As discussed in previous columns, financial crashes have similar statistical properties as earthquakes, craters on the moon, and many other natural phenomena, in that they follow what is known as a scale-free, power-law distribution. This means in effect that there is no typical size or scale: only the rule that, the larger an event is, the less likely it is to happen.

For example, if you double the size of an earthquake, it becomes about four times rarer, so earthquake frequency depends on size squared, or size to the power two – hence “power law.” There is no such thing as a “normal” pattern, and there is the ever-present possibility of extreme events.

This may sound like depressing news. What is the point in planning for the future if we can’t know what it will look like? However, there is a flip-side, for it isn’t just financial disasters which follow a power-law. The same thing holds for financial opportunities. The distribution of company sizes, for example, is scale-free: there are many tiny firms consisting of just a few people at one end of the scale, and a handful of global behemoths at the other. Opportunities are constantly bubbling up from the bottom. Most go nowhere, but a few will become massive.

The inverse question to the risk of disaster, is therefore the possibility of outlandish success – the bio-tech company that invents or accidentally discovers a blockbuster drug, the rash entrepreneurial gamble that pays a million to one. These are events that don’t register in the ‘normal’ economy of mainstream economics, exactly because they are statistically unlikely. And yet it is just such extraordinary events that define the economy.

Who would think that a small hamburger restaurant could grow into a global fast-food chain, or that a new author could produce a huge bestseller? Such events always come as a surprise, which is why we should always be open to new opportunities.

Individual wealth also follows a power-law distribution, at least for the top 50 percent of the population. The graph to the right combines two sets of data. The solid line on the right is from a UN report on the world wealth distribution (I have excluded those with less than $10,000 because the power law breaks down beyond that point).

The UN data only extends up to the richest one percent of the adult population, ie those with about half a million dollars or more of assets. The world’s billionaires are shown by the solid line on the left. The difference in magnitude between these two data sets is obviously enormous, and one might think there could be no connection between a billionaire and the rest of the world. However, a feature of scale-free distributions, is that they appear as a straight line when plotted on log-log scale (on this scale, 104 is 10,000, 106 is a million, 109 is a billion, and 1011 is a hundred billion).

I have interpolated the missing data with a dotted line, but data from individual countries show a similar pattern, and it is reasonable to conclude that the power-law distribution holds remarkably well over this range. As with “extreme events” on the stock market, billionaires are not random freak events or black swans, but part of the same picture as everyone else.

Power-law distributions are ubiquitous in nature, and are the signature of complex systems which have evolved to the edge of chaos – they are neither completely ordered, nor completely chaotic, but are on the boundary between the two.

This points to a major problem in economic theory. Most mainstream ideas, including chestnuts such as the ‘law of supply and demand’ or the ‘efficient market’ only make sense if you assume that there are a large number of essentially identical consumers, investors and firms. Drop those assumptions, and the theories don’t stand up.

However, while power-law distributions cannot easily be accommodated by the tools of mainstream economics, they do come easily in the framework of agent-based models. These are an important tool in life sciences such as ecology and biology, and are increasingly being used to model the economy.

One of the earlier examples of an agent-based model was a kind of computer game called Sugarscape, which involves agents moving over an imaginary grid looking for sugar. The rules for each agent are simple. To stay alive, they need food, so at each time step, they look around them, move towards any sugar they see, and eat it. Like a crop, the sugar grows back, but slowly. The agents have diverse characteristics, so some can see further or move more quickly, and can also reproduce through a process in which a couple’s attributes are randomly mixed up to produce a new agent.

While the rules are simple, the emergent behaviour is quite life-like. The agents swarm around like ants, and soon organise into separate tribes based on the geographical distribution of sugar. The more successful find mates and start large families. Some become exceedingly well-off, and hoard their excess sugar. The wealth distribution is seen to follow a scale-free power law, just as in the real world.

The system can occasionally lapse back to a disorganised state of conflict, as when population pressures result in a sudden shortage of food. The program has been used to study everything from economics to warfare.

Sugarscape has been criticised by feminist economists as being design to reflect male values. As Hazel Henderson points out, “if they had programmed half of their ‘agents’ with the behaviour females so often exhibit… they might have produced different results.” As with any model of society, it reflects our own biases and beliefs. But perhaps one day more realistic versions of such agent-based models will help us understand how wealth is distributed in the economy – and even spread the sugar a little more fairly.