Against the Gods: The Remarkable Story of Risk

by Peter L. Bernstein

Bayes’s theorem focuses on the frequent occasions when we have sound intuitive judgments about the probability of some event and want to understand how to alter those judgments as actual events unfold.

Those games were generally referred to as “hazard,” from al zahr, the Arabic word for dice.

Time is the dominant factor in gambling. Risk and time are opposite sides of the same coin, for if there were no tomorrow there would be no risk. Time transforms risk, and the nature of risk is shaped by the time horizon: the future is the playing field.

Probability theory seems a subject made to order for the Greeks, given their zest for gambling, their skill as mathematicians, their mastery of logic, and their obsession with proof. Yet, though the most civilized of all the ancients, they never ventured into that fascinating world. Their failure to do so is astonishing because the Greeks had the only recorded civilization up to that point untrammeled by a dominating priesthood that claimed a monopoly on the lines of communication with the powers of mystery.

The word calculate stems from calculus, the Latin word for pebble.

intransigence

The journalist William Hoffer has remarked, “The great golden spiral seems to be nature’s way of building quantity without sacrificing quality.”2

Priests were the first astronomers, and from astronomy came mathematics. When people recognized that nicks on stones and sticks no longer sufficed, they began to group numbers into tens or twenties, which were easy to count on fingers and toes.

The Greeks were able to reframe the ultimate questions because theirs was the first civilization in history to be free of the intellectual straitjacket imposed by an all-powerful priesthood.

The centerpiece of the Hindu-Arabic system was the invention of zero—sunya as the Indians called it, and cifr as it became in Arabic.9 The term has come down to us as “cipher,” which means empty and refers to the empty column in the abacus or counting frame.c

askance

movable type

Cardano’s great book on mathematics, Ars Magna (The Great Art), appeared in 1545, at the same time Copernicus was publishing his discoveries of the planetary system and Vesalius was producing his treatise on anatomy.

dissolute

Hacking asserts that Pascal’s line of analysis to answer this question is the beginning of the theory of decision-making. “Decision theory,” as Hacking describes it, “is the theory of deciding what to do when it is uncertain what will happen.”19 Making that decision is the essential first step in any effort to manage risk.

His relationship with his father was so poor that he took as his motto Invito patre sidera verso—”I am among the stars in spite of my father.”5

promulgated

This procedure of revising inferences about old information as new information arrives springs from a philosophical viewpoint that makes Bayes’s contribution strikingly modern: in a dynamic world, there is no single answer under conditions of uncertainty. The mathematician A.F.M. Smith has summed it up well: “Any approach to scientific inference which seeks to legitimise an answer in response to complex uncertainty is, for me, a totalitarian parody of a would-be rational learning process.”24

A famous mathematician and economist of a later age, Francis Ysidro Edgeworth, coined the term “Quetelismus” to describe the growing popularity of discovering normal distributions in places where they did not exist or that failed to meet the conditions that identify genuine normal distributions.20

Galton demonstrated this idea with a gadget he called the Quincunx to the Royal Society around 1874.33 The Quincunx looked a lot like an up-ended pinball machine. It had a narrow neck like an hour-glass, with about twenty pins stuck into the neck area. At the bottom, where the Quincunx was at its widest, was a row of little compartments. When shot were dropped through the neck, they hit the pins at random and tended to distribute themselves among the compartments in classic Gaussian fashion—most of them piled up in the middle, with smaller numbers on either side, and so on in diminishing numbers.

As Frank Knight observed fifteen years before Keynes published The General Theory, “At the bottom of the uncertainty problem in economics is the forward-looking character of the economic process itself.”30 Because the economic environment is constantly changing, all economic data are specific to their own time period. Consequently they provide only a frail basis for generalizations. Real time matters more than time in the abstract, and samples drawn from the past have little relevance. What was 75% probable yesterday has an unknown probability tomorrow. A system that cannot rely on the frequency distribution of past events is peculiarly vulnerable to surprise and is inherently volatile.

Kahneman and Tversky use the expression “failure of invariance” to describe inconsistent (not necessarily incorrect) choices when the same problem appears in different frames.

Two medical researchers, David Redelmeier and Eldar Shafir, reported in the Journal of the American Medical Association on a study designed to reveal how doctors respond as the number of possible options for treatment is increased.17

In a 1992 paper that summarized advances in Prospect Theory, Kahneman and Tversky made the following observation: “Theories of choice are at best approximate and incomplete . . . Choice is a constructive and contingent process. When faced with a complex problem, people . . . use computational shortcuts and editing operations.”

waywardness.

Opposed though he was to mechanical applications of the laws of probability and the quantification of uncertainty, Keynes recognized that this body of thought had profound implications for humanity: The importance of probability can only be derived from the judgment that it is rational to be guided by it in action; and a practical dependence on it can only be justified by a judgment that in action we ought to act to take some account of it. It is for this reason that probability is to us the “guide of life,” since to us, as Locke says, “in the greatest part of our concernment, God has afforded only the Twilight, as I may so say, of Probability, suitable, I presume, to that state of Mediocrity and Probationership He has been pleased to place us in here.”10