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July 29 - August 13, 2018
Occupational guilds in both Greece and Rome maintained cooperatives whose members paid money into a pool that would take care of a family if the head of the household met with premature death.
In Italy, for example, farmers set up agricultural cooperatives to insure one another against bad weather; farmers in areas with a good growing season would agree to compensate those whose weather had been less favorable. The Monte dei Paschi, which became one of the largest banks in Italy, was established in Siena in 1473 to serve as an intermediary for such arrangements.
Activity in the insurance business gained momentum around 1600. The term “policy,” which was already in general use by then, comes from the Italian “polizza,” which meant a promise or an undertaking.
In 1738, the Papers of the Imperial Academy of Sciences in St. Petersburg carried an essay with this central theme: “the value of an item must not be based on its price, but rather on the utility that it yields.”
Nicolaus lived a long life, from 1623 to 1708, and had three sons, Jacob, Nicolaus (known as Nicolaus I), and Johann. We shall meet Jacob again shortly, as the discoverer of the Law of Large Numbers in his book Ars Conjectandi {The Art of Conjecture). Jacob was both a great teacher who attracted students from all over Europe and an acclaimed genius in mathematics, engineering, and astronomy. The Victorian statistician Francis Galton describes him as having “a bilious and melancholic temperament. . . sure but slow.”4 His relationship with his father was so poor that he took as his motto Invito
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When Daniel won a prize from the French Academy of Sciences for his work on planetary orbits, his father, who coveted the prize for himself, threw him out of the house.
Daniel Bernoulli had a brother five years older than he, also named Nicolaus; by convention, this Nicolaus is known as Nicolaus III, his grandfather being numberless, his uncle being Nicolaus I, and his elder first cousin being Nicolaus II.
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.
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.
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.
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.”
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 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.”
These investment managers defined the risk in the Nifty-Fifty, not as the risk of overpaying, but as the risk of not owning them:
VCs
In December 1972, Polaroid was selling for 96 times its 1972 earnings, McDonald’s was selling for 80 times, and IFF was selling for 73 times; the Standard & Poor’s Index of 500 stocks was selling at an average of 19 times.
For truly patient individuals who felt most comfortable owning familiar, high-quality companies, most of whose products they encountered in their daily round of shopping, an investment in the Nifty-Fifty would have provided ample utility.
Bernoulli introduced another novel idea that economists today consider a driving force in economic growth—human capital.
Utility provided the underpinnings for the Law of Supply and Demand, a striking innovation of Victorian economists that marked the jumping-off point for understanding how markets behave and how buyers and sellers reach agreement on price.
In 1738, when Bernoulli’s paper appeared, Alexander Pope was at the height of his career, studding his poems with classical allusions, warning that “A little learning is a dangerous thing,” and proclaiming that “The proper study of mankind is man.”
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.
In a mathematical sense a zero-sum game is a loser’s game when it is valued in terms of utility.
Bernoulli uses his example to warn gamblers that they will suffer a loss of utility even in a fair game. This depressing result, he points out, is: Nature’s admonition to avoid the dice altogether . . . . [E]veryone who bets any part of his fortune, however small, on a mathematically fair game of chance acts irrationally . . . . [T]he imprudence of a gambler will be the greater the larger part of his fortune which he exposes to a game of chance.
The first person to consider the linkages between probability and the quality of information was another and older Bernoulli, Daniel’s uncle Jacob, who lived from 1654 to 1705.
Jacob Bernoulli had first put the question of how to develop probabilities from sample data in 1703. In a letter to his friend Leibniz, he commented that he found it strange that we know the odds of throwing a seven instead of an eight with a pair of dice, but we do not know the probability that a man of twenty will outlive a man of sixty. Might we not, he asks, find the answer to this question by examining a large number of pairs of men of each age?
The past, or whatever data we choose to analyze, is only a fragment of reality.
All the law tells us is that the average of a large number of throws will be more likely than the average of a small number of throws to differ from the true average by less than some stated amount.
Jacob concludes that “we can determine the number of instances a posteriori with almost as great accuracy as if they were know to us a priori.”6 His calculations indicate that 25,550 drawings from the jar would suffice to show, with a chance exceeding 1000/1001, that the result would be within 2% of the true ratio of 3:2. That’s moral certainty for you.
