Casanova's Lottery: The History of a Revolutionary Game of Chance 9780226820781

Table of contents :
Contents
Introduction
One  Casanova
Two  The Genoese Lottery
Three  The Establishment of the Loterie in 1757
Four  Problems and Adjustments in the Early Drawings
Five  Antoine Blanquet and the Great Expansion of 177
Six  The Introduction of Bonus Numbers: Les Primes Gratuites
Seven  The Spread of the Loterie in Europe
Eight  Data Security: The Design of the Tickets
Nine  The Loterie and the Revolution
Ten  Was the Loterie Fair?
Eleven  Dreams and Astrology: The Bettors and the Loterie
Twelve  The Number 45 and the Maturity of Chances
Thirteen  How Much Did They Bet, and Where?
Fourteen  Muskets, Fine- Tuned Risk, and Voltaire
Fifteen  The Loterie in Textbooks and Manuals
Sixteen  The Suppression of the Loterie in 1836
Conclusion
Acknowledgments
Appendix One  Probability
Appendix Two  Laplace’s Lottery Theorem
References
Index

Citation preview

C a s a n o va’s L o t t e r y

Casanova’s Lottery The History of a R e v o l u t i o n a r y G a m e of C h a n c e S T E P H E N M. S T I G L E R

The University of Chicago Press C hic ag o and L ond on

The University of Chicago Press, Chicago 60637 The University of Chicago Press, Ltd., London © 2022 by The University of Chicago All rights reserved. No part of this book may be used or reproduced in any manner whatsoever without written permission, except in the case of brief quotations in critical articles and reviews. For more information, contact the University of Chicago Press, 1427 E. 60th St., Chicago, IL 60637. Published 2022 Printed in the United States of America 31 30 29 28 27 26 25 24 23 22

1 2 3 4 5

ISBN-13: 978-0-226-82077-4 (cloth) ISBN-13: 978-0-226-82079-8 (paper) ISBN-13: 978-0-226-82078-1 (e-book) DOI: https://doi.org/10.7208/chicago/9780226820781.001.0001 Library of Congress Cataloging-in-Publication Data Names: Stigler, Stephen M., author. Title: Casanova’s lottery : the history of a revolutionary game of chance / Stephen M. Stigler. Description: Chicago ; London : The University of Chicago Press, 2022. | Includes bibliographical references and index. Identifiers: LCCN 2022003384 | ISBN 9780226820774 (cloth) | ISBN 9780226820798 (paperback) | ISBN 9780226820781 (ebook) Subjects: LCSH: Lotteries—France—History—18th century. | Lotteries— France—History—19th century. | Risk—Social aspects—France. Classification: LCC HG6195 .S75 2022 | DDC 795.3/80944—dc23/eng/20220201 LC record available at https://lccn.loc.gov/2022003384 ♾ This paper meets the requirements of ANSI/NISO Z39.48-1992 (Permanence of Paper).

To Virginia, my winning ticket in the lottery of life

Contents

Introduction One



Two

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

1

9

Casanova

19

The Genoese Lottery Three

23



The Establishment of the Loterie in 1757 Four

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Five

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28

Problems and Adjustments in the Early Drawings 42

Antoine Blanquet and the Great Expansion of 1776 Six

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50

The Introduction of Bonus Numbers: Les Primes Gratuites Seven



55

The Spread of the Loterie in Europe

Eight

61

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Data Security: The Design of the Tickets Nine

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67

The Loterie and the Revolution Ten



77

Was the Loterie Fair? Eleven

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Twelve

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90

Dreams and Astrology: The Bettors and the Loterie 110

The Number 45 and the Maturity of Chances Thirteen

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Fourteen

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132

How Much Did They Bet, and Where? 143

Muskets, Fine-Tuned Risk, and Voltaire Fifteen



Sixteen

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Conclusion

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156

The Loterie in Textbooks and Manuals 172

The Suppression of the Loterie in 1836 197

Acknowledgments  211 Appendix 1. Probability  213 Appendix 2. Laplace’s Lottery Theorem  216 References  229 Index  239

Introduction

T h i s i s t h e h i s t or y of a n unusual lottery, one that flourished in France from 1758 until 1836 with a three-year hiatus during the French Revolution. It certainly was not the first state lottery: the modern history of state lotteries goes back at least to the sixteenth century. Nor was it the first based upon the same plan for carrying out the drawing, a plan that was an ancestor of modern lotto and was introduced in the seventeenth century on a small scale in a few Italian cities. But this lottery was unique in one intriguing aspect: In adopting its framework, the French state took on risk in a way no other state lottery has done, before or since. At each drawing, the state was at risk of losing a large amount; what is more, that risk was precisely calculable, generally well understood, and yet taken on by the state with little more than a mathematical theory to protect it. At its peak, it provided up to 4 percent of the French national income. France by the 1750s was familiar with financial innovation. The sixty-four-year reign of the “Sun King,” Louis XIV of France, ended with his death in 1715. His was the longest reign in European history, and he left a treasury that was essentially bankrupt, principally due to debts accumulated in a long series of wars. The French economy, predominantly agricultural, was doing well, but the nearly panicked Finance Ministry could not find much solace in that, and this led to

2  Introduction

financial experimentation and one of the most improbable careers in history. John Law was born in Scotland in 1671. By 1694 he was living in London, where he killed a man in a duel with swords over a woman’s affection, and was sentenced to death. But he managed first to get out on appeal and later to escape and flee to the continent, where he traveled between Amsterdam and Paris, and by 1715 caught the ear of the Duke of Orléans. The duke became the regent of France at the death of Louis XIV and was central to the Finance Ministry, and he was receptive to a plan Law had been pushing: a managed currency based on instruments of credit rather than stocks of precious metals. In effect, Law invented the basis for most nations’ currency of the present day, but in his case it was based upon volatile investments involving North American fur trade and land contracts, the beginning of what came to be called the “Mississippi Bubble.” At first it went well: from 1715 to 1720 he founded the Banque générale, which then became the National Bank. By 1720 he was the French controller-general of finances (essentially the prime minister), and he became the wealthiest man in the world, at one time or another owning twenty-one estates in France. But the volatility, and a few unwise actions in coping with it, undermined public confidence.1 The swings in value of the Mississippi shares were mild in comparison to those that would come with the later currency “assignats” introduced by the Revolutionary government in 1790. Unlike the assignats, Law’s new currency never became valueless, but confidence in it was shaken to the point where the currency was eventually returned to a base in gold and silver.2 The episode brought ruin to many people in the already stressed French and British economies. Law was pushed out of his appointment, and left France. After a few more adventures, he died in Venice in 1729. The French treasury recovered, but the episode left conflicting lessons. It opened eyes to the possibility of financial experimentation, and it alerted the ministers to the high risks and potential costs involved with such experimentation. As would be expected in human affairs, these lessons were heeded, but not for long. Within a decade of John Law’s departure, a peculiar scheme was devised with the aim of eliminating some of the accumulated debt, but with a defect

Introduction  3

in conception that allowed the French philosopher Voltaire and the French explorer-mathematician La Condamine to make millions of francs with essentially no risk (see chapter 14). This and many other such projects involved lotteries; most of these were a variation on a plan that may have originated in the Netherlands in the 1400s. Those state lotteries were set up as raffles. A large number of tickets were to be sold, and a fraction of the proceeds— often at least a half— would be set aside for the state, with the remainder allocated as prizes. A drawing would be held, tickets would be picked in an apparently random way, and the prizes given in different amounts to the winners. That scheme was attractive to the state because once the tickets were sold, the state was guaranteed to make a profit. But that certainty was illusory. The ticket buyers were not given any guarantee, and the sale could be a hard sell indeed. It might take years to sell the tickets, and the sale might never be completed, leading the state to retrospectively change the rules and lower the payout: a practice that not only generated rebellious ill will but also made the launching of a second lottery a nearly impossible task. In 1567, Queen Elizabeth of England, through her ministers, announced a lottery to help support her navy. A total of 400,000 tickets were to be sold for ten shillings each, gaining a net profit of £100,000 for the Crown.3 But after two years, less than a tenth of the tickets had been sold. The lottery was held with prizes proportionately reduced, and with the gain to the Crown eaten up by expenses. On paper, there was no way such lotteries could lose money; in practice they were cumbersome in administration, and could take months or years to complete. And if the sales failed, so did the lottery and the reputation of the state. The new French plan instituted in 1758 was based upon an earlier lottery believed to have originated in Genoa (see chapter 2). The “Genoese” lottery had significant advantages. It could be offered frequently: at one time in France, there were fifteen drawings a month. It could scale up geographically: at one time there were more than a thousand offices in France alone. It allowed the clientele to shape their own bets, to take on as much or as little risk as they wished. But it also presented new and difficult challenges. The raffles lotteries had the comfort of knowing that the number

4  Introduction

of winners was strictly limited, and that the policing of awards was relatively easy. Indeed, they did not need to worry greatly if the prize went to the right person; they paid out the same amount whether the award was accurately given or not. With the French Loterie (I will call it by its French name, to distinguish it from other species), the number of winners was unknown in advance since many people could hold bets on the same numbers, perhaps yielding many winners of the largest prizes. The opportunity for fraud was immense, either through counterfeited tickets or through crooked agents selling tickets after the drawing was announced. The administrative burdens were correspondingly immense: all of this oversight was needed everywhere, in a system of a thousand offices at one time, spread over an empire. The French at that time tended to “farm out”—or “outsource,” in modern terminology— the administration of other tax-generating efforts, but not so with the Loterie. Apparently the risks were such that closer control was deemed necessary. The response to these challenges was the development of novel schemes for what we now call data security, some of which bear uncanny resemblance to schemes employed in the Internet age (see chapter 8). There were also innovations in bookkeeping that went far beyond the simple “double entry” that had been introduced in the 1400s, where two entries— assets and liabilities— gave a simple check in sums; the Loterie officials needed full accuracy in every entry of a bet, with links to the bettor’s name. Admittedly, they also had, and used, some deterrents we do not employ today, such as the dungeon. The challenges that came with the scale of the enterprise were many. The sales offices required agents with some knowledge of mathematics beyond simple arithmetic— skills not taught universally in that era, and which thus required job training. Communication at a distance was slow, and some of the distances were quite large. There was competition from foreign lotteries eager to sell in France as well. While, as in the present day, a large amount of the public comment was directed against the Loterie, the only governmental actions taken were instead aimed to prohibit foreign entry, and were less than successful. As the Loterie administration learned what worked and what

Introduction  5

F ig ur e 0 . 1 . Menut’s 1834 book which initiated this investigation: the Almanach Romain sur la loterie de France.

was unnecessary, and as methods of fraud advanced as well, there was a need for minor changes in data security methods. Understanding how the lottery worked and how it was used by the betting public presents a difficult problem for a historian. Existing histories are plentiful but superficial; they are not concerned with details of operation, and those details are not easy to obtain. Yet the details were crucial for success. My investigation was begun in 1994 when I purchased a small book by mail, sight unseen, from a French bookseller for $350. It was dirty, worn, and falling apart at the seams (figure 0.1).4 The book, the Almanach Romain sur la loterie de France, was of a species of literature referred to as ephemera. It was published in 1834 and intended to be consumed rather than read or archived. Its mul-

6  Introduction

tiple roles in this story will be revealed as the story progresses, but my immediate reaction on seeing it was skepticism. It was a lottery almanac with information for gamblers. But what soon caught my eye was the rather large amount of data: not just occasional facts, of which there were many, but also a list of all the winning numbers for the entire history of the Loterie since 1758, as well as data on prizes actually awarded. It included data through 1834, some written by hand by an early owner, and thus covered almost the entire history of the Loterie. Several questions immediately came to mind. Were the drawings fair (see chapter 10)? Were there any trends in behavior (see chapters 13 and 17)? Converting the data to digital form for modern analysis was not simple then or now— the dirty pages and old fonts were not accessible by 1994 optical character recognition programs. I started to type the data in, very slowly. And then one day a bright University of Chicago undergraduate student named Teresa Ging stopped by my office and asked if I had any suggestions for her senior honors paper. She wrote a fine paper, and with her help I got a digital archive. That old book and others in the same spirit served as unusual sources, opening a look at how the Loterie was carried out, who bought tickets, and what size bets they placed. In fact, as I later realized, the Almanach gave the results of a precisely randomized survey of the French betting public in the early 1800s, more than a century before randomized surveys were invented. (chapter 11) This realization helped answer a number of these questions, as we shall see. Another source was an extensive collection of French laws and decrees I accumulated, some in bound form, many as originally printed for distribution. These were documents that shed light by their extremely detailed description of procedures, and by their choice of which topics to detail and which to omit, all of this information conspicuously absent in many histories.5 As I accumulated these and many other books and pamphlets on lotteries from 1619 to the present, a broad picture of the Loterie and its position in society began to emerge, complementary to the narrow details of operation that held their own fascination. This was helped by my acquisition of three very large scrapbooks of lottery ephemera from England and the conti-

Introduction  7

nent, including tickets, advertising bills, notices, contracts, and even manuscript material from the 1600s to 1900. The broader picture of the Loterie derived from these varied sources helps us understand its place in the history of risk. Embarking on an endeavor with uncertain hopes for gain and the risk of loss has always been an unavoidable part of life, for the earliest huntergatherer as for the modern pedestrian crossing a busy street. Perhaps as an evolutionary consequence of the ubiquity of risk, an actual taste for risk can also be found in all societies of all times. This can be clearly seen in the presence of gambling throughout recorded history. Dice appear in archeological digs associated with cultures at least as far back as that of the Akkadians in 2300 BCE: yes, cubical dice numbered one to six on the sides, a pattern pointing to the appearance of gambling being as old as the appearance of writing. By about 600 BCE, dice began to be widely made with opposite sides adding to seven, just like modern casino dice and surely for the same reason: to allow easy detection of false dice.6 For a century prior to the founding of the Loterie, there were great changes in how people viewed risk or chance. The philosopher Ian Hacking identified the period from the 1660s to the 1710s as the “Emergence of Probability,” when earlier vague concepts based on tendencies became a precise mathematical science— with laws of large numbers and the beginnings of a full calculus of chances— through the works of Pascal and Fermat in the 1650s, and those of Jacob Bernoulli, Abraham de Moivre, and Pierre Rémond de Montmort from 1700 to 1720.7 This work was brought to a much wider readership first by Christian Huygens, whose deceptively simple tract of 1657 was reprinted often over a sixty-year period, and later by Richard Seymour, Thomas Simpson, Edmond Hoyle, and many other authors of books for gamblers. At roughly the same time, whole industries offering annuities and insurance were growing and profiting from the realization that in some cases, as Ian Hacking later termed it, chance could be “tamed.”8 That term was a bit overly optimistic. What was unpredictable in individual cases could be reliably predicted in aggregates of individuals; but, as investors repeatedly learn,

8  Introduction

chance is never really tamed. The law of large numbers was a mathematical theorem; it was also an empirically demonstrated fact. But as with civil laws, violations of the law do occur at inconvenient times. By the 1750s, the soil was fertile for more French financial experimentation, this time with the Loterie.9 Of course, this raises other questions. Why did other nations not try the same experiment? And once the French had a huge success, one that furnished large amounts for the national treasury, why did they close the Loterie down in 1836? The answer to the first question is that several countries did try the same experiment, at least in much smaller-scaled and more tentative enterprises, as we shall see. These included many German cities, where adoption soon ran into the challenges the French had overcome only with difficulty. This was true even in the case of Frederick the Great in Prussia, who uncharacteristically lacked the spine to persevere. Still others planted different crops— different lotteries thought to be safer— in that same soil. As for the second question—Why was the Loterie “suppressed” in 1836?— that is the subject of the final chapter of this book. A note on French currency: Over the centuries discussed here, there were many changes in the use of names for monetary units — and, to a lesser degree, in the units themselves. Before 1641, and after 1795 until the euro, the basic unit was popularly known as the franc. In 1641, the value of one franc was determined by one livre tournois— literally one Tours pound of silver, the standard in the city of Tours— and the common term in use for a unit became “livre.” Before 1795, the year in which the metric system was introduced, the livre or franc was divided into twenty sol, or twenty sous. After 1795, it was divided into one hundred centimes. To try to lessen confusion, the term franc is here used consistently as the unit, even for the period when it was not in common use. The smaller units (sol, sou, or centime) are used as is appropriate to the time discussed.



