Against the Gods: The Remarkable Story of Risk

by Peter L. Bernstein

The revolutionary idea that defines the boundary between modern times and the past is the mastery of risk: the notion that the future is more than a whim of the gods and that men and women are not passive before nature.

The capacity to manage risk, and with it the appetite to take risk and make forward-looking choices, are key elements of the energy that drives the economic system forward.

In 1703, Gottfried von Leibniz commented to the Swiss scientist and mathematician Jacob Bernoulli that “[N]ature has established patterns originating in the return of events, but only for the most part,”1 thereby prompting Bernoulli to invent the Law of Large Numbers and methods of statistical sampling that drive modern activities as varied as opinion polling, wine tasting, stock picking, and the testing of new drugs.b Leibniz’s admonition—”but only for the most part”—was more profound than he may have realized, for he provided the key to why there is such a thing as risk in the first place: without that qualification, everything would be predictable, and in a world where every event is identical to a previous event no change would ever occur.

Whenever we make any decision based on the expectation that matters will return to “normal,” we are employing the notion of regression to the mean.

“[O]ur knowledge of the way things work, in society or in nature, comes trailing clouds of vagueness. Vast ills have followed a belief in certainty.”3

The word “risk” derives from the early Italian risicare, which means “to dare.” In this sense, risk is a choice rather than a fate. The actions we dare to take, which depend on how free we are to make choices, are what the story of risk is all about. And that story helps define what it means to be a human being.

The earliest-known form of gambling was a kind of dice game played with what was known as an astragalus, or knuckle-bone.

Losing streaks and winning streaks occur frequently in games of chance, as they do in real life. Gamblers respond to these events in asymmetric fashion: they appeal to the law of averages to bring losing streaks to a speedy end. And they appeal to that same law of averages to suspend itself so that winning streaks will go on and on.

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.

Time matters most when decisions are irreversible. And yet many irreversible decisions must be made on the basis of incomplete information.

The more uncertain the outcome, the greater may be the value of procrastination. Hamlet had it wrong: he who hesitates is halfway home.

I do not mean to suggest that the Greeks gave no thought to the nature of probability. The ancient Greek word εικος (eikos), which meant plausible or probable, had the same sense as the modern concept of probability: “to be expected with some degree of certainty.” Socrates defines εικος as “likeness to truth.”10

“It is a double doubt,” declares the Talmud. If it is established (method unspecified) that the bride came to the marriage bed no longer a virgin, one side of the double doubt is whether the man responsible was the prospective groom himself—whether the event occurred “under him . . . or not under him.” As to the second side of the doubt, the argument continues: “And if you say that it was under him, there is doubt whether it was by violence or by her free will.” Each side of the double doubt is given a 50–50 chance. With impressive statistical sophistication, the philosophers conclude that there is only one chance in four (1/2 × 1/2) that the woman committed adultery before marriage. Therefore, the husband cannot divorce her on those grounds.

The prospect of getting rich is highly motivating, and few people get rich without taking a gamble.

There is more to that blunt statement than meets the eye. Trade is a mutually beneficial process, a transaction in which both parties perceive themselves as wealthier than they were before. What a radical idea! Up to that point, people who got rich had done so largely by exploitation or by plundering another’s wealth. Although Europeans continued to plunder across the seas, at home the accumulation of wealth was open to the many rather than the few. The newly rich were now the smart, the adventuresome, the innovators—most of them businessmen—instead of just the hereditary princes and their minions.

Without numbers, there are no odds and no probabilities; without odds and probabilities, the only way to deal with risk is to appeal to the gods and the fates. Without numbers, risk is wholly a matter of gut.

The unique quality of the Greek spirit was the insistence on proof. “Why?” mattered more to them than “What?” 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. This same set of attitudes led the Greeks to become the world’s first tourists and colonizers as they made the Mediterranean basin their private preserve.

The algebraic equations we are all familiar with today—equations like a + bx = c—are known as Diophantine equations.

It was al-Khowârizmî who was the first mathematician to establish rules for adding, subtracting, multiplying, and dividing with the new Hindu numerals. In another treatise, Hisâb al-jabr w’ almuqâbalah, or “Science of transposition and cancellation,” he specifies the process for manipulating algebraic equations. The word al-jabr thus gives us our word algebra, the science of equations.

sixteenth-century physician named Girolamo Cardano was just such a person. Cardano’s credentials as a gambling addict alone would justify his appearance in the history of risk, but he demonstrated extraordinary talents in many other areas as well. The surprise is that Cardano is so little known. He is the quintessential Renaissance man.

fact, the German word, wahrscheinlich, captures this sense of the concept well: it translates literally into English as “with the appearance of truth.”

Probability has always carried this double meaning, one looking into the future, the other interpreting the past, one concerned with our opinions, the other concerned with what we actually know. The distinction will appear repeatedly throughout

In reality, the odds are substantially less than 5%, because, unlike craps, horse racing cannot take place in somebody’s living room. Horse races require a track, and the owners of the track and the state that licenses the track all have a priority claim on the betting pool. If you restate the odds on each horse in a race in terms of probabilities—as the 20-to-l shot has a probability of winning of 4.8%—and add up the probabilities, you will find that the total exceeds 100%. The difference between that total and 100% is a measure of the amount that the owners and the state are skimming off the top.

