I recently ran across a fascinating article by Brian Burke at Advanced NFL Stats on the power law. He uses it specifically with respect to such things as coaching tenure and distribution of Pro Bowl selections in the NFL, but along the way he uses such things as the financial crisis that hit last fall to illustrate and explain the power law, and that’s what makes the article interesting (at least to me). For instance, Burke writes,
Our current financial crisis was in part caused by a fundamentally wrong assumption about risk distributions in the debt markets. An oversimplified explanation is that investment companies made lucrative but risky investments, and then hedged against their failure by buying insurance in the form of complex derivatives in case they went bust. These companies thought that they had cracked the code and solved the problem of risk once and for all. (One of the reasons the company AIG is central to the problem is that it’s the company that led the selling of all that insurance.)
The problem was that the insurance was priced based on an assumption of bell curve distributions of market risk. A model known as the Correlated Gaussian Copula was developed by a Chinese mathematician named Li, and it was widely used throughout the financial industry for measuring and pricing risk. Unfortunately, financial markets act more like earthquakes than normally distributed phenomena like rainfall or human height. There are lots of minor fluctuations but occasionally the bottom drops out. The power law distribution has a ‘fatter tail’ at the extremes than the normal distribution, meaning extreme outcomes are considerably more likely.
As Burke explains, power law distributions tend to arise with networks, especially complex, self-organizing ones; thus, he writes,
Power law distributions are noteworthy because they are the signatures of mature self-organizing complex systems. It’s also a feature of ‘rich-get-richer’ systems. So when we see power law distributions, we can make some qualitative inferences about the system we’re observing. For example, the BCS system is certainly a rich-get-richer organization. We can even quantify just how hierarchical it is and how difficult it is for second-tier teams to break into the elite.
The problem with the BCS isn’t just that it’s a rich-get-richer system. That’s just the natural way of the world. Even in supposedly ‘egalitarian’ systems like socialism, the rich still get richer. The difference is that initial outcomes in socialist systems are based primarily on one’s political connections, where in a free market they tend to be based on how productive or innovative one is. The problem is that the elite ‘nodes’ of the BCS have colluded to preserve their status on top, preventing a natural churn in who the elite are.
This is, among other things, an excellent succinct explanation of why socialism doesn’t produce the beneficial equality it promises: it actually increases the opportunities for elites to collude to preserve their status on top. The freer the market, the freer the society, the fewer levers they have to do so and the more opportunity there are for upstarts to upstage them and push them out of the way. The more controlled the market, the more controlled the society, the more levers the elites have, and the more ways and opportunities they have to use that control to keep anyone from breaking into their circle and taking their place.