One



Casanova

G i ac omo C as anova was b or n i n V e n ic e in 1725. His parents were an actress and an actor. In his early years he showed all the traits that would mark his later life: an inquisitive intelligence and broad interests, without the capacity to pursue any one for very long. By the time he was twenty-one he had studied philosophy, mathematics, medicine, law, religion, and music, and had explored careers in religion, the military, music, and gambling. He was a magnet for scandal, whether through his talent for seduction, his interests in mystical religion, or his gambling losses; and he had an uncanny ability to recover, often after having fled to another city or country. Early on, he met and impressed powerful people: cardinals and the pope in Italy, philosophers and royalty abroad. He became acquainted with Voltaire, Goethe, and Mozart. In the 1790s, having worn out his welcome in most of Europe, he found refuge as a guest in the Castle of Dux (now Duchkov, in the Czech Republic) where he wrote (in French) his extensive memoirs, the chief basis of his posthumous fame. They were not published until the 1820s, decades after his death in 1797, and for many years they were only known for his accounts of romantic adventures. One edition had the added note, “These memoirs were not written for children; they may outrage readers also offended by Chaucer, La Fontaine, Rabelais and the Old Testament.” More recently, they have been recognized as a unique social history of his

10  C h a p t e r O n e

time, and have been praised by literary critics as diverse as Edmund Wilson and Stefan Zweig as one the great literary achievements of the eighteenth century.1 The manuscript now resides in Paris, and a definitive edition has reached its third volume, with more to come.2 Even early in his life, Casanova had made innumerable enemies, most consequentially the Venetian Inquisition. In 1753 in Venice he had helped save the life of the powerful Senator Bragadin, thus acquiring an important patron. But two years later, even his patron was insufficient protection when his enemies managed to arrange for the Inquisition to catch him with forbidden books on the mystical sect of cabalism, and with evidence of engagement in Freemasonry. On 12 September 1755 he was sentenced to five years imprisonment in the Piombi (Leads), a high-security prison on the top floor of the doge’s palace in the center of Venice. Its name derived from the slabs of lead that covered the roof, and the lead efficiently passed cold in winter and heat in summer to the cells just below the roof, thus adding to the prisoners’ misery. Casanova’s own misery was only slightly alleviated when after five months his patron was able to provide him access to books, some warm bedding, and better food. But life with little hope of release grew heavy on him, and he plotted to escape. On a rare chaperoned walk he managed to secretly pick up a spike of iron, and after weeks of sharpening, he started to cut through the wood floor beneath his bed, planning to escape into the chamber below when it was empty some night. To his great disappointment, he was moved to a different cell three days before his intended escape. Soon he came to a different plan. He smuggled the sharpened iron to a priest in an adjacent cell, who used it to access the crawl space above the ceiling and then cut through to Casanova’s cell, eventually allowing both of them to escape to the slippery, pitched lead roof. After a harrowing traverse in a heavy fog, they used a rope Casanova had made of bedsheets to lower themselves to an unguarded part of the palace. They bluffed their way out, and then escaped by gondola on the canal. After a year and a half in prison, Casanova was free. There were many other frightening details, making it a riveting story that he told for many years to

Casanova  11

entranced audiences, and finally published in 1787. He escaped from the city-state of Venice and made his way to Paris, but the story of his unparalleled escape traveled even faster. Casanova arrived in Paris on 5 January 1757. The Seine, he tells us, was frozen to the depth of one foot, and due to his hasty departure from Italy, he had only a few material resources from his patron. But he had lived in Paris in 1751 and 1752, and knew many people who could serve as useful contacts to help him build a new reputation. Since news of his dramatic escape from the Piombi had preceded his arrival, there was no shortage of well-placed people eager to hear the details from him in person— a request he granted frequently and at great length. Casanova was soon able to establish himself in a respectable manner, and he gained audiences with a few powerful government officials, including the Count de Boulogne, who was the controller-general, a position charged with directing the king’s finance ministry. At all critical junctures in his life, luck— often good luck— had presented Casanova with opportunities that, with skill and audacity, he managed to exploit for at least temporary success. When he called upon the Count de Boulogne, he found he was in the company of Joseph de Pâris-Duverney, who since 1751 had served as the first intendent (director) of the École militaire. The count was under the impression that Casanova had some grand plans for making money that the count might support to their mutual benefit— plans that Casanova had not yet thought of, though he had led others to believe they existed. The count introduced Duverney, saying that Duverney “needs twenty millions for his Military School. The thing is to find them without expense to the State or burdening the royal treasury.” Casanova tells us in his memoirs that with the overconfidence that had served him so well on earlier occasions, he replied, “I realize that everything has become more difficult, however, I have in mind a plan that would yield to the king the interest on a hundred millions.”3 Du v er n e y : And how much would this cost the king? C as anova : Only the expense of collecting it.

12  C h a p t e r O n e

D: Then it is the nation that would provide the revenue? C : Yes, but voluntarily. D: I know what you have in mind. C : I would be surprised, Monsieur, for I have not communicated my idea to anyone.

Casanova had been bluffing, but he was curious to learn what plan Duverney thought he had, and he eagerly accepted an invitation to dinner the next day at Duverney’s Chateau Plaisance in NeuillyPlaisance, about twelve kilometers east of Paris, where others involved in the project would also be present. The discussion Casanova was joining was an important one in the history of the École militaire. The school had been established by royal edict of Louis XV in January 1751. It was a grand plan, with the political backing of the king’s council and with the ardent support of his mistress and trusted confidant, the Madame de Pompadour, who also had played a role in initiating the plan. But financial support was needed as well. Presumably the Crown could bear some immediate costs, but the costs here would be continual and probably growing. Initially the ministers had cast their eyes on a peculiar “sin tax”: a long-established government levy on the sale of playing cards. Such a tax had been established, also by royal edict, in February of 1581. It applied to the sale of all playing cards and tarot cards, with a set amount per card to go to the Crown. On 13 January 1751, nearly the 170th anniversary of the introduction of the tax, the original edict was revised to bring the money to the support of the École royale militaire. While this served well for a while, it is clear from the frequency of amendments to the law that it was difficult to administer. Apparently, smuggling and tax evasion were well developed by that time, and by the mid-1750s the government had tried to deal with this by restricting the supply points to sixty-four locations in France permitted to manufacture cards, with the further requirement that each of the country’s twenty-nine districts could obtain cards only from a specified set of two to four designated points of manufacture.

Casanova  13

Cards in Paris were required to come from manufacture within Paris, or from Versailles or Beauvais. Clearly this was an attempt to combat tax evasion, but one can imagine what a nightmare enforcement of this tax law must have been. When Casanova arrived at Duverney’s Chateau Plaisance at the appointed time that January day, he found Duverney and six or seven treasury officials in discussion, gathered around a large fireplace for warmth. Casanova was announced and introduced as a friend of the minister of foreign affairs and of the controller-general. For an hour and a half, Casanova listened in silence to a wide-ranging technical discussion that he mostly did not understand and found boring, until the group finally adjourned for dinner. After an excellent hourand-a-half-long meal, during which Casanova continued his role as a silent listener, Duverney invited him and two of the treasury officials to his study, where they were joined by another man of about fifty (Casanova was then thirty-one).The newcomer, also Italian, was named Giovanni Calzabigi. At this point Duverney handed Casanova a notebook, announcing, “There is your plan.” The title page of the notebook began, “Lottery of ninety lots, of which the winning lots, drawn each month at random, can only fall on five numbers.” Casanova instantly recognized what the subject was: namely, a form of lottery he had known in Venice, which had been introduced in Genoa in the seventeenth century. He knew its rules; he knew the chances of outcomes and forms of possible bets; and he lost no time in stating that, yes, indeed this was his plan. Casanova reported the ensuing conversation: Duv er ne y : Monsieur, you have been anticipated; the project is Monsieur de Calzabigi’s. C as anova : I am delighted to find that Monsieur and I think alike. But if you have not adopted it, dare I ask the reason? D : Several plausible arguments have been presented against the plan, and the answers to them have been weak. C : I know of only one argument in all of nature which would silence me. That would be if the king would not allow his subjects to gamble.

14  C h a p t e r O n e

D: That argument is irrelevant; the king will permit his subjects to gamble. But will they gamble? C : I am astonished that there is doubt; if the people are sure that they will be paid if they win. D: Then let us suppose that they will gamble if they are certain that there is a fund sufficient to pay. How is that money to be raised? C : The royal treasury. A council decree. It suffices that the king is thought able to pay a hundred millions. D: A hundred millions? C : Yes, Monsieur. One must dazzle. D: Then you think the king could possibly lose that sum? C : I suppose it is possible over a long period, but only after taking in a hundred and fifty millions. Knowing the implication of this political calculation, you need only start from there. D: Monsieur, I am not the only one involved. Will you agree that at the first drawing the king can lose an immense sum? C : Between that possibility and reality there is an infinite gap, but I admit it. If the king loses a large sum at the first drawing, the success of the lottery is assured. It is a misfortune to be desired. Moral forces can be calculated like probabilities. You know that all insurance companies are rich. I will prove to you before all the mathematicians in Europe that, if God is neutral, it is impossible that the king will not take in one franc profit for every five collected by this lottery. That is the secret. Will you admit reason must surrender to mathematical demonstration? D: I admit that. But why cannot the rules on betting balance the wagers so as to guarantee the king an unfailing profit? C: No such system can do that; it could at most give a temporary restriction for one number or two or three that were unusually overbet and (should that number occur) give the backer of the lottery a large loss. It could declare those numbers closed. But it could only ensure a profit by putting off the drawing until all the chances were equally bet, and then there would be no lottery, for the drawing might not take place for ten years. And then the lottery would become no more than a fraud. What saves it from that dishonorable name is the scheduled drawing every month; then the public is sure that the backer can lose. D: Will you be kind enough to speak to the full council?

Casanova  15

C : With pleasure. D: And answer all objections? C : All. D: Will you bring me your plan? C : I will not, Monsieur, until its principle is accepted and I am certain that it will be adopted and that I will receive the benefits I will ask for. D: But your plan can only be the same as this one. C : I doubt it. In my plan I will indicate roughly how much the king will profit each year, and I will demonstrate it. D: We could then sell it to a company which would pay the king a fixed sum. C : I beg your pardon. The lottery can only prosper under an essential precondition. I would not wish to be involved if it were not managed by the Crown. A separate management would operate for their own bureaucratic profit, increasing the operation to that end and diminishing the eagerness of customers to participate. Of that I am sure. The lottery, if I am to be involved, must be royal or nothing. D: Monsieur Calzabigi thinks the same as you.

In fact, Casanova realized immediately that a partnership with Calzabigi would be the best way to proceed, and he was happy when Calzabigi approached him three days later with exactly that proposal. Calzabigi had been trying to sell the project to the council for two years, and in a single meeting Casanova had all but sealed the deal. There was one new surprise: the plan was indeed due to a Calzabigi, but not the gentleman at the chateau, Giovanni Calzabigi. The author of the plan was his elder brother, Ranieri Calzabigi, who was responsible for all the mathematics and most of the organization. Ranieri suffered from a leprosy-like skin disease that left a fine mind intact, but a scab-covered body and a need to constantly scratch himself that made it impossible for him to appear in public or meet with governmental officials. When Casanova met with Ranieri a few days later he was greatly impressed, describing him as “very intelligent— a great arithmetician, thoroughly acquainted with theoretical and practical finance, a wit, and a poet. His brother was also very clever, but inferior to him in every way.” Later, after Ranieri left Paris in 1760

16  C h a p t e r O n e

for Vienna, he would become famous as an opera librettist in collaboration with Christoph Gluck. Casanova needed the Calzabigis, who clearly had the managerial skill that that Casanova lacked and had no interest in. When at the chateau Casanova was asked, “Have you capable people for the lottery offices?” he had answered, “All I need is intelligent machines, of which there must be plenty in France”— hardly a reassuring answer to the question of recruiting a skilled labor force in France at that time, even if it was then common to refer to human calculators as “computers.” The elder Calzabigi was well qualified to handle this properly. And the Calzabigis needed Casanova. They had failed in two years of effort. Casanova had the contacts, and the intellectual skills to convince those contacts. He had met every other objection that had been raised by the council, overriding their deep risk adversity with uncommon confidence. Duverney also saw the virtue in the partnership; he wanted Casanova as the salesman to the council, and he trusted Ranieri Calzabigi as a steady hand for the directorship. Duverney invited Casanova to Versailles to meet with Madame de Pompadour and with the minister of foreign affairs. Casanova had met the madame when he was previously in Paris and by chance had had a seat at the opera immediately below her private box. She welcomed him in Versailles by recalling the exchange of banter they had. She expressed interest in his escape from the Piombi, and offered a sympathetic comment without asking for the story. On this occasion she was accompanied by the Prince of Soubise, a high official in the military, and she offered her support in the project. The meeting with the minister of foreign affairs went well also, and he too issued a general statement of support. The next day, Casanova was summoned to a meeting of the council at the École militaire. Just prior to that meeting he met again with Giovanni Calzabigi, who brought with him his brother’s detailed calculations. The council had also invited Jean d’Alembert, who was a member of the Académie des sciences as well as an accomplished mathematical scientist and, later, permanent secretary of the Académie Française. It was before this group that Casanova pre-

Casanova  17

sented the plan, sticking closely to that of the Calzabigis in all details. Casanova spoke for a half hour, followed by the director of finances for the school, who summarized the issues. During the question period that followed, Casanova dealt effectively with the objections, all of which he had expected to hear. He described his own summary in his memoirs in these words: I said that if the art of calculation in general was properly the art of finding the expression of a single relation arising from the combination of several relations, the same definition applied to moral calculation, which was as certain as mathematical calculation. I convinced them that without that certainty the world would never have had insurance companies, which, all of them rich and flourishing, laugh at Fortune and at the weak minds which fear her. I ended by saying that there was not a man at once learned and honorable in the world who could offer to be the head of this lottery on the understanding that it would win at every drawing, and that if such a man should appear with the temerity to give them that assurance they should turn him out, because either he would not keep his promise or, if he kept it, he would be a scoundrel.

The council, including the distinguished visitor d’Alembert, endorsed the plan. D’Alembert would be famously skeptical in his later writings on probability.4 Why was he quiet on this occasion? Was he calmed by Casanova’s use of exactly the definition of “art of calculation” found in the article “Calcul” in volume 2 of the Encyclopedie, which had appeared in January 1752, edited by Diderot and d’Alembert? Or was this wording a later creation, drawn from the Encyclopedie in the 1790s, when his memoirs were written? D’Alembert himself wrote the article on “Loterie,” but that volume would be published only in 1765, and while it would show knowledge of the odds in lotteries, it would make no specific reference to the Genoese or École militaire lotteries. In two other places where d’Alembert mentioned lotteries, he showed no interest in this Loterie, addressing only simple raffles under that name.5 In any event, apparently there was full governmental agreement about the projected lottery within a week, but

18  C h a p t e r O n e

it took several months before the full decree was publicly issued, with a detailed description of how the lottery was to be organized and which bets would be paid at what odds. The date of the decree was 15 October 1757, and it stated that the drawings could commence in November. But after further delays, it was only in April 1758 that the first drawing occurred.



