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  *Amos Tversky, who plays an important role in Chapters 16 and 17, died unexpectedly just as this book was about to go into print.

  *The scientist who developed the Saturn 5 rocket that launched the first Apollo mission to the moon put it this way: "You want a valve that doesn't leak and you try everything possible to develop one. But the real world provides you with a leaky valve. You have to determine how much leaking you can tolerate." (Obituary of Arthur Rudolph, in The New York Times, January 3, 1996.)

  *Chapter 7 describes Jacob Bernoulli's achievements in detail. The Law of Large Numbers says in essence that the difference between the observed value of a sample and its true value will diminish as the number of observations in the sample increases.

  *Peter Kinder has pointed out to me a great historical irony in all this. The Vikings and other Norsemen who laid waste to Roman civilization and destroyed the repositories of learning in the ninth century reappear in history as the Normans who brought back to the West the achievements of Arabic learning in the twelfth century.

  *One of those odd quirks that numbers can produce reveals that you can derive 0.618 if you take the square root of 5, which is 2.24, subtract 1, and then divide by 2; this result is the algebraic proof of Fibonacci's sequence of numbers.

  In technical terms, the formula for the Fibonacci ratio is as follows: the ratio of the smaller part to the larger part equals the ratio of the larger part to the whole.

  *The Arabic term survives even in Russian, where it appears as tsifra, which is the word for number.

  *Readers who are not interested in the technicalities of this discussion can skip to page 53 without any loss of continuity.

  *The full text of this correspondence, translated into English, appears in David, 1962, Appendix 4.

  *Mathematicians will note that what Pascal has really provided here is the binomial expansion, or the coefficients of
each successive multiplication of (a + b) by itself. For example, the first row is (a + b)° = 1, while the fourth row is (a + b)3 = 1a3 + 3a2b + 3abz + 1 b'.

  *At this point, Pascal anticipates Daniel Bernoulli's epochal breakthrough in decision analysis in 1738, which we explore in detail in Chapter 6.

  The Latin title for this book was Ars Cogitandi. See Hacking, 1975, pp. 12 and 24.

  *The information on the quantity of bread a penny could buy provided a standard for estimating the cost of living. In our own times, a package of goods and services is used as the standard.

  *The word "venery" descends from the Middle-French word vener, to hunt (from which also comes the word "venison") and from Venus (from which comes the word "venereal)." A venerable word indeed!

  *Lloyd's, in short, is the ancestor of the huge Bloomberg business news network of our own time.

  *The trust business in Boston was founded by Nathaniel Bowditch in the 1810s to serve the same market.

  tThis principle applied to life insurance as well. The debts of a soldier who died in battle were forgiven and did not have to be repaid.

  In the United States it survived into the twentieth century. Here it was known as "industrial insurance" and usually covered only funeral expenses. My father-in-law had a little book in which he kept a record of the weekly premiums he paid into such a policy.

  `As usual, the essay was published in Latin. The Latin title of the publication in which it appeared was Commentarii Academiae Scientiarum Imperialis Petropolitanae, Tomus V.

  *Newman is not easy to characterize, although his The World of Mathematics was a major source for this book. He was a student of philosophy and mathematics who became a highly successful lawyer and public servant. A one-time senior member of the editorial board of Scientific American, he was an avid collector of scientific documents of great historical importance. He died in 1966.

  *Daniel's uncle Jacob, who will play a major role in the next chapter, once wrote that "the value of our expectation always signifies something in the middle between the best we can hope for and the worst we can fear." (Hacking, 1975, p. 144.)

  *With the assistance of Richard Sylla and Leora Mapper, the best information I have been able to obtain about the value of ducats in the early 18th century is that one ducat could have purchased the equivalent of about $40 in today's money. Baumol and Baumol, Appendix, provides an approximate confirmation of this estimate. See also McKuster, 1978, and Warren and Pearson, 1993.

  *Bernoulli's solution to the paradox has been criticized because he fails to consider a game in which the prize would rise at a faster rate than the rate Nicolaus had specified. Nevertheless, unless there is a point where the player has zero interest in any additional wealth, the paradox will ultimately come into play no matter what the rate is.

  *A theoretical exploration into this question appears in Durand, 1959, which anticipated the events described in the paragraphs immediately following.

  *This is an oversimplification. The utility of any absolute loss depends on the wealth of the loser. Here the implicit assumption is that the two players have equal wealth.

  *He did have sufficient poetry in his soul to request that the beautiful Fibonacci spiral be engraved on his tombstone, claiming that the way it could grow without changing its form was "a symbol of fortitude and constancy in adversity: or even of the resurrection of our flesh." He went on to ask that it be inscribed with the epitaph "Eadem mutata resurgo" [However changed it is always the same]. See David, 1962, p. 139.

  *At a later point in the correspondence with Jacob, Leibniz observed, "It is certain that someone who tried to use modern observations from London and Paris to judge mortality rates of the Fathers before the flood would enormously deviate from the truth." (Hacking, 1975, p. 164.)

  *The franc/dollar exchange rate has been remarkably steady over the years at around 5:1. Hence, 2,000 francs was the equivalent of $400 dollars of 1807 purchasing power. A dollar in 1807 bought about twelve times as much as today's dollar.

