# Adapting to the entirely unpredictable: black swans, fat tails, aberrant events, and hubristic models

Since the spring of 2007, economic theorist **Peter Hammond** has been working on a Marie Curie research project called "Adapting to the Entirely Unpredictable." Here, he comments on some aspects of a well-known book apparently on the same topic – Nassim Nicholas Taleb’s *The Black Swan: The Impact of the Highly Improbable*, published in 2007.

In 82 AD, Juvenal wrote of "rara avis in terris nigroque simillima cygno" (a rare bird upon earth, and exceedingly like a black swan), but that was imaginative irony. In the year 1697, Willem de Vlamingh was the first European to record seeing a real live black swan in its native Australian habitat.

Later, John Stuart Mill wrote: "No amount of observations of white swans can allow the inference that all swans are white, but the observation of a single black swan is sufficient to refute that conclusion." It became a classical example in elementary philosophy.

Taleb’s book provides many vivid examples of events, often related to finance or economics, which he sees as meeting his characterisation of a "black swan" event as an "outlier" with "an extreme impact" for which "human nature makes us concoct explanations after the event."

The book was written before the recent crisis in global financial markets. Nevertheless, it does discuss several earlier crises, such as the stock market crash of October 1987, which are often plausibly blamed on faulty statistical models.

Truly aberrant "black swan" events are those with no probabilities are attached because our models do not even contemplate their possibility

Indeed, at an early stage of his book, Taleb defines a "special case of 'gray' swans," which are rare but expected. More precisely, they have probability distributions described by "Mandelbrotian randomness," a particular class of "fat-tailed" probability distribution following a power law.

These distributions put so much weight on outliers, or extreme values of a random variable *v* that, for large enough *k*, the expectation of the *k*th power of *v*, otherwise known as the *k*th moment of the distribution, becomes infinite. This is in stark contrast to the normal or Gaussian distribution, for which the tail of the distribution is so "thin" that all moments exist.

Yet this is typically not the issue with the random value of an asset, especially a derivative security. For these, there is a positive probability of losing everything. This kind of extreme risk cannot be captured by a Gaussian distribution or by any "smooth" alternative such as a power law.

But there is little really new here, since statisticians and financial economists, along with decision and game theorists, have been coming to terms with probability distributions that do not correspond to a smooth density function.

Much more challenging than these gray swans, however, are the true black swans, which effectively break our existing scientific models. The eponymous example is when Europeans first became aware of the (black) swan species now called *cygnus atratus*, since that broke any of their previous biological models of the genus *cygnus*.

Taleb does recognise that such events could occur, but regards them as "totally intractable," scientifically speaking. Nevertheless, biologists have formulated statistical models intended to forecast probabilistically the likely number of new species that one might expect to find in a poorly explored habitat. And economists have developed many models of economic growth with technical progress, which may be approximately treated as the accumulation of many small but favourable surprises.

More generally, any practical model, especially in the social sciences, must have bounded scope and so must ignore some real possibilities that could occur and have a noticeable impact. Recent examples include several bank failures and the first UK bank run in over 100 years. So could important new scientific discoveries relating to climate change or its mitigation.

These could be described as "aberrant" events which, by definition, lie outside the current model and the occurrence of which effectively breaks the model. Indeed, aberrant events should be distinguished from events within the model, which, like Taleb’s gray swans, are recognised but given extremely low or even zero probability.

In his classic book *The Foundations of Statistics*, Leonard Savage discusses "small worlds" and contrasts the proverb "cross your bridges as you come to them" with the almost contradictory "look before you leap."

The first of these recommends a model that temporarily leaves out future bridges, so encountering a river that cannot be forded becomes an aberrant event. What I like to call "hubristic" models may well do this, and so recommend taking short cuts whose lack of viability becomes clear only as a river bank comes into view. A more cautious route would be along well-marked paths that lead to a useable bridge whenever a significant river is encountered.

Less "hubristic" modelling could help forestall economic crises or deal better with climate change

As the statistician George Box wrote: "Essentially, all models are wrong, but some are useful." Like engineers, it might be wiser to allow some safety margins rather than lurching between successive small-world and insufficiently imaginative hubristic models that never even consider the possibility of an aberrant event.

So while all useful statistical decision models are no doubt incompletely specified, it would be wise to allow for the possibility that they are sure to need serious re-specification at some time, possibly in the near future.

Meanwhile, despite its title, Taleb’s book is mostly about how statistical models, especially in finance, should pay more attention to low-probability gray swans. It would be much more interesting – though undoubtedly challenging – to discuss truly aberrant black swan events, to which no probabilities are attached because the model we use does not even contemplate their possibility.

As for whether less hubristic modelling could help forestall economic crises or deal better with climate change, it seems indisputable that we should at least try. But that is a topic for later discussion.

## Further details

The full title of Peter Hammond’s Marie Curie research project is "Adapting to the Entirely Unpredictable, and Other Aspects of Dynamic Behaviour: Beyond the von Neumann Standard Paradigm in Games and Economics." The title alludes to John von Neumann’s pioneering 1928 paper, which offers a "game in extensive form" as a complete mathematical description of many single or multi-person decision problems.

In theory, computers can play chess perfectly; in practice, computers can currently guarantee perfect play only when no more than six pieces remain on the board. Similar limitations apply to all difficult decision problems, including how to model economic and financial systems. So any decision model should be flexible enough to allow graceful adaptation to potential changes that any practical model must otherwise ignore.

## The author

Peter Hammond FBA is Marie Curie Professor in the Department of Economics at the University of Warwick.