Two



The Genoese Lottery

Lot t er i e s h ad f l o ur i s he d i n E ur ope from the early sixteenth century, but with one exception they were all of a sort we might now call a raffle, in which tickets are sold and then a drawing distributes a part of the income from the sale or other prizes to some of the ticket holders. For such lotteries the operator was guaranteed a profit, as long as all the tickets were sold. The sale and drawing could take months or even years. The exception was a lottery that began in Genoa by the 1600s, and which by 1757 had attracted a small number of imitators, mostly in Italy. The date of its origin is uncertain. Some accounts say that it was begun by a man named Benedetto Gentile in 1620, but there seems to be no hard evidence of this origin.1 Most accounts say that it was inspired by the practice in some Italian citystates in that era, including Genoa, of choosing members for a governing council by lot from a list of eligible noblemen, some accounts even stating that five men would be selected from a list of ninety each year. This inspired betting on the election, a practice Felloni documents in Genoa back to at least 1374; and, as the story goes, at some point an entrepreneur realized that one could hold the lottery without having the election.2 The Harvard Library holds one piece of evidence that supports this story. It is a broadside proposal dated 28 April 1662 by a London merchant named Thomas Harnage and others for a lottery that is similar

20  C h a p t e r T w o

in spirit to that of Casanova and the Calzibigis. It differed in that five numbers are drawn from one hundred, not ninety. The payoff scheme provided rewards that were smaller than those in the later version; we will discuss that later. The broadside was titled “Il Guoco di Genoua; or, The most Delightful and Profitable Chance of Fortune.” As part of the sales pitch, there was testimony by five gentlemen stating, We whose Names are here under written, do Certifie upon our knowledge, having lived in Italy, That in the City of Genoua there is a Play called Il Guoco delli Senatori, which is an Extraction made every six Moneths before the Senate of five Gentlemens Names from out of one Hundred or thereabouts, which five so drawn out arrives to the Honour of being Senators. And this Play is there Exercised by the Permission and Licence of said Senate.3

At another place in the broadside, the authors state of the gambling version, “That in Genoua, they most commonly make their Extraction from out of 120, whereas we make it but out of 100.” This both supports the lottery’s political origin and testifies to its early practice in the 1600s in Genoa, but with rules slightly different from those commonly accepted after 1700, where the draw was from ninety numbers with a comparable set of payouts. There is no indication that the scheme described in the broadside was successful, although a handwritten note suggests it may have been tried out in June 1662. An alternative explanation is that the lottery was derived from a more ancient ancestor. A simple version of this type of lottery had in fact been introduced in China during the Second Han Dynasty, 25 to 220 CE. It was called the Hua-Hoey Lottery, or the Game of ThirtySix Animals. Thirty-six cards were prepared, each bearing the image of a different animal. We do not know how the game was conducted originally, but by the 1800s each card showed the animal and a person reincarnated from that animal, and the people were related in various ways that made it easy for superstitious gamblers to relate the cards to recent dreams, as suggestions to help them choose their bets. Bets could be placed upon any card, and the gamblers would then assemble

The Genoese Lottery  21

Figur e 2.1. A Venetian lottery broadside from 1736. For each of the numbers 1 to 90, it lists the name of a woman who would benefit as a charity case if the number were drawn, and the name of a saint with the number. These were intended to help the bettor choose numbers. The sentence just below the title announced a 20 percent increase in payouts for ambes, and an 80 percent increase for ternes, for the 8 March 1736 drawing. Some possible payouts are listed at the bottom.

to witness one of the cards being selected at random. Payoff odds apparently varied. Sometimes the operator would wait until all bets were placed, and would then himself bet on the animal with the smallest number of bets, the pool of all bets then to be divided evenly among all the winners. At other times the operator simply paid the winners at thirty-to-one odds (thirty-five to one would have been an even bet). The lottery persisted in some Asian locations at least through the nineteenth century, and it was even imported to Chinese communities in Europe and America.4 Might Marco Polo have brought it back to Italy among his other treasures, and thus influenced the development of the Genoese lottery? He had, after all, been held prisoner from 1297 to 1299 in Genoa, where he famously recounted his Asian adventures. And if the lottery was tried in Genoa at that

22  C h a p t e r T w o

much earlier date, it might even have served as an inspiration for the Genoese election system. In any event, by the year 1700 the lottery was well established in Genoa. By 1757 versions of it were to be found in Venice (see figure 2.1 for an example from 1736), Rome, Naples, and Milan, and it had even spread to Vienna. The operational rules from Genoa were adopted by the Calzabigis and Casanova, and were reported in the decree of October 1757.



Three



The Establishment of the Loterie in 1757

A F r e nc h de c r e e of 15 O c t ob e r 1 7 5 7 declared the establishment of the new Loterie; the decree served as both an advertiser and an adjudicator. It announced the need for financial support for the École militaire, and it stated that to that end, a lottery was being created and operated by the Crown as a concession to the École for the next thirty years. Assuming a starting date in November, it would then be delivering its proceeds exclusively to the École through September 1787. The lottery was to be “composed on the same principles as those which are established in Rome, Genoa, Venice, Milan, Naples & Vienna in Austria.” The decree noted that such a lottery had operated successfully in Genoa for about sixty years. But this was more than a simple announcement; it was intended to serve as a rule book, defining the game explicitly, with a clarity intended to resolve all future questions and disputes on payments. The decree had remarkably little to say about the physical operation of the Loterie. It simply stated that the drawing required a “Wheel of Fortune,” a rotatable cage to hold ninety hollowed balls; in each of these would be a number between 1 and 90. There was to be a monthly drawing in which five balls would be drawn without replacement in the rooms of the Arsenal of Paris, in the presence of the members of the king’s council and others. The substance of the decree was in the discussion of the betting.

24  C h a p t e r T h r e e

Prior to each drawing, bets could be placed at sales offices around the city in three different ways: 1. A bet that a specified number would be among the five drawn (called a bet on an extrait). 2. A bet that two specified numbers would be among the five drawn (a bet on an ambe). 3. A bet that three specified numbers would be among the five drawn (a bet on a terne.

The names of the bets were taken from the Italian names: estratto, ambo, terno. At later times and in a few jurisdictions, other bets would be allowed; but initially, in Paris as in Genoa and Rome, these three were the only allowed bets. It was possible to place multiple bets. For example, the bettor might specify the three numbers 14, 27, and 68, and pay for three separate bets on the three numbers as extraits, as well as for three bets on the three possible ambes (14 and 27, 14 and 68, and 27 and 68), and also one bet on the set of three as a terne. The bettor might even risk different amounts on the different types. This freedom to tailor bets to individual preferences was new to France with this Loterie. After the drawing, the winning numbers would be printed and posted publicly at the lottery offices, as seen in figure 3.1. Winners could claim their prizes according to the following schedule: One unit bet paid for a winning extrait would return 15 units. One unit bet paid for a winning ambe would return 270 units. One unit bet paid for a winning terne would return 5,200 units. Of course all bets favored the state: The calculations are fairly easy (see appendix 1); and if the bets were to be fair, the payoffs would have been 18, 400.5, and 11,748 units. But, as the decree (a public document) proudly pointed out, the new lottery was more generous than the foreign competitors: “It now remains only to observe that the conditions offered to the public in this plan are superior to those in Italy and Vienna.” The decree offered table 3.1 as evidence for this claim. Aside from the need to guarantee the Crown a profit, what calcu-

Figur e 3.1. A young woman asks an upset man to check her Loterie ticket against the posted list: “Have I won? Have I lost? Read!” From the novel by Lebrun-Tossa (1801), discussed in chapter 11.

26  C h a p t e r T h r e e

Tabl e 3.1. Loterie payoffs in Paris compared to those in Italy and Vienna, from the October 1757 decree Payment for one unit bet on

extrait

ambe

terne

in Italy

13⅓

266⅔

5,142⅞

in Vienna

12

255

3,000

in Paris

15

270

5,200

lations led to these payoffs in particular? Casanova’s memoirs provide an interesting explanation, one that played a key role in convincing the ministers of the plan’s viability. Imagine a different lottery— like this one, but one in which six balls were to be drawn instead of five. What would the fair payoffs be for these same bets with six drawn? Clearly, since you have more draws to match your numbers, the chance of winning would be higher, so the payoffs would be lower. Again a simple calculation (appendix 1) gives the answer: the payoffs would be 15, 267, and 5,875. The designated payoffs for the three types of bets were 15, 270, and 5,200, suggesting either an approximate version of the six-ball fair payoffs or an error in the calculation for the six-ball terne. Why is this a useful argument? Because Casanova used this analogy to make a rhetorical point: The king’s risk was likened to a fair six-ball lottery, but one in which only five balls were drawn. He wrote, “This made it scientifically certain that the King’s advantage would always be one in 5, which came to 18 in 90, which was the total number of lots in the lottery.” As he had described the advantage to Duverney at the chateau, “Whoever brings the King six francs will receive five”— a statement that was only literally true on average for bets on the extrait. Of course, the king would expect to profit even more on the ambe and the terne. Casanova’s arguments may have carried the day when he made his presentation, but between that meeting and the edict of October 15, the king’s ministers developed a case of cold feet. Casanova had allowed that there would be a temptation for the Loterie to remove risk by balancing the portfolio of bets; that is, by refusing to accept further

The Establishment of the Loterie in 1757  27

bets on numbers that had attracted great attention, and ideally having equal amounts bet on all ninety numbers. That, he declared, would be a mistake. Even if it could be done, it would cost dearly in reputation. The people must believe that the king could lose money in a drawing. But at some point in the summer of 1757 other arguments prevailed, and the October edict included a short section placing limits on bets that was directly contrary to Casanova’s advice. Section 5 of that edict stipulated that the Loterie office should ensure that the total of all bets on any specific extrait should not exceed 6,000 francs, the total on each specific ambe should not exceed 300 francs, and the total on each specific terne was limited to 150 francs. This limit would thus go beyond a limit on all individual bets that was placed at 24 francs, and would be an attempt to guard against consortia, where by plot or by social contagion, heavy bets were placed on a very limited set of choices. Thus, the total amounts the Loterie risked on bets on each specific number or combination were 6,000 × 15 = 90,000 francs, 300 × 270 = 81,000 francs, and 150 × 5,200 = 780,000 francs. The edict also granted the Loterie the authority “to extend the bets” if they found that appropriate and gave notice to the public. This statement presumably would have allowed them to shrink the limits as well as enlarge them. Even in the first few drawings the limits would have been hard to implement, as they would require a continuously updated account in the central office of all actions taken in every single agency. That task might have been feasible for the extrait in the early years, but would have been increasingly more difficult for combinations, and generally infeasible when the number of agents reached several hundred. There seems to be no direct evidence that these limits were ever invoked, and in any case they soon disappeared from sight, and did not recur in any later versions of the Loterie or in later edicts. One can imagine that the fact that the Loterie was quite profitable from the beginning would have soon eliminated the temptation to limit bets beyond the existing limit to twenty-four francs on any single bet by a given individual, a limit that would be considerably increased in 1776 (see chapter 5).



Four



Problems and Adjustments in the Early Drawings

The f irst draw i ng was he l d on 18 April 1758, and the five numbers drawn were 83, 4, 51, 27, and 5. For the rest of that year, the drawings were held every other month, becoming monthly after that. Giovanni Calzabigi had been installed as director of the Loterie with Casanova’s support, and he received an annual salary of 4,000 francs, with another 3,000 for each drawing. Casanova himself had received the same salary, and rights to manage six lottery collector’s offices, five of which he promptly sold for 2,000 francs each. He opened the sixth office on Rue Saint-Martin, with his valet as chief clerk. The Loterie was apparently off to an auspicious start, but unexpected organizational challenges must have immediately become apparent. The Loterie’s organizers had known that continued success required public confidence in the fairness of the draw and in the certainty that winners would be promptly paid. When the first drawing was held, it was generally advertised that the winners could collect at the main office a week later. Evidently that was not soon enough. There was a crush of eager winners demanding to be paid immediately, and Casanova quickly posted that he would personally redeem all winning tickets that he had signed from his office, and that they would be paid twenty-four hours after the draw at his office. This generated a large amount of business for Casanova, and a considerable number of complaints from others’ offices. The director told the complainants they could match his service if they wished.

Problems and Adjustments  29

Casanova’s receipts from the sales for the first drawing totaled 40,000 francs; from this he had to pay out about 17,000 or 18,000, nearly all on ambes, and the remainder to the Loterie. That drawing gave the Loterie a total profit of 600,000 francs on a total received of two million. There were, Casanova stated, at that drawing about eighteen or twenty winning ternes. As Casanova had predicted, the large payoffs helped build the lottery’s reputation. At the second drawing, he himself had to pay a terne winner 40,000 francs. The receipts were 60,000, but he had to borrow briefly to make the short-term payment, again to the benefit of his reputation. The problem of public confidence had faced all public lotteries since the 1500s. You can, as Abraham Lincoln said, fool some of the people all of the time, and a lottery with secretly predetermined outcomes might be successful once or twice. But for its establishment as a continuing institution, a strong assurance was needed that the numbers being drawn were all treated equally. The decree of 15 October 1757 had stated the procedure for conducting the drawing in a scant three sentences: A wheel of fortune will contain ninety identical balls; in each of these balls will be a number from 1 up to & including the number 90, & in each case there will be appended the name of a Saint, or some other name, at the choice of the Administrators. On the day of the drawing, these numbers, before being placed in their balls, will be displayed successively under the eyes of assistants. After this formality, we will mix the ninety balls in the wheel of fortune, and we will draw five only: it will be these five numbers that determine the amount of the gain that will result from the bets marked on the tickets of those who take part in the Lottery.

That first drawing in April 1758 must have not gone entirely smoothly. Something must have occurred that convinced the ministers that a much more detailed and transparently fair method was required. The earlier “blanks” or raffle-style lotteries had for more than a century addressed this concern by extensively mixing the material for the drawing in large tubs, and by having the drawing itself be

30  C h a p t e r F o u r

Fig ur e 4.1. The ceremony of drawing at a 1751 English “blanks” Lottery.

done by blindfolded orphans in a public setting. Figure 4.1 shows the setup in a 1751 “blanks” lottery in London. There were two tubs, one for the numbered slips corresponding to the tickets sold (70,000 in that 1751 lottery), and the other with slips that either showed a prize or were blank, indicating no prize (10,000 prizes and 60,000 blanks mixed together in London in 1751). Each numbered ticket would be paired with a simultaneously drawn slip from the other tub. The two orphans, one for each tub, would have been blindfolded at the start of the drawing. The theory behind this was that an orphan would not be directed by a parent to cheat, and children are viewed as generally more innocent than adults. The blindfolds were evidence that even orphans cannot be trusted entirely; in the 1770s there were sufficient attempts to bribe them that elaborate countermeasures, including random choice of orphans, were introduced for British state lotteries.1 A professional magician today would be skeptical that this would guarantee a fair draw, but the data that are available support the fairness of the draw (see chapter 10).

Problems and Adjustments  31

Fig ur e 4.2. William Hogarth’s 1721 depiction of a lottery drawing, including “Fraud tempting Despair with Mony at a Trap-door in the Pedestal.”