The next step was to tackle the question of how human beings recognize and respond to the probabilities they confront. This, ultimately, is what risk management and decision-making are all about and where the balance between measurement and gut becomes the focal point of the whole story.

The basic mathematical concept behind this geometric algebra had been recognized long before Fermat and Pascal took it up. Omar Khayyam had considered it some 450 years earlier. In 1303, a Chinese mathematician named Chu Shih-chieh, explicitly denying any originality, approached the problem by means of a device that he called the “Precious Mirror of the Four Elements.” Cardano had also mentioned such a device.

“the first thing which we must consider is that the money the players have put into the game no longer belongs to them . . . but they have received in return the right to expect that which luck will bring them, according to the rules upon which they agreed at the outset.”

Making that decision is the essential first step in any effort to manage risk.

God is not, whether you lead your life piously or sinfully is immaterial. But suppose that God is. Then if you bet against the existence of God by refusing to live a life of piety and sacraments you run the risk of eternal damnation; the winner of the bet that God exists has the possibility of salvation. As salvation is clearly preferable to eternal damnation, the correct decision is to act on the basis that God is. “Which way should we incline?” The answer was obvious to Pascal.

“Fear of harm ought to be proportional not merely to the gravity of the harm, but also to the probability of the event.”

the idea that both gravity and probability should influence a decision. We could turn this assertion around and state that a decision should involve the strength of our desire for a particular outcome as well as the degree of our belief about the probability of that outcome.

The word “statistics” is derived from the analysis of quantitative facts about the state. Graunt and Petty may be considered the co-fathers of this important field of study.

Halley’s prediction that the comet would reappear in 1758 electrified the world when the comet arrived right on schedule. Halley’s name is celebrated every 76 years as his comet sweeps across the skies.

What does Canopius’s coffee have to do with Graunt or Halley or with the concept of risk? Simply that a coffee house was the birthplace of Lloyd’s of London, which for more than two centuries was the most famous of all insurance companies.18 Insurance is a business that is totally dependent on the process of sampling, averages, independence of observations, and the notion of normal that motivated Graunt’s research into London’s population and Halley’s into Breslaw’s.

The gambling spirit of that prosperous era fostered rapid innovation in the London insurance industry. Underwriters were willing to write insurance policies against almost any kind of risk, including, according to one history, house-breaking, highway robbery, death by gin-drinking, the death of horses, and “assurance of female chastity”—of which all but the last are still insurable.20 On a more serious basis, the demand for fire insurance had expanded rapidly after the great fire of London in 1666.

The first company to be owned by stockholders was the Insurance Company of North America in Philadelphia, which wrote policies on fire and marine insurance and issued the first life-insurance policies in America—six-term policies on sea captains.d21

“Fear of harm ought to be proportional not merely to the gravity of the harm, but also to the probability of the event.”

Both authors build their arguments on the proposition that any decision relating to risk involves two distinct and yet inseparable elements: the objective facts and a subjective view about the desirability of what is to be gained, or lost, by the decision. Both objective measurement and subjective degrees of belief are essential; neither is sufficient by itself.

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

A sense of balance and respect for reason were hallmarks of the Enlightenment. It was in this setting that Bernoulli transformed the mysticism of the Fort-Royal Logic into a logical argument addressed to rational decision-makers.

Bernoulli finds this hypothesis flawed as a description of how people in real life go about making decisions, because it focuses only on the facts; it ignores the consequences of a probable outcome for a person who has to make a decision when the future is uncertain. Price—and probability—are not enough in determining what something is worth. Although the facts are the same for everyone, “the utility . . . is dependent on the particular circumstances of the person making the estimate . . . . There is no reason to assume that . . . the risks anticipated by each [individual] must be deemed equal in value.” To each his own.

And that’s a good thing. If everyone valued every risk in precisely the same way, many risky opportunities would be passed up. Venturesome people place high utility on the small probability of huge gains and low utility on the larger probability of loss. Others place little utility on the probability of gain because their paramount goal is to preserve their capital. Where one sees sunshine, the other sees a thunderstorm. Without the venturesome, the world would turn a lot more slowly.

Think of what life would be like if everyone were phobic about

lightning, flying in airplanes, or investing in start-up companies. We are indeed fortunate that human beings di...

he suggests a systematic approach for determining how much each individual desires more over less: the desire is inversely proportionate to the quantity of goods possessed.

On many occasions the facts are not the same for everyone. Different people have different information; each of us tends to color the information we have in our own fashion. Even the most rational among us will often disagree about what the facts mean.

If the satisfaction to be derived from each successive increase in wealth is smaller than the satisfaction derived from the previous increase in wealth, then the disutility caused by a loss will always exceed the positive utility provided by a gain of equal size. That was my client’s message to me.

In only rare cases does life replicate games of chance, for which we can determine the probability of an outcome before an event even occurs—a priori, as Jacob Bernoulli puts it. In most instances, we have to estimate probabilities from what happened after the fact—a posteriori.

Games of chance reduce everything to a hard number, but in real life we use such measures as “a little,” “a lot,” or “not too much, please” much more often than we use a precise quantitative measure.

Rather, the law states that increasing the number of throws will correspondingly increase the probability that the ratio of heads thrown to total throws will vary from 50% by less than some stated amount, no matter how small. The word “vary” is what matters.

“Probability,” he declares, “is degree of certainty and differs from absolute certainty as the part differs from the whole.”