Highlights of a Career Grounded in Major Disasters

BOOK REVIEW: Six Sources of Collapse: A Mathematician’s Perspective on How Things Can Fall Apart in the Blink of an Eye. By Charles R. Hadlock, Mathematical  Association of America, Washington, DC, 2012, 221 pages, \$50.00.

Charles Hadlock is a mathematician who spent much of his career at the consulting firm Arthur D. Little. His assignments included post-event analyses of several front-page disasters, including those at Bhopal, Three Mile Island, and Prince William Sound. The expertise developed in such projects brought the firm,  originally headquartered in Boston and now a multinational, an increasing flow of commissions from corporations—many of them among the Fortune 500—seeking to identify potential disasters lurking alongside their projected paths of expansion. The teams assigned to conduct such studies operated at the highest corporate levels, often reporting directly to the firms’ boards of directors, rather than to management. In his book and in a related public lecture titled “Sustainability or Collapse? An Exploration of Key Dynamics That May Determine Our Future” that he gave at the MAA Carriage House in May 2013, Hadlock described memorable moments in his career, and a few of the methods he found useful.

The book is organized around two tables, one from the first chapter, the other from the eighth and final chapter. The former lists and briefly describes nine categories of entities vulnerable to collapse; the latter names and (with equal brevity) describes the titular “six sources of collapse.” The more technical chapters between present tales of collapse brought on by one or more of the six sources; each chapter is devoted to a particular source of collapse and—because Hadlock intends the book to be read by non-mathematicians—to the mathematics needed for the analysis thereof. Each chapter $$I \in \{2,\ldots,7\}$$ fills (at least partially) row $$I-1$$ of his imaginary matrix with stories explaining, for various $$J \in \{1,\ldots,9\}$$, how a source of type $$i-1$$ can contribute to the collapse of an entity of type $$J$$. No attempt is made to fill in the entire matrix, or even to convince the reader that it could be done.

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One chapter examines collapses resulting from apparently unlikely events, such as the June 2011 power outage in Montana caused by a deer carcass draped across a power line some thirty feet above ground level. How, Hadlock wonders, is one to compute the odds against that? Because he is convinced that most people—horseracing handicappers and insurance writers naturally excepted—are inept estimators and interpreters of probabilities, the chapter begins with a primer on statistics stressing extreme-value statistics. In it he identifies an extreme-value index (EVI) $$\gamma$$ that, when positive, indicates a complementary cumulative distribution function (CCDF =1 – CDF) asymptotic to $$x^{-r}$$ as $$x \rightarrow \infty$$. Thus, when $$\gamma > 0$$, it is customary to say that the associated probability density function has heavy tails, meaning that it imputes far greater likelihood to large deviations from the median—there might not be a mean—than would any garden variety distribution, such as the normal, logistic, or exponential. Examples include the Cauchy distribution $$(\gamma=1)$$, the Fréchet distribution $$(\gamma=1)$$, the Pareto distribution $$(\gamma=1/r)$$, and the Student’s $$t$$ distribution $$(\gamma=1/n)$$.

A team from Little that included Hadlock was called on to estimate the probability of an admittedly unlikely event as part of an evaluation of a proposal for burying radioactive wastes in the numerous salt domes that underlie much of the Gulf Coast region, both on shore and off. The likelihood that groundwater might leach the wastes away from the domes into more sensitive areas seemed minimal—the salt was still there millions of years after deposition, whereas it would surely be gone if the domes had not remained dry throughout the intervening millennia. Yet the Little team worried that drilling for petroleum or other minerals might accidentally puncture a storage site, letting water in and (eventually) radioactive waste out. Advocates of the plan were adamant that the probability of such an accident was negligible, as the sites would be clearly marked on every official map and drillers, in any case, are well regulated. Their claims were corroborated, moreover, by operators of several salt mines located within such domes. In short, the probability of such an accident was widely agreed to be effectively zero! Nada!

Informed opinion did not change until November 20, 1980, when a Texaco rig drilled a hole, 14 inches in diameter, directly into an operating salt mine at Jefferson Island, Louisiana. The entire content of the shallow lake in which the rig was standing—including the rig itself and several nearby barges—disappeared forever through the rapidly expanding hole into the mine below. Apparently, the location of the salt mine had been miscalculated—a fact that made it into the Little team’s final report. Amazingly, nobody was killed in the accident, as it played out slowly enough for everyone on the rig, and in several nearby fishing boats, to scramble to safety. In this case, the miscalculation of a small probability was costly in money and machines, but not in lives. It might easily have been otherwise.

Miscalculated probabilities also played a role in the demise of the hedge fund Long Term Capital Management, which almost precipitated a worldwide financial crisis in 1998, and in the crash of 2008 following the collapse of the US housing industry. Hadlock explains that LTCM, like many in the finance industry, was performing risk estimation in what is known as a Black–Scholes environment, wherein all random processes are presumed to be Wiener processes $$w(t), o \leq t < \infty$$, driven by Gaussian white noise. An important assumption concerning such processes is that distinct increments $$\Delta w(t) = w(t+h) - w(t)$$ and $$\Delta w(s) = w(s+h') - w(s)$$ are stochastically independent as long as the intervals $$[t,t+h]$$ and $$[s,s+h']$$ do not overlap. When that assumption is violated, the processes of interest cease to be Wiener processes and risk estimates obtained by assuming otherwise lose whatever validity they may once have had.

What happened to LTCM in 1998, and to many financial institutions in 2007–08, was that successive increments in the values of certain assets suddenly became preponderantly negative, so that accounts expected to fluctuate in value around roughly stationary means began to drift steadily southward. And because those accounts tended to be highly leveraged—meaning that they had been purchased mainly with borrowed money—a host of nominal owners found themselves in bankruptcy, owing far more than their total net worth. In each case, the outcome had been made to seem, by inappropriate risk assessment procedures, no more than a vanishingly remote possibility.

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The remaining five “source” chapters follow a similar pattern: a motivational paragraph or two, followed by a thumbnail sketch of some useful branch of mathematics—statistics, the Monte Carlo method, game theory, differential equations, catastrophe theory, network analysis—followed by a series of war stories explaining how a particular source could cause or at least contribute to the collapse of some important entity, be it a bridge, a communications network, or a global financial system.

It would be interesting to know how Hadlock’s analysis applies to the most worrisome collapse of all, the one of which even chronic doom sayers hesitate to speak, namely the collapse of modern civilization predicted by the 1972 Club of Rome report, “The Limits to Growth.” That oft vilified document identified three possible paths of global development, corresponding to three general classes of policy alternative: effective political action, misguided political action, and continued political inaction. Forty years later, the predictions corresponding to the third alternative are right on track as to population growth, food production, and several other fundamental quantities. They imply, among other things, that global food production will peak around the year 2030, and that (amid untold pain and suffering) global population will do likewise a few years later. Could clear thinking of the sort Hadlock describes prevent such an outcome?

Hadlock’s writing is invariably clear and concise, yet informal and appealing, as befits a mathematical memoirist. The book is by no means a textbook, although it could be regarded as a compendium of modules suitable for presentation to undergraduates at various stages of development by instructors seeking to enhance their lectures with a smattering of genuine applications. For that reason alone, every teaching mathematician should own a copy of this book!

James Case writes from Baltimore, Maryland.