  *The standard deviation was the device that de Moivre discovered for measuring the dispersion of observations around their mean. Approximately two-thirds of the observations (68.26%) will fall within a range of plus or minus one standard deviation around the mean; 95.46% will fall within two standard deviations around the mean.

  *Richard Price's experience reminds us that the data themselves must be of good quality. Othe.-wise, GIGO: garbage in, garbage out.

  Readers skilled in statistics will protest that I should have used the lognormal analysis in the discussion that follows. For readers not so skilled, the presentation in this form is much more intelligible and the loss of accuracy struck me as too modest to justify further complexity.

  *Galton would surely have classified Cardano as eminent, but what would he have thought of Cardano's disastrous progeny? Gauss, also an eminent man, scored better. He produced five surviving children, of whom one was a distinguished engineer and two of whom emigrated to the United States to run successful businesses (but also to escape from their father's domineering influence); one of them was also a brilliant linguist, a gambler, and a skillful mathematician as well.

  *1 am not related to Sanford Bernstein, by the way.

  *Opposite tendencies are apparent in the historical record of interest rates, which reflect "aversion" to the mean. A trend once in place has a higher probability of continuing than reversing. Over two-year periods, the variance of the yield on 90-day Treasury bills is 2.2 times annual data; over eight-year periods, the variance is nearly 32 times the annual data; for longer-term interest rates, the pattern is similar but muted.

  *They called depressions "panics" in those days; "depression" was a euphemism coined for the occasion. Later, "recession" became the accepted euphemism. One can only speculate on how deep a recession would have to be before the experts would decide to call it a "depression."

  *Florence Nightingale was described by Edward Cook, one of her biographers, as a "passionate statistician." A compulsive collector of data in the tradition of Galton, she was also an enthusiastic admirer of the work of Quetelet, which inspired her pioneering work in medical and other social statistics. See Kendall and Plackett, 1977, pp. 310-327.

  *It is conceivable, however, that the opposite might occur. Risk often serves as a stimulant. Without risk, a society might turn passive in the face of the future.

  *Knight rarely uses such arcane words. "Apodeictic" means incontestable, necessarily true because logically certain.

  *The feeling appears to have been mutual. Morgenstern took a dim view of Samuelson's knowledge of mathematics. Complaining that "[von Neumann] says [Samuelson] has murky ideas about stability," he predicted that "even in thirty years he won't absorb game theory!" See Leonard, 1994, p. 494n.

  tRopke, also a Christian, was far more emphatic than Morgenstern had been about his reasons for leaving Hitler's Germany.

  *The word has a Latin root, from portare, to carry, and foglio, leaf or sheet. Portfolio has thus come to mean a collection of paper assets.

  *The standard deviation of the Mexican market alone leapt from 8% to 10% a month (four times the monthly volatility of the S&P 500) to better than 15% a month during the first half of 1995.

  *For an extended and informative discussion of these issues, see TheJournal of Investing, Fall 1994.

  *Jack Benny had a routine on a Sunday radio show in which he remained silent when confronted by a mugger demanding "Your money or your life." After a long pause, the mugger cried, "Come on!" "I'm thinking it over," Benny predictably responded.

  *Kahneman has described his introduction to experimentation when one of his professors told the story of a child being offered the choice between a small lollipop today or a larger lollipop tomorrow. The child's response to this simple question correlated with critical aspects of the child's life, such as family income, one or two parents present, and degree of trust.

  *An excellent review of this matter appears in "The Triumph of Indexin
g," a booklet published by the Vanguard Group of mutual funds in May 1995. This controversial subject receives more detailed treatment later in this chapter.

  *In a speech to the National Association of Realtors in May 1995, none other than the Chairman of the Federal Reserve Board, Alan Greenspan, confirmed the piggy bank metaphor: "It is hard to overestimate the importance of house price trends for consumer psyches and behavior.... Consumers view their home equity as a cushion or security blanket against the possibility of future hard times." As a consequence of the growth in borrowing in the form of home equity loans, home equity has shrunk from 73% of home value in 1983 to around 55% at this writing, provoking what the July 10, 1995, issue of Business Week describes as "a major deterrent to buoyant spending."

  tWe exclude financial corporations from these calculations to avoid double-counting. Banks and other financial organizations re-lend to the nonfinancial sector most of the money they borrow.

  BAs usual, Shakespeare got there first. In Act 1, Scene 1, lines 168-171 of Timon of Athens, the jeweler says to Timon, "My Lord, tis rated/As those which sell would give; but you well know/Things of like value differing in their owners/Are prized by their masters."

  *In Chapter 7 of Thaler, 1987, in fact, Thaler declared that von Neumann-Morgenstern utility had failed in psychological testing. See p. 139.

  This bald assertion should be interpreted broadly. Cross-cultural problems and concerns for the health of the home country add to the value of domestic securities and detract from the value of foreign securities.

  *The bond even offered protection against the possibility that one pound sterling might subsequently buy more or less than 25 francs. The French went off gold in 1870, at which time one pound sterling could buy substantially more than 25 francs.

  *This is an oversimplification to make the basic point. Most individual home mortgages are packaged with other mortgages and sold off in the open market to a wide variety of investors. In effect, the bankers have traded off the risks of prepayment to a market more willing to bear that risk; these mortgage-backed securities are complex, volatile, and much too risky for amateur investors to play around with.