Generally a single point estimate is rarely enough to describe a distribution. It seems that the “Nakamoto coefficient” is essentially the weighed 51st percentile (the weights being entity proportions) and as such suffers from the same drawbacks as other point estimates.
The three stated benefits of measuring decentralization are valid but the authors have not demonstrated that the proposed single point estimate and apples to oranges comparison(“Minimum Nakamoto”) provides a suitable measure of such decentralization.
A few technical comments: calculating the coefficient as currently defined does not yield the answer that is expected. Instead of N_s it would give K-N_s+1. In order to get N_s, as discussed elsewhere in the text, the proportions should be sorted in the opposite direction (i.e., p_1>p_2>…>p_K).
Lastly, something is up with the Bitcoin address example. It’s easy to check that the statement “171 addresses (controlled by exchanges or individuals) are sufficient to get to 51% of Bitcoin ownership” is not true. The source used by the authors (bitinfocharts.com) states on its stats page that the distribution of BTC is: Top 10/100/1,000/10,000 addresses => 5.02% / 17.73% / 34.65% / 57.33% Total. So at something in between 1000 and 10000 top addresses contain 34 to 57% of BTC. Scraping the original data and calculating the weighted 51st percentile (or Nakamoto coefficient) gives 5213 as the number of addresses that contain 51% of BTC. The top 171 addresses only have some 21.3% of BTC (graph here http://i.imgur.com/MU1ptww.png). Also, considering there is nothing preventing the same individuals controlling multiple addresses it is not immediately clear how this result can be interpreted as a proxy for ease of asserting control.