This type of drawing had been used for many years before this. Already in 1721 the artist William Hogarth had produced a satirical picture in this spirit (figure 4.2). Indeed, the need to have the actual selection made by a disinterested party must have been evident from the beginning: the earliest known description of the drawing of a Genoese lottery, namely the 1662 English broadside mentioned in chapter 2, stipulated that the numbers should be “drawn or taken from out of the said Hundred by the hand of some Child or other person unconcerned.”2 By the end of May 1758 the Loterie adopted essentially this approach, albeit with a smaller wheel of fortune instead of the large tub. In quick reaction to whatever had caused concern in that first drawing, they rushed into print a notice dated 31 May, describing in great

32  C h a p t e r F o u r

detail what procedure should be followed, starting with the second drawing: Notice on the Second Drawing of the l’École Royale Militaire Loterie. The second Drawing of the lottery of the l’École Royale Militaire will be on the 27th of this month of June, between four and five o’clock in the afternoon, at the Arsenal, in the hall called the Magazin Royal. Gentlemen of the Council of the l’École Royale Militaire and the Lieutenant General of Police of the City, Prevôté and Vicomté de Paris, will be present at the Drawing, which will be attended by all who wish to come. The room will be open for everyone. Here is the way in which the Drawing will be executed. At the end of the Hall, on the side facing the Cour de la Bastille, a platform will be raised, below which the gentlemen of the Council of the l’École Royale Militaire will be seated. The Secretary of the Council will have his place at the front and on the right side of the platform. There will be in the middle a Crystal Wheel of Fortune, open and empty; to the left of the platform, on a table prepared for this purpose, will be the 90 Numbers that make up the Loterie, from and including no. 1, to and including no. 90. Not only will these 90 numbers be marked each in large size on a vellum square, eight inches long and five inches wide, but they will also be written in large letters on the other side of the same square, to avoid any ambiguity. In addition, a Fleur de Lys, placed on the top of the square, will clearly show the direction in which the Number must be read. On the same Table, there will be 90 small boxes of white cardboard, in each of which one of the 90 numbers will soon be secured, to be then placed in the Wheel of Fortune. The order to proceed with the Drawing having been given by gentlemen of the Council of the l’École Royale Militaire, a Deputy Clerk for the Drawing will advance to the middle of the platform, and will first show to all the assistants that the Wheel of Fortune is empty. Then, approaching the Table on which the 90 Numbers and their boxes rest, the Deputy Clerk will place himself between the Table and the Wheel of Fortune; and taking the vellum square in which the Number 1 has been marked and written, it will be shown to the public by raising

Problems and Adjustments  33

and turning it several times on the side of the writing and on the side of the Number. He will then speak out the Number loudly and intelligibly; and finally after rolling up the square of vellum with his fingers, he will secure it in one of the boxes and throw it into the Wheel of Fortune. He will follow this same operation for all the other Numbers until the ninetieth and last; and then all the said 90 Numbers will thus have been placed in the Wheel. Immediately afterwards, the Wheel of Fortune will be closed by the Deputy Clerk, who will then turn it for a few minutes, after which an Orphan, who will have been chosen to extract the five Numbers, will advance towards the Wheel to begin the drawing. As soon as the members of the Council of the L’École Royale Militaire have ordered the extraction of the five Numbers, the Wheel of Fortune will be turned again for some time, after which it will be stopped; and the Deputy Clerk at the Drawing having opened it, the Orphan, blindfolded, will select a first box, which he will hand over to the Clerk. The Clerk having raised his hand holding the box, will open it with the other hand: he will hold the vellum square on which the Number is marked and written; and having unrolled it, he will speak said Number aloud. He will then show it to the public, by displaying both the side where the number is marked in numerals, and the side where the number is written in letters. The Secretary of the Council of the l’École Royale Militaire, who will be charged with keeping the minutes of the Drawing, will record the Number in the minutes; and this formality having been completed, the square of vellum, on which said Number is written and marked, will be by the Deputy Clerk displayed in the middle of the Hall with the box which enclosed it. The same operation will be repeated for each of the five Numbers which must be extracted from the Wheel of Fortune, and all five having been thus drawn, announced, and registered, the Deputy Clerk will close the Wheel; and taking the minutes from the hands of the Secretary of the Council, he will read them aloud; and the reading being made, the said minutes will be signed by gentlemen of the Council of the l’École Royale Militaire. A copy shall be issued to the district Chief Receivers [sales agents]; and the order will be given at the same time to proclaim and display the five Numbers that had been selected in the Drawing.

34  C h a p t e r F o u r

At any time when the Wheel of Fortune has been turned, all the assistants will be permitted to ask that it be turned again; and that will be done, but we warn the public to use this permission with moderation so as not to unnecessarily prolong the Drawing. The Receivers of the Loterie are spread among the different districts of Paris, and they will close their Registers on Thursday, the twentieth day of the month of June; however, for the convenience and satisfaction of the public, they will continue to accept all the requests that will be made to them; they will provide acknowledgments, and they will obtain the signed tickets from the Receiver General Office, which will itself also receive directly all the bets from those who will want to take part in the Loterie, until Monday morning 26th day of the same month, to its Office, rue Montmartre, opposite the Crescent and next to the General Office. The Public is also warned that, because of the extent and multiplicity of the operations of the management of this Loterie, it is impossible, especially near the beginning of a Drawing, to pay the winning Tickets until fifteen days after the date of Receiver’s receipts; if these Tickets can be paid, they will be paid. But especially near a Drawing, we will work to deliver them as soon as possible: those holding winning Tickets must be patient in this respect, and their receipts from the Receivers are evidence in place of tickets. Several persons who have won Lots in the previous Drawing did exchange those receipts, even at the moment that they were to receive their winnings. There will never be lost tickets; they are recorded with the Receivers, or with the General Bureau of the Loterie, and during the six months granted by the Judgment of the Council for the payment of the Lots it is impossible that we will not find them there. Although they cannot be paid on the simple receipt alone, that receipt is however an indication to find the Ticket, and a title to claim it, and it will be delivered. The public seemed so pleased with the form of the first drawing of this Loterie, that we will never depart from it, and that if it happens that we make some changes, it would be only as much as they tend to make this form even clearer and more demonstrative: the General Administrator of this Loterie will receive, even with pleasure, all the opinions that one wishes to give him in this respect; it will be easy to send them in an envelope to his address at the General Office. Permit to print, in Paris on 31 May 1758. BERTIN

Problems and Adjustments  35

Fig ur e 4.3. The results of the second drawing of the Loterie in June 1758, as officially posted. From Rouault de la Vigne 1934.

Given the extraordinary attention to every trivial detail, this must have been an exercise in damage control. It may have been a reaction to complaints about ambiguous draws (6 or 9?), to perceived insufficient mixing, or to a lack of transparency in how the separate containers were loaded. If so, it must have been successful. There seems to have been no variance in the announced practice over the next sixty years of the Loterie. Figure 4.4 is a generic picture of an early drawing; it could depict the second drawing on 27 June 1758, as well as the 1,250th in 1816 (as shown in figure 10.1 in chapter 10). Figure 4.3 shows the results of the second drawing as posted at the time. There were other steps taken in 1758 to meet challenges. Running the Loterie required the recruitment and education of a sales force that could be trusted to handle the king’s money. Suppose, to take a realistic example, that a customer came in with a list of five favorite numbers from the ninety possible, and asked to place a bet of twelve sous (the minimum allowed bet, there being twenty sous to a franc) on each of the possible extraits, ambes, and ternes that could be found from the five numbers. How much should he or she be charged

36  C h a p t e r F o u r

Fig ur e 4.4. The ceremony of the drawing at the Loterie in the early years.

for the bets? And after the draw, how much should be paid? For a given five numbers, there are in fact five possible extraits, ten possible ambes, and ten possible ternes: a total of twenty-five bets, totaling 25 × 12 = 300 sous = 15 francs. If the five favorite numbers were the five drawn, all these bets would win, paying a total of 32,865 francs. How do you train people to make these calculations correctly, where errors could be costly to the Crown? Indeed, how do you bring to the public’s attention the possibility that an investment of 15 francs could return as large a sum as 32,865 francs, not bothering to tell them that for this bet the expected payoff was much lower, only 9.2 francs? A part of the answer was to provide a small book containing the results of the calculations as needed by the sales force, and at the same time to make those calculations available to the public. The book, Tables de la Loterie de l’École Royale Établie par arrêt du conseil du 15 octobre, was published in 1758 (probably early in the year, before the first drawing) by the commercial publisher Quillau.3 It contained seventy-nine tables.

Problems and Adjustments  37

For example, table 12 (figure 4.5) supposed each bet was for twelve sous, and each row in the table corresponded to the number of favorite numbers the customer brought in (from 2 to 15), with columns answering these questions: How many ambes could be bet with those numbers? What would the total price be? How much would the payout be if two, three, four, or five of the numbers should be drawn? The tables gave the answers for different bet sizes (from the minimum allowed bet of twelve sous to the maximum allowed, twenty-four francs each) and each of the different bets (extraits, ambes, or ternes), as well as tables for bets for all extraits and ambes in the list of favorites, all extraits and ternes, all ambes and ternes, and all extraits, ambes, and ternes. For example, from table 12, if eight numbers are bet, there are twenty-eight possible ambes, and if twelve sous are bet on each of the twenty-eight bets, this amounts to sixteen francs and sixteen sous. If two of the eight are drawn, you win 162 francs; if five of the eight are drawn, you win 1,620 francs. If, having chosen eight numbers, instead of betting only ambes you bet twelve sous each on all extraits, ambes, and ternes, table 67 tells you it will cost you fifty-five francs and four sous. If two of your eight are drawn, you win 180 francs; if five of the eight are drawn, you win 32,865 francs. The book had a ten-page introduction explaining the structure. A plausible guess is that the tables were constructed by Ranieri Calzabigi, and the introduction could have been written by Casanova. The final passage repeats an argument Casanova used with the king’s council, pointing out the ability to make what we might today call personalized bets: “Of the many reflections which could be added to these tables and the lottery in general, we restrict ourselves to one. This point is that, since here the customer is the master of the choice of numbers and of the size of the bets (which weighted by the probability give the importance of the bet), this lottery admits all manner of options for the bettor, and these may be varied at will. It must be granted that this is a point of view very different from that of the other lotteries, where everything is constant and determined. The only consideration for other lotteries is the price of the tickets, the size of the lots being arbitrarily established by the entre-

38  C h a p t e r F o u r

F ig ur e 4 . 5 . Tables 12 and 67, explaining to the Loterie’s sales agents the prices of various combinations of bets, to insure they were accurately charged. The table uses the older names for units; the “sol” would later become a sou, and a “livre” would become a franc. Twenty sols equaled one livre.

preneurs, and never scaled to the views or the ambition of the Individuals who place the bet. Here these can instead be easily adapted to all degrees of comfort and risk; to each his lottery is made up to his liking; this lottery, as it were, admits a different approach relative to each person who is interested in it, and can be accommodated to individual wishes; thus, after having proposed a sum to wager, the bettor has the opportunity of focusing on specific numbers, perhaps with less risk but larger wagers, or alternatively in a less specific and less expensive way, by forming a general bet on several numbers, or perhaps by combining the one with the other. These different compositions can at the same time in different ways represent the lottery ticket of all of society.4

Problems and Adjustments  39

Fig ur e 4.5. (cont.)

Notwithstanding the efforts made, there are indications that some improvements in procedure were needed. In 1759 there were two new edicts, one to impose a strict penalty of up to three thousand francs on lottery offices that were slow in delivering the proceeds of a drawing to the king, and another to permit smaller bets on ambes and ternes (three sous and six sous, respectively, while retaining the lower limit of twelve sous for the extrait). A year later, there was a sign that organizational procedures needed to be tightened. From the beginning there had been a requirement that a winning ticket must be checked against the office register to ensure that the numbers had indeed been properly registered before the drawing. This was necessary to prevent fraud. and may have been successful by and large; apparently there were no major scandals. But

40  C h a p t e r F o u r

there was another reason for the registration— namely, to protect the integrity of the lottery by paying all legitimate winners. An edict issued in September 1760 suggests that in some cases the office had been lax in its recording, leading to a complaint by an apparently legitimate winner of a bet on a terne. The edict introduced a redundant system of recording to provide a double check. The case that precipitated this change involved a bet by a gentleman named Boucher who bet the numbers 30, 38, 89 on the twenty-fourth draw of the lottery in September 1760. When he presented his winning printed ticket, no registered record could be found. He must have been of impeccable integrity, for on 27 September the edict was issued, proclaiming the new procedures and citing his case by name. Could “Boucher” have been the famous painter, François Boucher, who had painted Madame de Pompadour’s portrait several times? Presumably he was paid well on all occasions. By the end of 1760, Casanova and the Calzabigis had departed from France and their roles in the lottery. The reasons for their departures were not stated. Casanova’s reason may have simply been his roving eye for adventure; the Calzabigis had become dispensable once the French had learned the details and adapted the procedures to the local customs. Casanova subsequently pursued all manner of affairs, including one as an agent in a strange plot to enlist Sweden’s help in a war against England (it did not go well).5 The Calzibigis went first to Brussels, to sell the Loterie to the agents of Maria Theresa there, with some success. But soon thereafter, Ranieri Calzabigi was off to Vienna and a future in opera, and Giovanni Calzabigi moved on to a role as a financial advisor in Brussels in 1760 and then in Berlin in 1763, again offering the Loterie plan to the Prussian king, Frederick the Great. The paths of Giovanni and Casanova did cross again in Berlin in 1764. Giovanni Calzabigi initially had convinced Frederick to back a lottery on the Parisian model in 1763, with the king as guarantor; but Frederick soon became worried about potential losses, and even when in the following year Giovanni had Casanova try to reassure the king as he had the ministers in Paris, Frederick threatened to withdraw. Fearing loss of royal sponsorship, Casanova and Giovanni

Problems and Adjustments  41

hatched a plan that required a number of other stockholders willing to make substantial individual commitments, but they were unable to generate the necessary enthusiasm; that plan died an early death. Frederick found a way to reduce his risk by pushing responsibility to a firm independent of the king, without royal guarantee, and the Berlin version of the Loterie survived until 1806, without Calzabigi’s involvement after 1766.6 The success of the École militaire loterie continued unabated. A decree dated 3 November 1770 stated that the king, in recognition that success depended upon the citizens’ access to lottery offices, announced that henceforth there would be eighty offices in Paris and its suburbs, separated sufficiently far apart so that they would not harm each other’s business.7 And there matters stood, without further royal comment, until June of 1776.



