The fundamental mystery of power law distributions

From Inequality in nature and society by Marten Scheffera, Bas van Bavelb, Ingrid A. van de Leemputa, and Egbert H. van Nes. Abstract:

Most societies are economically dominated by a small elite, and similarly, natural communities are typically dominated by a small fraction of the species. Here we reveal a strong similarity between patterns of inequality in nature and society, hinting at fundamental unifying mechanisms. We show that chance alone will drive 1% or less of the community to dominate 50% of all resources in situations where gains and losses are multiplicative, as in returns on assets or growth rates of populations. Key mechanisms that counteract such hyperdominance include natural enemies in nature and wealth-equalizing institutions in society. However, historical research of European developments over the past millennium suggests that such institutions become ineffective in times of societal upscaling. A corollary is that in a globalizing world, wealth will inevitably be appropriated by a very small fraction of the population unless effective wealth-equalizing institutions emerge at the global level.

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Prospects

Our analysis suggests that even if all actors are equivalent, in the absence of counteracting forces, there is an intrinsic tendency for significant inequality to arise from multiplicative chance effects. Although the surprising similarity between inequality of species abundances and wealth may have the same roots on an abstract level, this does not imply that wealth inequality is “natural.” Indeed, in nature, the amount of resources held by individuals (e.g., territory size) is typically quite equal within a species. While wealth inequality may have emerged as far back as the Neolithic era (31, 36), the relative amount of wealth appropriated by the richest has increased as societies have scaled up. One explanation for this effect is scale itself. Put simply, one can accumulate less wealth in a village than across the globe. However, as we have argued, another explanation is that installing effective institutions to dampen inequality becomes more challenging as scale increases. Excessive concentration of wealth is widely thought to hamper economic growth, concentrate power in the hands of a small elite, and increase the chance of social unrest and political instability (1, 2, 4, 37⇓–39). This raises questions about the prospects for current societies. Phases of upscaling of governance successfully curbed unconstrained growth of inequality first in the communities of late medieval Europe and later in the nation states of the 20th century, but in both cases, this was a lengthy and painful process. Whether scaling up of effective governance can now be done at the global level and, if so, what this new form of governance might look like, remains unclear.

So the researchers are ideologically committed to equality and are finding that inequality is a natural condition, both in nature and in human societies.

Their commitment towards "equality" should not obscure that the issue is puzzling and interesting.

I have no inherent objection to inequality as long as everyone's quality of life is improving and as long as there are safeguards to ensure that there are checks and balances to prevent any particular group from coercively exerting power over other citizens. Our history is marked by a long run of improvement in human quality of life based on Age of Enlightenment principles (reason, evidence, scientific method, property, open markets, natural rights, rule of law, equality before the law, constrained government, consent of the governed, etc.). Our history is also marked by disastrous efforts to achieve improvement by displacing the Enlightenment Package with coercive ideologies, many of them seeking a puritanical equality which always comes at the expense of quality of life improvement. You end up with worsening living standards and worsening inequality at the same time.

The researchers allude to "upscaling of governance successfully curbed unconstrained growth of inequality first in the communities of late medieval Europe and later in the nation states of the 20th century, but in both cases, this was a lengthy and painful process." Their explanation of these episodes of upscaling of governance are not informative and entail chasing links to links. I am unconvinced by the claim. The only persistent re-equalizations of which I am aware being rigorously documented are those occasions when countries are visited by the four horsemen of the Apocalypse, Conquest, War, Famine, and Death. Plague reduces inequality. War reduces inequality. Famine reduces inequality. Basically, population decimation reduces inequality. These are not desirable policies but they are the only ones currently known to reliably reduce inequality.

From my perspective, it is better to focus on improving societal and personal productivity which enhances quality of life for everyone but that is not an attractive proposition for those chasing the chimera of equality.

But what is really interesting to me is the researcher's demonstration that the power laws underpinning inequality are as evident in nature as they are in social conditions. "Surprisingly, Gini indices for our natural communities are quite similar to the Gini indices for wealth distributions of 181 countries (data sources listed in SI Appendix, section 1)."

They are hinting at the real underlying mystery which has fascinated me for a long time and which I still cannot explain, the prevalence of power law distributions in so many natural systems.

In societies, inequality is also found for other units besides the wealth of single actors or households. For instance, power law-like distributions characterized by high inequality are found in statistics on city sizes, number of copies sold of bestseller books, number of adherents of religious bodies, and number of links to web sites (9). In addition, firm size typically varies widely, with a few companies dominating the market (10, 11). At first glance, firm size may seem comparable on an abstract level to the wealth of households. Indeed, firms may grow and shrink depending on vagaries of markets and other factors. However, there are also important differences. For instance, firms are relatively ephemeral entities that are linked through a global web of shareholders (12) and may be fused or split depending on shareholders’ decisions and antitrust legislation. In this paper, we limit our discussion to the wealth of households for our comparison of nature and society.

Much of the paper is an acknowledgement of the prevalence of power laws and how that prevalence undermines the researcher's desire for equality among everyone. It does not really get at the real mystery. Why are there power laws everywhere?

Some specific applications:

Pareto Distribution - The crude vernacular description is that 20% of X produces 80% of Y. 20% of criminals commit 80% of the crimes; 80% of the wealth in a society is owned by 20% of the individuals; 20% of the producers create 80% of the supply, etc.

Zipf's Law - "For example, Zipf's law states that given some corpus of natural language utterances, the frequency of any word is inversely proportional to its rank in the frequency table. Thus the most frequent word will occur approximately twice as often as the second most frequent word, three times as often as the third most frequent word, etc.: the rank-frequency distribution is an inverse relation."

Lotka's Law - "It describes the frequency of publication by authors in any given field. It states that the number of authors making {\displaystyle x} x contributions in a given period is a fraction of the number making a single contribution, following the formula {\displaystyle 1/x^{a}} {\displaystyle 1/x^{a}} where a nearly always equals two, i.e., an approximate inverse-square law, where the number of authors publishing a certain number of articles is a fixed ratio to the number of authors publishing a single article."

An example: Articles published in a field are likely to follow a Lotka distribution where 10 authors write one article each, 9 authors write 2 articles each, and so on up to the most productive author, of which there is only one who writes ten articles. In sum, there are 294 articles by 155 authors writing an average of 1.9 articles each. But with Lotka's Law distribution, the top 10% (roughly fifteen authors) produce 87 of the 155 articles, i.e the top 10% of authors produce 56% of the work.

Price's Square Root Law - Price’s square root law or Price’s law pertains to the relationship between the research literature on a subject and the number of researchers in the subject area stating that half of the publications come from the square root of all contributors. Thus, if 100 papers are written by 25 authors, five authors will have contributed 50 papers.

These are all specific applications of power law distributions. Power law distributions are at the heart of what Scheffera, et al are researching.

Natural distributions in nature and society are astonishingly frequently described by a power law distribution. Small percentages of the population are responsible for large percentages of the outcomes.

But why? Why is this so frequently the case? We don't know. And without knowing the causal mechanisms, we have to be very careful about interfering in processes because we can't foresee what the likely outcomes will be.