Jacob had started with the probability that the error between an observed value and the true value would fall within some specified bound; he then went on to calculate the number of observations needed to raise the probability to that amount. Nicolaus tried to turn his uncle’s version of probability around.
De Mensura Sortis is probably the first work that explicitly defines risk as chance of loss:
By de moivre
The shape of de Moivre’s curve enabled him to calculate a statistical measure of its dispersion around the mean.
Sd
Bayes was a Nonconformist; he rejected most of the ceremonial rituals that the Church of England had retained from the Catholic Church after their separation in the time of Henry VIII.
Most serious, he appears to have underestimated life expectancies, with the result that the life-insurance premiums were much higher than they needed to be. The Equitable Society flourished on this error; the British government, using the same tables to determine annuity payments to its pensioners, lost heavily.20
During the last 27 years of his life, which ended at the age of 78 in 1855, Carl Friedrich Gauss slept only once away from his home in Göttingen.
Like many mathematicians before and after him, Gauss also was a childhood genius—a fact that displeased his father as much as it seems to have pleased his mother.
His father was an uncouth laborer who despised the boy’s intellectual precocity and made life as difficult as possible for him.
Gauss = Prahlad
In 1807, as the French army was approaching Göttingen, Napoleon ordered his troops to spare the city because “the greatest mathematician of all times is living there.”
A wealthy friend offered to help out, but Gauss rebuffed him. Before Gauss could say no a second time, the fine was paid for him by a distinguished French mathematician, Marquis Pierre Simon de Laplace (1749–1827).
Eric Temple Bell, one of Gauss’s biographers, believes that mathematics might have been fifty years further along if Gauss had been more forthcoming; “Things buried for years or decades in [his] diary would have made half a dozen great reputations had they been published promptly.”
He brushed off Fermat’s Last Theorem, which had stood as a fascinating challenge to mathematicians for over a hundred years, as “An isolated proposition with very little interest for me, because I could easily lay down a multitude of such propositions, which one could neither prove nor dispose of.”9
As Gauss analyzed the distribution of these estimates, he observed that they varied widely, but, as the estimates increased in number, they seemed to cluster around a central point. That central point was the mean—statistical
The more measurements Gauss took, the clearer the picture became and the more it resembled the bell curve that de Moivre had come up with 83 years earlier.
Francis Galton, whom we will meet in the next chapter, rhapsodized over the normal distribution: [T]he “Law Of Frequency Of Error”. . . reigns with serenity and in complete self-effacement amidst the wildest confusion. The huger the mob . . . the more perfect is its sway. It is the supreme law of Unreason. Whenever a large sample of chaotic elements are taken in hand . . . an unsuspected and most beautiful form of regularity proves to have been latent all along.13
As Galton suggested, two conditions are necessary for observations to be distributed normally, or symmetrically, around their average. First, there must be as large a number of observations as possible. Second, the observations must be independent, like rolls of the dice. Order is impossible to find unless disorder is there first.
People can make serious mistakes by sampling data that are not independent.
normal distribution is most unlikely, although not impossible, when the observations are dependent upon one another—that is, when the probability of one event is determined by a preceding event.
He classified the degree of attractiveness of girls he passed on the street, pricking a hole in a left-pocket card when a girl was comely and pricking a right-pocket card when she was plain. In his “Beauty Map” of Britain, London girls scored highest; Aberdeen girls scored lowest.
Galton
Unable to speak Hottentot and uncertain how to undertake this urgent piece of research, he still managed to achieve his goal: Of a sudden my eye fell upon my sextant; the bright thought struck me, and I took a series of observations upon her figure in every direction. . . . [T]his being done, I boldly pulled out my measuring tape, and measured the distance from where I was to the place where she stood, and having thus obtained both base and angles, I worked out the results by trigonometry and logarithms.
King Nagoro found it hard to believe that there were places in the world inhabited entirely by people with white skins. To him, Galton and his friends were rare migratory animals or some kind of anomaly. One of Galton’s companions had to undress repeatedly before the king to prove that he was white all over.
In 1796, when he was 65 years old, Erasmus published a two-volume work called Zoonomia, or the Theory of Generations. Although the book went through three editions in seven years, it failed to impress the scientific community because it was rich in theory but poor in facts.
Galton was 22 when his father died, leaving a substantial estate to his seven surviving children. Deciding that he could now do anything he liked, he soon chose to give up formal studies.