Five



Antoine Blanquet and the Great Expansion of 1776

I n t h e e a r ly 1 7 7 0 s, a period of falling prices in agricultural markets made an already precarious French financial system even more troubled. On 24 August 1774 the king appointed the remarkable economist-philosopher Anne Robert Jacques Turgot as controllergeneral of finances. It was expected that Turgot would reform the economic system, in part by working to reduce the heavy weight of past royal favors to well-placed individuals: pensions and sinecures that had ballooned to become a drain on the king’s finances. Turgot did set to doing that. Indeed, he approached the task with little attention to the political pushback, and that insensitivity would eventually lead to Turgot’s dismissal in May 1776. But other steps were taken as well, and one of these involved the Loterie. The Loterie of the École militaire fell under the charge of the Ministry of War, where it had been administered since 1757 and had been doing well, providing about two million francs a year to meet the needs of the École, making this a bright spot in the king’s budget. The Loterie had not changed since its beginning in any serious way, while foreign lotteries were increasingly encroaching on the French market with larger prizes and more betting opportunities. It was natural that the king would look to the Loterie as a potential source for increased revenue. Turgot’s papers give an indication of how the matter proceeded.1 Evidently all parties agreed that for the Loterie to con-

Antoine Blanquet and the Great Expansion  43

tribute to general finances beyond the École, a significant expansion would be required: many more betting offices situated more widely, and more betting options. The minister of war was not enthusiastic about adding the supervision of such an expansion to his portfolio, and he made a proposal: The Loterie would be transferred entirely to the Finance Ministry, but in recognition of the 1757 promise of a thirty-year concession to the École militaire, the École would be granted a two-million-franc payment each year for the next decade as compensation for forfeiting its concession, approximately what it was then clearing as profit from the Loterie. The Finance Ministry would take over sole administration of the Loterie. Turgot might have been expected to welcome this offer, but the papers indicate that his reaction was less than enthusiastic. The Finance Ministry’s in-house analysis of the plan of the minister of war is preserved in Turgot’s papers; it was highly critical. The summary document indicated that Turgot regarded the Loterie as undesirable on moral grounds; it was an “indecent project” that should not be expanded. The bets on ambes and ternes were already quite unfavorable for the bettor, and adding even worse options would only increase the toll it would take on the people. The report included a short analysis of the projected business plan, suggesting that after accounting for the costs of administration, the profits from expansion would be meager. The report’s assessment of the business plan would turn out to be much too pessimistic, but in any case the resistance in the Finance Ministry faded as it became clear that the king wanted the expansion, and that if it was going to be enacted, it would be better for Finance to retain control, since it would in any event have to make up the loss to the École’s budget. The timing was such as to suggest that the resistance may have contributed to Turgot’s dismissal in May 1776. The Crown granted the requested concession to the École and seized the lottery for general purposes, expanding its scope considerably. Formally this was accomplished by an edict on 30 June 1776 that suppressed the Loterie of the École militaire and created a new lottery, the Loterie royale de France. In practice this was simply an expansion

44  C h a p t e r F i v e

of the former structure, using the same bureaucratic structure, the same offices, and many of the same people— built on the same plan, but with a few new rules. The expansion would require a larger administration, and that is where Antoine Blanquet enters the picture, quite literally. Every printed ticket in the Loterie from this expansion to the Revolution carried Blanquet’s signature as a guarantee of authenticity, and a pledge that the winning tickets would be paid promptly, even with the larger prizes offered (figure 5.1). The tickets show several differences from the Loterie for the École: more possible bets, some changes in design (which we will consider in chapter 8), Blanquet’s signature, and now only the first names of women were given. The last of these represented a policy change (which we will also discuss in chapter 8). But Blanquet’s name carried a very different message: a significant increase in the administration to operate the significantly expanded Loterie. This was a curious sign, for the simple reason that Antoine Blanquet did not exist; he was a fictional character. There was no Antoine Blanquet! The edict was quite clear on this. It stated that the top administrator would be an intendant to be appointed by the king, and that the main operations would be overseen by a group of twelve administrators drawn from different districts in France. A year later, the post of intendant was eliminated and the number of administrators was reduced to six.2 The administrators were named individually in the edict, but it was also stated that they would be known collectively as “Antoine Blanquet.” Initially the administrators were required to contribute equally to a total bond of 3,600,000 francs, for which they would receive 5 percent interest on their portion; but a year later the king eliminated this requirement, refunded those who had paid, and offered two million francs of his own in its place.3 This would stand as a reserve to help assure the bettors and make tangible the king’s guarantee as backing. The use of the corporate name “Antoine Blanquet” would mean that there was no need to change the name as time went on, and it tied the reserve fund to the bets clearly and unambiguously. Antoine Blanquet was a fund, not a man. The change in

Figur e 5.1. A ticket from the second drawing in April 1783, with both Blanquet’s printed signature and an agent’s handwritten signature. A total of six francs, ten sous were bet on all possible simple bets involving the five given numbers, from the ambes to the quine, omitting the extraits. None of these numbers were drawn on this day.

46  C h a p t e r F i v e

the size of the administration between June 1776 and July 1777 indicated that the king had difficulty recruiting administrators willing to put up such funds, and so he decided to simply take the reserve fund from the Treasury. The 1776 edict announced some significant changes in bets offered. The number of drawings per month was doubled to two, and four new bets were introduced, two of which offered substantially larger prizes: The extrait déterminé permitted a bet on a specific number in a specific slot, such as “The number 53 is drawn as the fourth of the five draws.” This was to pay seventy units for each unit bet. The ambe déterminé permitted a bet on a specific pair of numbers in specific slots, such as “The number 53 is drawn as the fourth of the five draws and the number 27 is drawn as the fifth.” This was to pay 4,900 units for each unit bet. The quaterne permitted a bet on four specific numbers without specifying order or slots, such as “The numbers 16, 22, 48, 53 are drawn among the five draws, in any order.” This was to pay 70,000 units for each unit bet. The quine permitted a bet on five specific numbers without specifying order or slots, such as “The numbers 16, 22, 48, 53, and 77 are the five drawn, in any order.” This was to pay 1,000,000 units for each unit bet. There is a sense in which the new déterminé bets made the simpler versions redundant: If, for example, you bet one unit on the number 62 as an extrait déterminé for each of the five possible slots— that is, five separate one-unit bets on (62, x, x, x, x), (x, 62, x, x, x), (x, x, 62, x, x), (x, x, x, 62, x), and (x, x, x, x, 62)— that was equivalent to betting five units on 62 as an extrait simple, in the sense that either both schemes would win or both would lose. You might then suppose that the payoffs would be the same in both cases, but they were not. Since the drawing was without replacement and 62 could come in only one of the five slots, equality would require the extrait déterminé to pay five times what the extrait simple paid, namely 5 × 15 = 75; but in nearly all Loteries it paid only 70. Similarly, an ambe simple could be mimicked by betting all possible ambes déterminés with the same two numbers, and there were 5 × 4 = 20 such ambes déterminés. Again,

Antoine Blanquet and the Great Expansion  47

T a b l e 5 . 1 . The changes in bets offered and payoffs (per unit bet) in the French Loterie over its history Chance

1757– Aug 1776

Sept 1776– Jan 1777

Feb 1777– 93; 1797– 1836

Extrait

1/18

15

15

15

Extrait determiné

1/90

–

70

70

Ambe

1/400.5

270

270

270

Ambe determiné

1/8,010

–

4,900

5,100

1/11,748

5,200

5,200

5500

1/511,038

–

70,000

75,000

1/43,949,268

–

1,000,000

1,000,000*

Bet

Terne Quaterne Quine

*The quine was discontinued in 1804, and was probably not always available at all offices before that.

one might expect the ambe déterminé then to pay twenty times what an ambe simple would pay, or 20 × 270 = 5,400, rather than the 4,900 it paid in 1777. To judge from the surviving tickets, few if any people bet on the less favorable ambe déterminé, whether it was because they realized the bet was less advantageous or because going to the trouble of reconstructing an ambe simple through twenty other bets was too tedious to consider. Interestingly, the same was not true for the extrait bets, as we shall see in chapter 12. Table 5.1 summarizes the changes in bets. The latter two of these bets, the quaterne and the quine, had been considered by Casanova and the Calzabigis, but while they were very favorable to the Crown, the risk had been judged too high. Fair bets would have paid back 511,038 units (instead of the stated payoff of 70,000) and 43,949,268 units (instead of the stated payoff of 1,000,000), respectively. In 1757 the finance minister was already sufficiently nervous about possibly losing on a terne. And while there is no indication that questions of fraud came up— say, if a crooked salesperson sold a back-dated ticket after the draw— the possibility of fraud was greater than the probability of an honest win. By 1776

48  C h a p t e r F i v e

F ig ur e 5 . 2 . A small 1776 broadside for the Loterie, praising the king. The poem under the picture of Fortune states: While at the movement of a turning wheel, Five uncertain numbers shake a thousand hearts, France, not affected by the whims of fate; Louis, devotes his favors to your needs.

the confidence in the law of large numbers and in the oversight of the offices had grown enough so that the Crown was ready to take the chance, responding to competition from at least some foreign lottery payouts (figure 5.2). The limits on bets were also changed. To broaden the appeal, the limits on the sizes of bets were greatly widened. The new ranges (minimum to maximum bets) were: extrait extrait déterminé ambe

1 sou to 10,000 francs 12 sous to 1,000 francs 6 deniers (= .5 sou) to 400 francs

Antoine Blanquet and the Great Expansion  49

ambe déterminé terne quaterne quine

6 deniers to 180 francs 6 deniers to 150 francs 6 deniers to 12 francs 6 deniers to 3 francs

In addition, no ticket would be issued unless the total of all bets was at least twelve sous. While no explicit reasoning was given for these new limits, we can read into them a growing awareness of pricing so that even less wealthy gamblers could participate at the low end, and a nagging worry about maximum risk on the upper end. With the June 1776 published payouts, a maximum bet on each of these seven wagers could yield, in thousands of francs, for the first three bets (extrait, extrait déterminé, ambe), 150, 70, and 108, and for the last four, 882, 780, 840, and 3,000. Since these yields were very far from equally likely to occur, the administrators’ eyes must have been set on maximum loss, regardless of chance— a view that can be defended if the possibility of fraud is taken into account. That the lower three bets were frequently made as parts of combination bets— such as, “Bet on three numbers as three extraits [three bets] and all the ambes that would include them [three more bets]”— suggested placing a lower maximum per bet for these, since if all three numbers were drawn the bettor would win six bets returning on six units bet: 3(15) + 3(270) = 855 units. That is, the terne, quaterne, and quine were more often made as single bets, while the extrait and ambe were frequently multiples (as in “all the extraits using the five numbers”), which may explain the gap between the first group and the last group.



Six



The Introduction of Bonus Numbers: Les Primes Gratuites

One ot her nov elt y was of f er ed as an incentive for participants in the 1776 edition of the Loterie. It was described in the edict of 30 June 1776, where it was referred to as primes gratuites (free bonus numbers), and it received its first real trial in the drawing of 1 October 1776. It must have been judged a failure very soon, because on 3 December 1776 there was another edict, this time canceling primes gratuites and, as compensation, raising the payouts on the ambe déterminé, the terne, and the quaterne respectively to 5,100 (from 4,900), to 5,500 (from 5,200), and to 75,000 (from 70,000) units per unit bet. The stated reason for the cancellation was that the plan for primes gratuites required too much calculation, and customers preferred better payoffs instead. In fact, while the canceled plan was an interesting early experiment in the design of lotteries, it would have been sufficiently confusing both to customers and to the sales staff so that a major reason for the immediate action was likely that the lottery staff could not cope with it. The gamblers could have simply ignored it or learned to live with it. Primes gratuites required the lottery at each drawing to draw four additional sets of five numbers after the regular drawing, bringing the total draw to five sets of five numbers (table 6.1). After each drawing of five, the balls were replaced and mixed before the next drawing. The first five constituted the main draw and led

The Introduction of Bonus Numbers  51

T abl e 6 . 1. The primes gratuites were used in eight drawings of the Loterie, from 1 October 1776 to 16 January 1777. In each case the first line gives the main drawing, followed by the four primes gratuites (PG1 to PG4) 1 Oct 1776

61

31

66

4

70

2 Dec 1776

28

77

25

82

86

PG1

22

39

51

80

20

PG1

42

38

82

73

25

PG2

27

72

65

73

19

PG2

78

47

59

50

34

PG3

79

34

14

47

87

PG3

37

11

43

87

85

PG4

12

88

63

44

51

PG4

69

59

19

72

55

16 Oct 1776

90

4

15

14

35

16 Dec 1776

41

40

64

36

82

PG1

65

62

43

16

52

PG1

11

58

51

18

6

PG2

29

42

45

10

15

PG2

22

34

46

87

54

PG3

41

79

80

71

45

PG3

15

44

49

3

45

PG4

8

44

13

42

75

PG4

6

21

73

83

86

54

65

20

89

90

2 Jan 1777

5

89

36

85

23

PG1

4

20

72

44

85

PG1

68

16

28

87

20

PG2

77

7

45

90

3

PG2

49

2

14

10

58

PG3

64

73

18

37

31

PG3

11

26

79

7

39

PG4

89

31

90

79

20

PG4

71

74

11

76

40

16 Nov 1776

77

83

32

41

44

16 Jan 1777

10

52

49

84

35

PG1

62

29

8

52

57

PG1

83

10

64

38

73

PG2

75

36

63

56

35

PG2

61

52

90

16

8

PG3

4

90

26

57

59

PG3

81

14

35

11

52

PG4

42

17

44

22

37

PG4

80

51

34

73

61

31 Oct 1776

Source: Annonces, Affiches, et avis Divers (1, 16, and 31 October, 16 November, and 7 and 16 December 1776, and 2 and 16 January 1777

to prizes in the same way as had always been the case. But gamblers who qualified were offered more chances, based on the added draws. Part of the confusion was the complex rule for qualification: basically, that qualified bettors had to make a specified number of bets (a rule that encouraged gamblers to make more bets), and had to declare their interest in advance, paying a small surcharge depending

52  C h a p t e r S i x

Tabl e 6.2. Payoffs under the rules for the primes gratuites, ignoring the small buy-in required. For example, if a bettor won on a terne in the initial draw, a unit bet would return 5,200 units, but if he lost on the initial draw and on the first prime gratuite, but won on the second prime gratuite (his last chance with a bet on a terne), he would receive 300 units for a unit bet. Win with the initial five draws

Win with the first prime gratuite

Win with the second prime gratuite

Win with the third prime gratuite

Win with the fourth prime gratuite

Extrait

15

 

 

 

 

Extrait déterminé

70

 

 

 

 

270

 

 

 

 

Ambe déterminé

4900

500

 

 

 

Terne

5,200

500

300

 

 

70,000

15,000

9,000

6,000

 

1,000,000

80,000

60,000

40,000

20,000

Bet

Ambe

Quaterne Quine

upon how many bets they made. As an incentive to make more bets, the surcharge was low, varying from three deniers each for six bets (the smallest number covered) down to one-twelfth of a denier per bet for thirty-one or more bets. In the money of the time, twelve deniers equaled one sou, and twenty sous equaled one franc. The number of additional chances a bettor received varied by bet. For example, a single bet on a terne that was successful in the first drawing of five would win 5,200 units as before, and would be paid that amount. But if the bet was unsuccessful, there was a second chance with the second set of five (the first extra set), and if that set resulted in a terne it would win 500 units. And if this was unsuccessful in reaching a terne, the second extra set of five provided another chance: if it gave a terne, it would win 300 units. That would be the last chance for a terne. Bets on a quaterne would go one step further, allowing use of the third extra set, and bets on a quine could use all of the four extra sets. An ambe déterminé would get a single extra

The Introduction of Bonus Numbers  53

Tabl e 6.3. The expected return on ten units bet. The first column gives the expected return without the primes gratuites. The last column gives the total expected return with the primes gratuites, ignoring the small buy-in required. Win with the initial five draws

Win with Win with Win with Win with the first the second the third the fourth Total expected prime prime prime prime value for a gratuite gratuite gratuite gratuite ten-unit bet

Extrait

8.33

 

 

 

 

8.33

Extrait déterminé

7.78

 

 

 

 

7.78

Ambe

6.74

 

 

 

 

6.74

Ambe déterminé

6.12

0.62

 

 

 

6.74

Terne

4.43

0.43

0.26

 

 

5.11

Quaterne

1.37

0.29

0.18

0.12

 

1.96

Quine

0.23

0.02

0.01

0.01

0.00

0.27

set. In every case, wins that used the extra draws paid much less than the wins that did not, but they at least gave some reward. For the ambe déterminé, the net effect of the primes gratuites was to increase the expected winnings by about 10 percent: for ten units bet, the expected return improved from 6.12 to 6.74 (see table 6.2). For the terne, the corresponding increase was about 15 percent, from 4.43 to 5.11 (see table 6.3). The change did not make any of these an attractive bet: there was still only slightly more than a half unit paid, on average, for a one-unit gain in expected winnings. For the quaterne the increase was 43 percent, and for the quine 20 percent; both remained poor investments. Primes gratuites did give an added incentive for gamblers to increase their wagers in order to qualify, but the complicated rules for qualification and payout put a significant burden on the sales staff, and there was the added worry that agents might fail to distinguish between wins with and wins without extra draws, a slip that could

54  C h a p t e r S i x

lead to bankruptcy. And the need to perform five full drawings twice a month, presumably following the full procedures of the May 1758 notice, would have worn down the operators as well as the audience. When the primes gratuites were canceled, the payoffs on the first three of the four bets affected were increased by 4, 5.8 and 7.1 percent respectively, while the bet for the quine was unchanged. This was meager compensation indeed; the bettors had in all cases higher expected winnings with primes gratuites. Adding 200 to the unit bet payoff with the ambe déterminé (whose chance of winning remained at 1 in 8,010) was less than adding an additional opportunity at the same chance to win with a 500-unit payoff, as primes gratuites allowed. There were eight drawings that included the primes gratuites, extended over four months. Table 6.1 gives the results for all of these, as published in an official newspaper of the time. There seem to be no accounts of these other than the results. One can imagine the effort involved in reconstituting the wheel of fortune after each drawing of five numbers, and the temptation to draw without replacement. But the duplication in results testifies that any such temptation was resisted. Furthermore, statistical tests do not point to any evident lack of mixing, as might be expected if bored or exhausted personnel refilled the wheel carelessly time after time for the extra draws. The return to single drawings in February 1777 must have come as a relief for everyone.



