be theoretically derived, what will be the corresponding probability assumptions and "rules of a game" for obtaining α = log4(5) ≈ 1.16
(Or it was the case that this particular tail index value was "found" from simulations that adjusted to the x=20 and y=80 values)