Against the Gods: The Remarkable Story of Risk (21 page)

Daniel Bernoulli received an appointment at St. Petersburg in the
same year as Nicolaus III and remained there until 1733, when he
returned to his hometown of Basel as Professor of Physics and
Philosophy. He was among the first of many outstanding scholars
whom Peter the Great would invite to Russia in the hope of establishing his new capital as a center of intellectual activity. According to
Galton, Daniel was "physician, botanist, and anatomist, writer on
hydrodynamics; very precocious."9 He was also a powerful mathematician and statistician, with a special interest in probability.

Bernoulli was very much a man of his times. The eighteenth century came to embrace rationality in reaction to the passion of the endless religious wars of the past century. As the bloody conflict finally
wound down, order and appreciation of classical forms replaced the fervor of the Counter-Reformation and the emotional character of the
baroque style in art. 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.

Daniel Bernoulli's St. Petersburg paper begins with a paragraph
that sets forth the thesis that he aims to attack:

Ever since mathematicians first began to study the measurement of
risk, there has been general agreement on the following proposition:
Expected values are computed by multiplying each possible gain by the num
ber of ways in which it can occur, and then dividing the sum of these products by the total number of cases.
*'0

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.

The concept of utility is experienced intuitively. It conveys the sense of usefulness, desirability, or satisfaction. The notion that arouses Bernoulli's impatience with mathematicians-"expected value"-is more technical. As Bernoulli points out, expected value equals the sum of the values of each of a number of outcomes multiplied by the probability of each outcome relative to all the other possibilities. On occasion, mathematicians still use the term "mathematical expectation" for expected value.

A coin has two sides, heads and tails, with a 50% chance of landing with one side or the other showing-a coin cannot come up showing both heads and tails at the same time. What is the expected value of a coin toss? We multiply 50% by one for heads and do the same for tails, take the sum-100%-and divide by two. The expected value of betting on a coin toss is 50%. You can expect either heads or tails, with equal likelihood.

What is the expected value of rolling two dice? If we add up the 11 numbers that mightcome up-2+3+4+5+6+7+8+9+10+ 11 + 12-the total works out to 77. The expected value of rolling two dice is 77/11, or exactly 7.

Yet these 11 numbers do not have an equal probability of coming up. As Cardano demonstrated, some outcomes are more likely than others when there are 36 different combinations that produce the 11 outcomes ranging from 2 to 12; two can be produced only by doubleone, but four can be produced in three ways, by 3 + 1, by 1 + 3, and
by 2 + 2. Cardano's useful table (page 52) lists a number of combinations in which each of the 11 outcomes can occur:

The expected value, or the mathematical expectation, of rolling two
dice is exactly 7, confirming our calculation of 77/11. Now we can see
why a roll of 7 plays such a critical role in the game of craps.

Bernoulli recognizes that such calculations are fine for games of
chance but insists that everyday life is quite a different matter. Even
when the probabilities are known (an oversimplification that later mathematicians would reject), rational decision-makers will try to maximize
expected utility-usefulness or satisfaction-rather than expected value.
Expected utility is calculated by the same method as that used to calculate expected value but with utility serving as the weighting factor.11

For example, Antoine Arnauld, the reputed author of the PortRoyal Logic, accused people frightened by thunderstorms of overestimating the small probability of being struck by lightning. He was wrong.
It was he who was ignoring something. The facts are the same for
everyone, and even people who are terrified at the first rumble of thun der are fully aware that it is highly unlikely that lightning will strike precisely where they are standing. Bernoulli saw the situation more clearly:
people with a phobia about being struck by lightning place such a heavy
weight on the consequences of that outcome that they tremble even
though they know that the odds on being hit are tiny.

Gut rules the measurement. Ask passengers in an airplane during
turbulent flying conditions whether each of them has an equal degree
of anxiety. Most people know full well that flying in an airplane is far
safer than driving in an automobile, but some passengers will keep the
flight attendants busy while others will snooze happily regardless of
the weather.

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 differ in their appetite for risk.

Once Bernoulli has established his basic thesis that people ascribe different values to risk, he introduces a pivotal idea: "[The] utility resulting
from any small increase in wealth will be inversely proportionate to the quantity of
goods previously possessed." Then he observes, "Considering the nature
of man, it seems to me that the foregoing hypothesis is apt to be valid
for many people to whom this sort of comparison can be applied."

The hypothesis that utility is inversely related to the quantity of
goods previously possessed is one of the great intellectual leaps in the
history of ideas. In less than one full printed page, Bernoulli converts
the process of calculating probabilities into a procedure for introducing
subjective considerations into decisions that have uncertain outcomes.

The brilliance of Bernoulli's formulation lies in his recognition that,
while the role of facts is to provide a single answer to expected value
(the facts are the same for everyone), the subjective process will pro duce as many answers as there are human beings involved. But he goes even further than that: 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.

For the first time in history Bernoulli is applying measurement to something that cannot be counted. He has acted as go-between in the wedding of intuition and measurement. Cardano, Pascal, and Fermat provided a method for figuring the risks in each throw of the dice, but Bernoulli introduces us to the risk-taker-the player who chooses how much to bet or whether to bet at all. While probability theory sets up the choices, Bernoulli defines the motivations of the person who does the choosing. This is an entirely new area of study and body of theory. Bernoulli laid the intellectual groundwork for much of what was to follow, not just in economics, but in theories about how people make decisions and choices in every aspect of life.

Bernoulli offers in his paper a number of interesting applications to illustrate his theory. The most tantalizing, and the most famous, of them has come to be known as the Petersburg Paradox, which was originally suggested to him by his "most honorable cousin the celebrated Nicolaus Bernoulli"-the dilatory editor of The Art of Conjecture.

Nicolaus proposes a game to be played between Peter and Paul, in which Peter tosses a coin and continues to toss it until it comes up heads. Peter will pay Paul one ducat if heads comes up on the first toss, two ducats if heads comes up on the second toss, four ducats on the third, and so on. With each additional throw the number of ducats Peter must pay Paul is doubled.*
How much should someone pay Paul-who stands to rake in a sizable sum of money-for the privilege of taking his place in this game?

The paradox arises because, according to Bernoulli, "The accepted method of calculation [expected value] does, indeed, value Paul's prospects at infinity [but] no one would be willing to purchase [those prospects] at a moderately high price .... [A]ny fairly reasonable man would sell his chance, with great pleasure, for twenty ducats.`

Bernoulli undertakes an extended mathematical analysis of the problem, based on his assumption that increases in wealth are inversely related to initial wealth. According to that assumption, the prize Paul might win on the two-hundredth throw would have only an infinitesimal amount of additional utility over what he would receive on the one-hundredth throw; even by the 51st throw, the number of ducats won would already have exceeded 1,000,000,000,000,000. (Measured in dollars, the total national debt of the U.S. government today is only four followed by twelve zeroes.)

Whether it be in ducats or dollars, the evaluation of Paul's expectation has long attracted the attention of leading scholars in mathematics, philosophy, and economics. An English history of mathematics by Isaac Todhunter, published in 1865, makes numerous references to the Petersburg Paradox and discusses some of the solutions that various mathematicians had proposed during the intervening years.12 Meanwhile, Bernoulli's paper remained in its original Latin until a German translation appeared in 1896. Even more sophisticated, complex mathematical treatments of the Paradox appeared after John Maynard Keynes made a brief reference to it in his Treatise on Probability, published in 1921. But it was not until 1954-216 years after its original publication-that the paper by Bernoulli finally appeared in an English translation.