Seven



The Spread of the Loterie in Europe

B e f or e 1 7 5 7 t h e r e i s no r e c or d of lotteries on the Genoa plan other than in some Italian cities and in Vienna. In the 1760s that began to change. In 1761 drawings commenced in Brussels and Münich (figure 7.1), two years later in Berlin and Spain, and by 1771, twenty-four more German states had initiated their own versions.1 This was a phenomenal growth; with few exceptions these were independent operations, and it is likely that many were single offices. Still, in each case there was a complex operation to be set up anew, and arrangements for financial backing capable of withstanding ill fortune had to be found. The payouts varied slightly, and the bets that were offered varied as well. In two cities the structure differed from that of the French Loterie: in Ludwigsburg six numbers were chosen from one hundred, and in Wiesbaden six were drawn from ninety (table 7.1). All of these new lotteries allowed the bet on the quaterne, a bet that Paris only included with the 1776 expansion. Many of them permitted the bet on an extrait déterminé, mostly at a payoff of seventyfive, making the choice between betting five units on an extrait simple and betting one unit each on the same number as five extraits déterminés, a choice among equals, as opposed to the lesser payoff of seventy that Paris introduced in 1776. Four cities offered the quine. There were a number of other small variations.

56  C h a p t e r S e v e n

F ig ur e 7. 1 . A 1761 broadside for the new Munich lottery, closely following the form of the 1736 Venice broadside seen in figure 2.1, but omitting the saints’ names.

The source of most of these data cannot tell us how long these lotteries lasted.2 Some, such as the ones at Altona and Copenhagen, lasted at least through 1820, by which time they had joined with a new lottery at Wandsbek in a group around Hamburg; see chapter 10. The Brussels lottery was joined to the French in 1800, after the Revolution when Brussels became part of French territory. But after Napoleon’s 1814 defeat, the drawings were moved to the French city of Lille. Quite a few of these operations must have failed, as a different “Klasse” lottery of the raffle type came to dominate in Germany from the 1780 on. Amid all this growth, another question arises: What about England? As we saw in chapter 2, there was a brief example of a Genoese lottery in England in 1662. That lottery is known only through a single-sheet prospectus, and it may never have become established,

Tabl e 7.1. Paris, Brussels, and twenty-six German cities, and the years each introduced the Loterie, with the payout odds they offered for bets Year Extrait Extrait determiné Ambe Terne Quaterne Quine Paris

1758

15

–

270

5,200

–

–

Brussels

1761

15

75

270

5,300

60,000

–

Munich

1761

14

67

240

4,800

Berlin

1763

15

75

270

5,300

60,000

–

Mannheim

1764

15

75

270

5,300

60,000

–

Würzburg

1767

15

75

270

5,300

60,000

–

Augsburg

1768

15

–

272

5,350

60,000

100,000

Coburg

1768

15

75

270

5,300

60,000

100,000

Hildburghausen 1768

15

–

270

5,300

60,000

300,000

Ansbach

1769

15

70

270

5,300

60,000

–

Koblenz

1769

15

75

272

5,350

60,000

–

Mainz

1769

15

75

270

5,300

60,000

–

Stralsund

1769

15

–

270

5,350

60,000

–

Wiesbaden*

1769

13

70

250

5,000

50,000

Bonn

1770

15

75

275

5,400

60,000

–

Köln am Rhein

1770

15

75

272

5,350

60,000

–

Dillingen

1770

15

75

280

5,400

70,000

350,000

Gotha

1770

15

75

270

5,300

60,000

–

Hamburg

1770

15

75

270

5,300

60,000

–

Regensburg

1770

15

75

280

5,400

60,000

–

Neustrelitz

1770

15

75

275

5,400

62,000

–

Altona

1771

15

70

270

5,300

60,000

–

Braunschweig

1771

15

75

270

5,300

60,000

–

Copenhagen

1771

15

70

270

5,300

60,000

–

Eutin

1771

15

75

270

5,500

70,000

–

Friedberg

1771

15

75

280

5,400

60,000

–

Ludwigsburg*

1771

16

–

300

6,000

70,000

–

Wetzlar

1771

15

75

280

5,400

60,000

–

*Two states had variations in the structure. Ludwigsburg drew six numbers from a set of one hundred with different odds, but offered bettors an option of “game A,” in which they could bet on only the first five drawn from one hundred. The odds given are for game A. Wiesbaden drew six from ninety, and the odds it gave reflected the greater chance the bettor had to win. The data on the German states other than Bavaria is derived from May 1771, 138– 211. The data for Munich comes from figure 7.1.

58  C h a p t e r S e v e n

but the sheet gives some information about what was offered. It was consciously patterned after the Genoese lottery, with the change that it involved drawing five balls from 100 rather than from 120, as it claims was the custom in Genoa. There are clear typographical errors and some ambiguity, but the bets available appear to have included the extrait, the ambe, and the terne. The payoffs were very conservative. The extrait would pay 100 for a successful bet of 7 (or about 14.3 for 1), the ambe 200 for a bet of 1, and a terne 2,000 for a bet of 1. “Fair” bet payoffs for a 100-ball Genoese lottery would have been, respectively, 20 for 1, 495 for 1, and 16,170 for 1.3 The failure to establish a version of a Genoese lottery in 1662 may be taken as indicating English unwillingness to undertake this type of risk. There seems to have been no successful effort to do so before the twentieth century. The lack of success does not necessarily indicate a lack of effort; and there is a tantalizing bit of evidence that Casanova did try to bring the Loterie across the English Channel. Casanova visited England for nine months, from June 1763 to March 1764, and his memoirs tell us that he initially planned to bring to England what he had helped bring to France. He traveled with a letter of introduction to Lord Egremont, a likely contact at that time, and a lottery proposal in hand. But Egremont died in August 1763, possibly before Casanova could even make a pitch, and there is no indication in his memoirs that he pursued the project further.4 The clue comes from a large collection of ephemera that a British publisher named Andrew White Tuer assembled in the 1880s. English lotteries were among Tuer’s wide interests, and he collected bills, broadsides, tickets, pictures (including the Hogarth 1721 in figure 4.2), plans for new lotteries, newspaper stories, and more. His collection supplied the copious illustrations for Ashton’s History of English Lotteries (1893).5 A good part of this collection survives in a large scrapbook, and at one point in that scrapbook there is a three-page handwritten letter, in French.6 The letter is not a draft (or there would be some alterations or edits; there are none), but it is also not as sent, since it lacks a salutation, an address, and a signature. It appears to be a secretary’s fair copy of the text of a letter. Even without a salutation, the text indicates the

The Spread of the Loterie in Europe  59

target: “Mylord North.” The letter proposes a lottery, without details. A reasonable guess is that the letter was intended to go to Fredrick North, Lord North, during the years 1768 to 1769 when North was chancellor of the exchequer for King George III. Lord North (1732– 92) is better known as the British prime minister during the American War of Independence, and was christened ungenerously by one biographer as “the prime minister who lost America.” While chancellor, North used the existing system of state lotteries to help balance the budget, first by offering lottery tickets as incentives for financing loans, and then introducing a revenue lottery in 1769. Those lotteries were on the traditional British raffle or “blanks” plan: sixty thousand tickets to be sold for thirteen pounds each, and some of the proceeds then distributed as prizes in a drawing.7 North would have been known as being sympathetic to new lottery ideas, and an attractive target for anyone with a new idea to sell. The letter promises a lottery “susceptible to mathematical demonstration” that could give the Crown two million pounds with each drawing, offering bettors a great range of bets at prices for any level of pocketbook. The author says that the scheme “will appear absurd or incredible at first sight,” but offers to perform a demonstration that will reconcile the plan with reason and common sense. He offers, if invited, to meet with North and furnish this demonstration— saying, however, that before he will reveal the key idea, he will ask North to give a secure guarantee that his labors will be suitably rewarded should the plan be adopted. He also says that while the execution of the scheme will be easy, it has “cost years of work and study to a man who, without being what one calls a promoter of projects, has made his principal occupation the science of commerce.”8 But why attribute the letter to Casanova, especially since there is no detail that even establishes the proposal as a Genoese lottery? Casanova had been in England four years earlier, and could well have been following the financial situation there. Some parts of the letter seem to echo the terms and strategy Casanova used with success in Paris in 1757: a clear mathematical demonstration, a variety of bets such as the Genoese lottery could offer but others could not. The let-

60  C h a p t e r S e v e n

ter also withholds details until the author is assured of compensation. But other parts are at variance from a Genoese lottery: the letter seems to propose a sequence of large lotteries rather than a constant series, and its allusion to bettors being able to resell tickets is more in the spirit of a blanks lottery. It is also doubtful that Casanova would have claimed that his main occupation was the science of commerce. Two scholars to whom I have shown the letter have independently suggested that another possible author was Ange Goudar (1708– 91), a French-born adventurer and entrepreneurial fellow active at the time who was associated with the “science of commerce.” He had met Casanova in England in 1764, and he is a plausible alternative candidate. It seems impossible to identify the author based on current evidence; a search in the archives of Lord North for a copy of the letter as sent would, if successful, provide the answer. In any event, the letter is testimony to a continued interest in England in such schemes.



Eight



Data Security: The Design of the Tickets

In se t t ing up t he Lot er ie in 1 757 and then expanding it in 1776, the organizers faced a number of formidable challenges. Some of these, such as the need to establish public confidence in the fairness of the draw and in the certainty that winners would be promptly paid, had faced all public lotteries since the 1500s. Others were without precedent because of the national scale of the enterprise and the large number of offices, such as that of protecting against costly errors by novice employees handling complicated transactions, and guarding against a type of fraud peculiar to this type of lottery. These problems dwarfed those of all previous lotteries in both scope and complexity. Previous lotteries had the problem of selling a huge number of tickets and carrying out a weeks-long drawing, but they were protected by the structure of the game. The total pool of prizes was fixed, and while fraud could lead to the awards going to the wrong people, the total being paid out had a guaranteed upper limit. Not so with the Loterie; the total to be paid out was potentially without limit. Duverney had put the question to Casanova: “Will you not admit that at the very first drawing the king can lose an immense sum?” The operators of the Loterie were protected by the principles of probability theory, but fraud does not respect theory. The dangers included lack of proper procedure in the act of drawing, tickets being sold after the draw was completed, and forged tickets. The first

62  C h a p t e r E i g h t

of these was not new, but the procedures put in place for the second drawing in 1758 seem to have worked; see chapter 10. The second, namely tickets sold after the drawing, required vigilance and carefully monitored access to the Loterie’s registers, the authorative record of who bet and on what. This too seems to have worked well; but even so, some fraud did occur. In November 1798 the official newspaper Moniteur Universel (6 Brumaire, an 7) reported that one bettor named Bodin claimed a 814,000-franc prize for a quaterne, but that after investigation he and one Commeau, the agent in Angers who had sold him the back-dated ticket, were instead awarded twenty years in irons. The Directoire, it was announced, was taking new measures to prevent such frauds in the future. Of course, the number of undiscovered such frauds is not known. The third possibility, forged tickets, had more urgency than in previous lotteries. In “blanks” lotteries every entrant had a different number, and at worst a forged ticket would lead to the wrong person receiving a prize, not to the lottery losing money. With the results being widely advertised and winners being eager to claim prizes, such cases would be rare and could be dealt with individually. But with the Loterie, the office would not know how many winners there had been, and would be in the position of paying out to all claimants. For the major prizes an agent could go through the registers to check the bets in painstaking detail, but in the pre-computer age the check was not feasible for the majority of cases. This left the office with the challenge of designing tickets and a system for recording bets that made false claims very difficult. The plan adopted was this: A bettor could appear at any of the many bureaus and make a choice of bets and amount. The agent would give the bettor a receipt, and would also copy the details into the register. Figure 8.1 shows a receipt from 1786, signed by two Loterie agents as part of the guard against fraud. The receipt itself had little standing: the official document the bettor would need to claim a prize was a ticket, specially printed in the Loterie office, that could be claimed by the bettor a few days later on presentation of the receipt. It was the bettor’s responsibility to check the receipt and the ticket and note any discrepancies. If there

Data Security  63

Figur e 8.1. A receipt from a bet on the first drawing (premier tirage) of June 1786. The bet was placed at bureau no. 116 in Paris, specifically on five extraits déterminés— each on the number 45, but for each of the five possible slots. Each bet was for eight francs (huit livres), so the total cost to the bettor was forty francs.

were problems, the most he could reclaim was the price he had paid. The ticket itself was a substantial document, designed to eliminate the possibility of forgery. Figure 8.2 shows an example from 1775. In all cases, each number was paired with a unique person’s name, usually that of a young woman or occasionally a saint. The hope ini-

64  C h a p t e r E i g h t

F ig ur e 8 . 2 . Ticket from 1775, the 199th drawing of the Loterie. The tickets were substantial documents, measuring 5 × 8 inches.

tially was that the names would be attractive for sales: that news of help for the poor or regard for a saint would enhance the popularity of the Loterie and encourage betting, much like the modern announcements that proceeds from a state lotto will go to education, with no guarantee that there will actually be a net addition to the education budget. The young women whose names were used, some

Data Security  65

as young as fourteen, were suggested by charity organizations; and if their names were selected as paired with winning numbers in a drawing, they would be entitled to a few hundred francs to serve as a dowry, thus enhancing their chance of good marriages. The same practice had been followed in Italy earlier; see the list of names in the 1736 Venetian broadside in figure 2.1, where each number was associated with both a woman and a saint. When a woman’s name was selected as a winner, a new name would replace it. In at least the early drawings, the women were shown as examples of innocent virtue. But this did not work well; a number of winners either died before marriage or never married, and by 1772 nearly a half of the dowries awarded were unclaimed.1 In addition, this practice would also have added to the administrator’s tasks, given the constant need to change the names associated with winning numbers after each drawing; and with the expansion in 1776, the Loterie used only first names, and did not change them after winning draws. The example from April 1775 (figure 8.2) shows a bold pattern of bets. The bettor had bet a total of twenty-four francs: forty-eight sous on each of the five extraits simples, and twelve sous on each of the ten ambes and on each of the ten ternes, for a total of 480 sous, or twenty-four francs. The ticket has two official signatures. The five young charity cases whose names appeared on the ticket were, of course, unlucky at that particular drawing (they would receive no contribution, since their numbers were not drawn); the survival of the ticket is testimony to that. Three of them would have won later: Michel Dufresne (no. 76) in October 1775, Claude Vitry (no. 17) in December 1775, and Jeanne Daniel (no. 86) in March 1776. The other two were less fortunate; this part of the program, the charity feature of the Loterie, was eliminated with the September drawing of 1776. From 1758 to 1776, a complicated printed pattern was uniquely appended to each number bet; an official seal was also printed on the ticket. The complicated codes for the date and numbers would have been difficult for a forger to duplicate. The use of such patterns in currency has a long history, and while it no doubt was a deterrent, the banks had long known there would always be some forgers with

66  C h a p t e r E i g h t

the talent to attempt to imitate the pattern. Wennerlind has noticed an interesting example, from the British Royal Mint in 1693, where this awareness could have been exploited with regard to patterns on coinage. The Mint knew that if the forgery was sufficiently difficult and a false coin was found, at least the list of suspects would be much shorter than otherwise, and there would be fewer “usual suspects” to interrogate.2 The practice of placing a complicated coded pattern for every number on a ticket was also discontinued when the Loterie expanded in 1776. The number codes had been found to be unnecessary, as well as impossible to memorize and cumbersome to check. It was found that assigning only the woman’s first name to the number served the same purpose quite well. The simplest variety of fraud was to alter a printed number by hand after the drawing. That alteration was relatively easy, but completely changing the woman’s name was not, and clerks could soon develop sufficient memory to spot mismatches of names to numbers. On the other hand, in the 1770s the Loterie had introduced a complicated code for the date of the drawing to appear at the top of each ticket (see again figure 8.2). That code remained on tickets after 1776, as a guard against a legitimate but unsuccessful ticket being presented when its numbers had appeared again as winners in a later drawing. The Loterie had been expanded to two drawings a month in 1776, and the code for the drawing also captured that change: see the symbols surrounding the word “second” (for the second drawing of April 1783) in figure 5.1.



Nine



The Loterie and the Revolution

T h e e x pa n de d L o t e r i e wa s a g r e at s u c c e s s. By the end of the 1780s, tickets for the Paris drawings were sold in more than eighty district offices near Paris, and at many dedicated offices in Lyon, Bordeaux, Strasbourg, and Lille. A 1790 budget for supervisory and printing staff lists total salaries of 549,473 francs annually for more than three hundred people. The eleven men at the rank of director or director general received 5,000 or 6,000 francs a year. A woman who took care of the linen at the head office was the lowest paid, at 300 francs a year. The profound effect of the French Revolution on the country was initially unnoticed by the Loterie. In July 1789 the Bastille fell, but the Loterie did not miss a drawing. In January 1792 King Louis XVI was convicted, and in January 1793 he was executed; in neither case did the Loterie miss a drawing. In October 1793 Marie Antoinette was executed, and again the Loterie did not miss a drawing. But by then the nation was sinking more deeply into the period known as the Terror, and three days after Marie Antoinette’s death the National Convention issued a decree suspending all lotteries except what was now known as the Loterie de France. Casanova had predicted that the Loterie would be a great success as long as the people had faith that they would be paid if they won, and the Terror surely had shaken that faith. In November the Na-

68  C h a p t e r N i n e

tional Convention finally decreed that all lotteries were suppressed, surely more a recognition of the lack of business and a need to stop administrative expenditures than an independent action. The decree of 15 November 1793 was cryptic, as if written in great haste: I.

Lotteries, of whatever nature and under whatever name they exist, are suppressed. II. As of today, no other drawings can be made, other than those which should take place by reason of their bets being authorized for the current month. III. The finance committee is responsible for presenting without delay a draft decree on the measures to be taken to ensure special interests. IV. The insertion of this decree in the bulletin will take the place of promulgation.

The fact that they did recognize the validity of the bets made to that date essentially kept the door ajar for another day. After the second drawing in November 1793— the 628th drawing in its history— the Loterie closed all its offices for the first time since April 1758. Despite the official closure in 1793, there is at least one sign that not all was quiet unofficially. A surviving ticket shows that at least some drawings of the “Petite Loterie Parisienne” were held in 1795, offering prizes up to and including the terne (figure 9.1). The financial situation of the various governments worsened, and renewing the Loterie offered at least a partial solution. Old arguments about the moral value of the enterprise held back any action until late 1797, when the situation had somewhat stabilized and the new government, the Directoire, needed money to a degree that quieted such reservations.1 On 8 October 1797 (17 Vendémiaire, an 6, of the Revolutionary calendar) the Loterie was formally reestablished as the Loterie nationale. Other than the name, there were no visible changes: the same bets, the same payoffs, two drawings a month. The first drawing for the Loterie nationale was on 1 December 1797. The earliest drawings were priced in francs and sous; but soon the metric system was recognized and centimes replaced sous.

Fig ur e 9.1 (top) A surviving ticket from an unofficial but evidently sanctioned drawing in Paris, 8 October 1795. The bettor bet on the numbers 27, 61, and 81 as the extraits, three ambes, and a terne— a total of seven bets. (b o t t om ) A ticket from the first drawing of the Loterie after it was reestablished after the revolution in Frimaire, an 6 (1 December 1797):similar betting, but on 12, 36, and 76.

70  C h a p t e r N i n e

T abl e 9 . 1. Days of the month on which cities held Loterie drawings Lille

1

11

21

Bordeaux

2

12

22

Paris

5

15

25

Strasbourg

7

17

27

Lyon

9

19

29

By September 1800 the financial needs and ambitions had grown, and a significant expansion was announced (table 9.1). In November the number of drawings per month in Paris went to three, and separate drawings would now be held in Bordeaux, Brussels, Lyon, and Strasbourg, each also three times a month, but on staggered days, so that the total of fifteen drawings a month in all of France was evenly spaced across the calendar. Gamblers in Paris could buy tickets at any of the regional offices; regional bettors could also buy tickets in Paris, as before. Brussels began its operations in November; Lyon began its own drawings in the following month, but Bordeaux and Strasbourg only opened in May 1801. For the remainder of its lifetime after the 1800 expansion, the Loterie saw only two changes. The option to bet on the quine was eliminated in October 1804, possibly because of renewed worry about real or potential fraud. At that same time, in recognition of Napoleon’s self-proclaimed new title of emperor in December 1804, the name was changed from Loterie nationale de France to Loterie impériale. A decade later, after Napoleon’s defeat and abdication in April 1814 and the restoration of Louis XVIII to the throne of France, the name changed again, to Loterie royale, an ironic return to the 1780s. In recognition of the same change in political fortunes, now that Brussels was part of the Netherlands, the Brussels office was moved to the French city of Lille. The other change also was related to the expansion of the French empire, and also saw a quick reversion in 1814. Napoleon had been impressed by the profits the Loterie generated, and by 1810 had

The Loterie and the Revolution  71

added lotteries of the same type in Turin, Milan, Genoa, Rome, Florence, and Hamburg to the portfolio of France. In most cases this was simply a matter of taking over existing enterprises, as in the Italian cities that had preceded the French in establishing them. Hamburg, where lotteries under the same plan had flourished since the 1770s, required a new central office and associated expenses. Of course, these lotteries all returned to their previous directors in 1814. The effect of political turmoil on the Loterie was again merely transitory. In 1776 the Loterie had returned about two million francs a year to the École militaire. In the first year after it was restored after the Revolution, it showed a profit of more than five million francs. In the year after its expansion in 1801 the profit reached nearly thirteen million, and in 1810 it reached the peak of more than twentyfour million francs. Table 9.2 shows that in the years 1797 through 1828, the only deficit was a relatively small amount of 376,697 francs in 1814. One can imagine that there was added expense in moving an office and coping with general turmoil as the armies of Europe descended on France, but the accounts do not show the deficit as being due to expenses. Rather, in that year there was an unusually high amount paid out to winners, reaching 90 percent of the amount taken in, far above that for any other year. A cynic might wonder if, in the midst of turmoil, the checks on fraud had weakened significantly. By the 1820s the profits were consistently back around ten million francs a year. In 1757 Casanova had promised the king’s ministers a return in the range of one franc for every five collected, or 20 percent. The average return over the years covered by this accounting was about 28 percent, excepting the anomalous year 1814, when it was only 10 percent. As a poignant reminder that even large operations such as the Loterie had also to deal with small matters, which in turn could have unexpected political implications, I offer this surviving 1802 letter (figure 9.2) from a debt collector for the Loterie in Bordeaux, written to the head office in Paris, reporting on a lack of progress.

21,910,578.59

36,775,494.68

31,033,043.71

53,312,438.30

74,911,549.71

75,707,013.55

70,155,196.05

69,302,946.40

77,822,983.70

74,371,181.35

71,475,505.45

72,600,478.60

88,263,502.20

63,535,250.85

70,433,043.00

7

8

9

10

11

12

13

14&1806

1807

1808

1809

1810

1811

1812

Received

6

Year

50,820,879.50

40,465,787.50

56,760,277.00

52,948,263.50

52,289,254.25

56,018,705.25

58,433,460.85

50,474,273.50

49,113,742.00

54,173,477.25

56,192,651.75

39,765,546.75

20,075,041.60

25,174,781.94

14,224,620.73

Paid out

72.15%

63.69%

64.31%

72.93%

73.16%

75.32%

75.09%

72.83%

70.01%

71.56%

75.01%

74.59%

64.69%

68.46%

64.92%

Percentage paid out

19,612,163.50

23,069,463.35

31,503,225.20

19,652,215.10

19,186,251.20

18,352,476.10

19,389,522.85

18,828,672.90

21,041,454.05

21,533,536.30

18,718,897.96

13,546,891.55

10,958,002.11

11,600,712.74

7,685,957.86

Profit

6,717,058.92

6,326,820.15

7,303,304.97

6,644,350.52

6,383,379.55

6,150,211.78

7,115,868.32

5,857,101.46

5,883,552.60

6,187,864.84

5,784,434.68

4,716,193.91

3,546,424.25

3,331,728.40

2,330,031.18

Expense

34.25%

27.43%

23.18%

33.81%

33.27%

33.51%

36.70%

31.11%

27.96%

28.74%

30.90%

34.81%

32.36%

28.72%

30.32%

Exp/ profit

12,895,104.58

16,742,643.20

24,199,920.23

13,007,864.58

12,802,871.65

12,202,264.32

12,273,654.53

12,971,571.44

15,157,901.45

15,345,671.46

12,934,463.28

8,830,697.64

7,411,577.86

8,268,984.34

5,355,926.68

Net profit

T abl e 9 . 2 . Accounts for the Loterie from its reestablishment in 1797 through 1828. The Revolutionary calendar year began after the third week of September, so the year “14 & 1806” runs a bit longer than fifteen months. Excerpted from table 24 in the Moniteur Universel, 14 April 1830, with a few typographical errors corrected.

33,287,936.75

32,074,443.65

42,463,015.85

47,987,918.90

58,867,061.30

53,609,933.10

57,622,254.10

51,401,279.70

52,363,138.65

49,399,305.15

50,909,529.60

57,256,801.80

51,354,765.11

51,735,728.45

53,183,007.25

1,771,896,217.75

1814

1815

1816

1817

1818

1819

1820

1821

1822

1823

1824

1825

1826

1827

1828

Totals

Average

76,769,892.25

1813

1,280,267,415.57

38,313,456.10

40,429,392.45

39,452,959.35

41,799,217.95

38,240,664.00

33,200,253.15

35,148,330.55

37,458,713.35

35,147,150.30

42,970,763.50

43,461,504.50

37,966,845.25

29,411,107.00

21,338,363.50

29,978,929.00

59,019,002.25

72.25%

72.04%

78.15%

76.82%

73.00%

75.11%

67.21%

67.12%

72.88%

61.00%

80.15%

73.83%

79.12%

69.26%

66.53%

90.06%

76.88%

491,628,802.18

14,869,551.15

11,306,336.00

11,901,805.76

15,457,583.85

12,668,865.60

16,199,052.00

17,214,808.10

13,942,566.35

22,475,103.80

10,639,169.60

15,405,556.80

10,021,073.65

13,051,908.85

10,736,080.15

3,309,007.75

17,750,890.00

156,392,686.83

4,211,600.88

4,170,373.32

4,170,438.31

4,637,477.86

4,520,481.95

4,345,230.54

4,565,306.36

4,682,675.55

4,852,538.01

4,968,539.14

5,080,307.40

4,380,897.40

3,812,980.41

3,077,829.55

3,685,705.10

6,951,979.52

31.81%

28.32%

36.89%

35.04%

30.00%

35.68%

26.82%

26.52%

33.59%

21.59%

46.70%

32.98%

43.72%

29.21%

28.67%

111.38%

39.16%

10,814,068.24

335,236,115.35

10,657,950.27

7,135,962.68

7,731,367.45

10,820,105.99

8,148,383.65

11,853,821.46

12,649,501.74

9,259,890.80

17,622,565.79

5,670,630.46

10,325,249.40

5,640,176.25

9,238,928.44

7,658,250.60

–376,697.35

10,798,910.48

74  C h a p t e r N i n e

Bordeaux le 17 pluviose an 10 [6 February 1802] Monsieur It is indeed a very long time since I wrote to you, but you must attribute this delay to the desire I had to give you better news than what I have to offer you in relation to the matters on which you have engaged me to investigate. The situation I have learned is that Mr. de Kirwan living in Bordeaux is not the one you are looking for as having been an officer in the Guard of Monsieur and a debtor of 18,000 [francs]. Mr. de Kirwan in Bordeaux is the father of the man in question who lives in England; I was not told in which part, but only that he had married a wealthy woman. It seems that there is no correspondence between the son and the father. Mr. Séjourné has not yet paid me the 48 francs you claim and yet I have been to his house five or six times; at first he promised to hand them over to me shortly, but the last time I visited him he gave me a long lamentation of his losses in the court of the Revolution; those he had experienced through an administration in which you were formerly employed; I tried to make him understand the injustice of the wolf, in the fable of the wolf and the lamb; however I cannot give you the assurance that your hopes for the return of these funds will not be crushed. I promise you I will try again to touch his soul; I will go see him and give you an account of the fruit of my approach. . . . With my warm salutation, Bermond

The Kirwans can be identified. The debtor was François David de Kirwan (1769– 1827?), a part of the “Guard of Monsieur,” the guard of the elder brother of the recently executed King Louis XVI. That brother, also known as the Count of Provence, survived the Revolution, traveling through Europe and eventually on to exile in England, and would return to France as King Louis XVIII with the Bourbon Restoration in 1814. Clearly, attempts to collect that large debt would take Mr. Bermond into delicate political matters in 1802. Kirwan’s estranged father in Bordeaux was Mark Kirwan, famous for

The Loterie and the Revolution  75

Fig ur e 9.2. The first page of Mr. Bermond’s letter of 6 February 1802.

a very different achievement. Born in Galway, Ireland, he was in 1802 the proprietor of Château Kirwan, then and now one of the highly ranked vineyards in the Margaux region of Bordeaux. That vineyard had been founded by an Englishman, John Collingwood. Mark Kirwan took it over after marrying Collingwood’s daughter in 1768, and managed it until his death in 1805. Thomas Jefferson was among its

76  C h a p t e r N i n e

clients and visited Château Kirwan in 1787. The vineyard was rated as Third Growth Grand Cru Classé in 1855. Mr. Séjourné may not have been as politically involved, and his debt was small, but his memory of the Revolution had not faded. In the fable of the wolf and lamb, the wolf comes upon a lamb and tries to come up with an excuse for eating it. To every excuse he offers, the lamb gives a cogent response that undermines the excuse, until the frustrated wolf simply eats the lamb; a wolf needs no excuse. How Mr. Séjourné made out in the end is unknown.



Ten



Was the Loterie Fair?

The v iabil it y of t he Lot er ie depended upon the validity of the random selection. You might agree with that statement, thinking it meant that drawing numbers with equal probabilities was needed to maintain public confidence; but that is not what is meant, and it is quite possibly not even true. Public confidence depended upon consistency, not equal treatment of numbers; and the public might well prefer inequality, with the opportunity to discover and profit from it. Rather, it was the king who depended upon the draw being reliably random. The calculation of payouts used simple counting methods, all based on the assumption that all draws were equally likely. If this assumption failed, it might have been possible for gamblers to profit at the government’s expense without resorting to fraud. If it failed badly, that bias would surely have been noticed and pounced upon. It would have been noticed because the Loterie published the results regularly in national newspapers, and independent publishers collected these results and sold them in aggregated form to potential bettors, including tables showing the relative frequency of numbers over time. It was necessary that the choices be random— truly random. Choices that were either too unbalanced or too balanced would cause problems. Any method of selection that permitted careful watchers to get an edge in prediction simply would not do. Neither players who

78  C h a p t e r T e n

bet on numbers that had come up frequently nor players who bet on numbers that had come up rarely should be systematically rewarded. The wheel of fortune used, a cage in which the balls could be observed tumbling as it rotated, certainly gave the impression of randomness (figure 10.1). Its use was popular in lotteries well before 1757, and versions of that wheel are still used in many lotteries today. Appearances can be deceiving; the dynamics of tumbling balls is not susceptible to easy analysis even today, and there are examples of failure to randomize. In the 1960s, the United States used such a device in a “draft lottery” designed to give the order in which young men would be chosen for military service, based on their birthdays. The agency in charge of the draft used a wheel of fortune that held 366 balls, marked in correspondence to the 366 days of the year, including February 29. In a given year, balls would be selected from the wheel one by one until all were gone, and the order of choice would be the order of birthdays for conscription that year. The wheel tumbled the balls for what all involved thought was a sufficient amount of time; but the balls had been inserted in batches related to the days and months, in order to be sure that all days were included once and only once, and the batching was not destroyed even with a long tumble. When the birth dates were cross-classified by month and the third of the draft order in which they fell (the first group of 122, the second group of 122, or the final group of 122), an association became evident (table 10.1).1 For the French Loterie, we will take as the source of data the small book pictured in the introduction to this volume: the 1834 edition of Almanach romain sur la loterie de France, published by Menut de St. Mesmin, directed toward people interested in betting on the Loterie (figure 10.2; see also figure 0.1).2 This almanac was one of a series of works Menut offered on the Loterie. Menut listed more than twentyfive of his other publications, including mathematical tables and a Répertoire Cabalistique that may have been a descendant of the book that got Casanova into trouble in the first place. The almanac itself contains all manner of useful information: the rules for the game, the locations of more than 150 betting offices in the Paris area, tables of

Fig ur e 10.1. A picture of a Loterie drawing from 1816. A woman in the audience is celebrating the draw of an 80, as she holds a ticket for that number (Anonymous, 1816).

T a b l e 1 0 . 1 . In 1971 the United States used a lottery to determine the order in which men would be called up for military service, by drawing successively from a wheel of fortune with the 366 possible birthdates until it was empty, the order of drawing being the order called up. The table shows the distribution among the twelve months, separately each for the first 122 drawn, the second 122 drawn, and the third 122 drawn. For example, among the first 122 dates drawn, 17 dates fell in December but only 9 in January. Statistical tests support the conclusion that the pattern observed would have been unlikely from a random drawing. The chi-square statistic is 37.16, with 22 d.f., a value that would be exceeded only 2 percent of the time with truly random selection (Fienberg 1971). Jan Feb Mar Apr May June July Aug Sept Oct Nov Dec Total 1st

9

7

5

8

9

11

12

13

10

9

12

17

122

2nd

12

12

10

8

7

7

7

7

15

15

12

10

122

3rd

10

10

16

14

15

12

12

11

5

7

6

4

122

Total

31

29

31

30

31

30

31

31

30

31

30

31

366

F ig ur e 1 0 . 2 . The author and publisher Menut de St. Mesmin at his desk, from the frontispiece of the small book Almanach Romain sur la loterie de France (Menut 1834).

Was the Loterie Fair?  81

the numbers of possible combinations, and advice as to which numbers were propitious. Menut told his readers which numbers were associated with which seasons, signs of the zodiac, and women’s names, as well as which numbers were “sympathetique” with which other numbers. He reported also on which of the numbers in each regional Loterie had not been drawn in a considerable number of drawings, and which pairs of numbers had not been drawn. But more to the point, Menut gave a considerable amount of data: the winning numbers drawn in every Loterie since the first in April 1758 (figure 10.3). He gave results through the end of 1833, with spaces for the reader to record those for 1834; and the first owner of this book obliged by recording the drawings for Paris and Bordeaux through October 1834. In all, 6,606 drawings are recorded, and since the Loterie was finally suppressed in January 1836, these constitute most (96 percent) of the drawings ever held. A cross-check with some newspaper accounts and a few partial listings (e.g., in Parisot 1801) verifies the remarkable accuracy of what was given. No formal tests of randomness were available before Karl Pearson introduced the chi-square test in 1900. Even then, two difficulties remained in the application of that test to this sort of data. The first of these was that even for the simplest question— namely, does the frequency of individual numbers appear as if randomly drawn?— allowance needed to be made for the fact that the groups of five were drawn without replacement. Naively, this would involve calculating the chi-square statistic for these data, a relatively simple measure of how far the frequencies in figure 10.4 are from being perfectly equal to one another, but a correction was needed. The correction is quite simple: rescale the chi-square statistic, multiplying by 89/85 and comparing the result to a chi-square distribution with 89 degrees of freedom.3 If that result is larger than 112.0 (which with perfectly random selection would only occur 5 percent of the time), it would indicate suspicion of the randomness. If that result were larger than 122.9 (which should occur but 1 percent of the time), the suspicion would be very strong that something was amiss. Of course with many tests we should not be surprised if a very few are near or above these values.

82  C h a p t e r T e n

Fig ur e s 10.3. The title page of Almanach Romain sur la loterie de France and the first page of data, showing the winning numbers drawn at the Paris Loterie from April 1758 to September 1761 (Menut 1834).

The results suggest there is no pronounced bias, whether one looks at all the data together or more narrowly at individual lotteries (table 10.2). Only in the Loterie de France (the Paris Loterie from 1776 to 1793) does the value approach a suggestive level; and considering the number of tests being done, that is hardly surprising. The second difficulty is of a different sort: we wish to know also about the frequency of pairs, triplets, and groups of four and five. Theoretically this can be overcome, but not in practice. Think of the simple case for individual numbers: there are 90 possibilities, and 6,606 × 5= 33,030 pieces of data. But in the extreme case of groups of five, there are 43,949,268 different possibilities and only 6,606 pieces of data, and quite strong biases could easily go undetected. Suppose,

Was the Loterie Fair?  83

F ig ur e 1 0 . 4 . The frequency of each of the integers 1 to 90 appearing among the winning numbers in 6,606 drawings of the Loterie; data from 1758 to 1834.

for example, that some flaw in the process limited possible draws of groups of 5 to only 4,000,000 equally likely possibilities, instead of 43,949,268. Then, the chance that 6,606 drawings would show no repeats at all (and thus not even a hint of bias) would be better than 0.75. There are some other tests that look at more than single numbers. For example, Harry Joe developed a test looking at the frequencies of all possible pairs that can be applied to the full data set.4 For the 6,606 drawings it gives a version of a chi-square of 3,884 with 4,004 degrees of freedom, well in accord with randomness. Another test is related to the classical “birthday problem,” where the usual question is, for a group of n people, what is the probability that two or more share the same birthday? The standard observation is that if n is at least 23 and all birthdays are equally likely, then the

84  C h a p t e r T e n

T a b l e 1 0 . 2 . The results of a statistical analysis of the frequency of the numbers 1 to 90, testing for uniformity in the drawings by single number. The table gives chi-squares (scaled up by 89/85) a measure of uniformity appropriate for this purpose, for all drawings and for the regional Loteries and different time periods as well. Only the Loterie de France (the Paris Loterie, from 1776 to 1793) shows a mild departure from uniformity, which is not surprising given the number of tests. For comparison, for 89 d.f., the 5, 3, and 1 percent points are 112, 116, and 123. Loterie All together, 6,606 drawings Ecole Militaire, 215 drawings Loterie de France, 413 drawings

Adjusted χ2 100.07 89.1

115.1

Paris, 1,289 drawings

74.9

Strasbourg, 1,161 drawings

90.6

Lyon, 1,171 drawings

109.8

Brussels, 1,170 drawings

84.3

Bordeaux, 1,187 drawings

79.6

chance is above a half that two or more will agree. It is usually not emphasized that too many birthday coincidences can be taken as evidence that birthdates are not equally likely (which is in fact the case). Here we have 43,949,268 “birthdays” (the number of possible drawings, unordered). The standard calculation tells us that the chance of finding at least one matching pair would be just above a half, with 7,806 “people”; we have n = 6,606, and for that number the chance of one or more matching pairs of drawings is 0.39. In fact, there is exactly one pair of drawings that match, though the numbers were drawn in different orders. The two draws were (on February 9) 33, 19, 78, 9, and 46, and (on August 29) 33, 78, 9, 19, and 46. Thus we judge the Loterie to pass this test as well, though the fact that both of these drawings occurred in Lyon in 1820, seven months apart, might raise an eyebrow. The quine was not allowed as a bet at that time, but a second eyebrow might be raised by the fact that at the August 29 Lyon

Was the Loterie Fair?  85

drawing, some “lucky” bettor in Paris won 42,224 francs on a quaterne by betting on 9, 19, 46, and 78, and that in the year 1820 there were five quaternes in all drawn in Lyon between February 19 and September 19, four with Paris bets winning, while the average number of quaternes drawn per year at Lyon between 1801 and 1834 was only 27/33 = 0.82, including these five. This is suspicious, but insufficient for conviction, considering the number of drawings involved in all Loteries. If we look for fourth-order agreements— pairs of drawings that agree in four of their five numbers, we find 233 matches. The expected number of matching pairs for n = 6,606 is 211, and a mathematical result due to David Aldous, known informally as “Aldous’s Poisson Heuristic”, gives the approximate standard deviation in this case as 15. The number of matches is high, suggesting some clumpiness that was not picked up in the tests of lower-order interactions, perhaps; but the tendency, if real, is quite slight. What other tests might be made? The possibility of serial dependence was considered by grouping the numbers 1 through 90 into thirds, and into decades, and in each case testing for Markov dependence within drawings (where a drawn number was influenced by the number drawn immediately before). In both cases, no sign of dependence was found.

The Lottery at Hamburg and Copenhagen Altona and Wandsbek are now boroughs of the German city of Hamburg, but from the mid-1600s until the mid-1800s they were under Danish rule, with the exception of their brief period from 1804 to 1814 as part of the French empire. In 1770– 71, Altona and Copenhagen began lotteries along the same lines as the Genoa lottery that the French had adopted in 1757, with drawings beginning in April 1771 in Altona, and in Copenhagen three months later. In 1774, Wandsbek followed their lead. Separate drawings took place twice a month at each of the three locations. None of these drawings allowed the quine, and their payoffs for the terne and quaterne were slightly smaller than those of the French drawings. As was the case in France, there were

86  C h a p t e r T e n

F ig ur e 1 0 . 5 . The Danish lottery calendar: (left) the title page, (middle) the first page of data for the Altona lottery starting in 1817, and (right) the summary table of counts of number occurrences for Altona from 1771 to 1826.

publications directed at potential gamblers, and these are a source of data about the drawing. The particular source used here is Königl. Dänischer Lotto Calender für das Jahr 1827— a small book much like the one Menut published for Paris— which included the results for the three lotteries in two forms: a detailed listing of the 170 winning draws in each of the three lotteries from the beginning of 1817 to October 1826, and tabular summaries of the frequency of occurrence of each of the 90 numbers from the 1770s to October 1826 (figure 10.5). An analysis of the tabular summaries makes it clear that they must contain a number of computational and/or typographical errors, since the totals correspond only roughly with what the correct totals would be, and the sums are not divisible by five as they would have to be (table 10.3). Of course, the more detailed data may also have errors; but since no calculation was needed to prepare the summaries for printing, there was less opportunity to err, and the French data were quite accurate where they could be checked. Overall, the results are like those for the French data: no sign of

Was the Loterie Fair?  87

Tabl e 10.3. The uniformity of the drawings of the Hamburg and Copenhagen lotteries tested using the 1817– 26 data from 510 drawings in all. Chisquares are scaled up by 89/85 for testing for uniformity in the drawings by single number, as in table 10.2. These results are very close to those for the full data from 1771– 74 to 1826, given only in aggregate form in the right panel of figure 10.5, for a total of approximately 3,445 drawings. Lottery Three cities, 510 drawings Altona, 170 drawings

Adjusted χ2 94.9 84.7

Copenhagen, 170 drawings

89.6

Wandsbek, 170 drawings

81.0

bias in the wheel of fortune, and certainly none that would have been obvious to gamblers. Here as in France, the operators had a strong incentive to produce unpredictable draws, and they succeeded. In the early years of those lotteries there was a prejudice against the Wandsbek lottery in the other districts. The lotteries were run independently, and winning tickets in one district were not honored in others. An old Danish expression dates to the era. To imply sarcastically that something was unreliable or fraudulent, one might say, “Det gælder ad Wandsbek til” (This is valid in Wandsbek). This expression is said to have originated after people tried to cash in tickets in Copenhagen from the Wandsbek lottery. Whether or not Wandsbek had earned this reputation is a matter of conjecture. I have found and tested some shorter series of drawings: Belgium (69 drawings from 1761 to 1765), Anspach (133 drawings from 1769 to 1777), and Mannheim (23 drawings from 1765 to 1766). In every case there was no detectable indication of problems.

Duvillard One archival record testifies to official concern in Paris about the fairness of the Loterie in 1818. It is a carefully constructed tabulation of the numbers drawn, found in the archive of the Swiss actuary

88  C h a p t e r T e n

Emmanuel-Etienne Duvillard (1755– 1832). Born in Geneva, Duvillard moved to Paris in 1773 and worked with the French government from 1776, first with Turgot’s ministry, and later with the statistical office in the Ministère de l’intérieur. From September 1812 he was bureau chief for general administration.5 In history Duvillard is best known for a life table he published in 1806 that apparently served as an important guide for insurance companies in France and England.6 He was an extremely careful analyst, numerically accurate and theoretically well-informed. He was just the sort of high-level civil servant the Loterie would have approached to ask for reassurance that the drawings were being correctly conducted. The single surviving document suggests that this is exactly what happened. The two-page document gives a detailed accounting of the draws of all the French Loterie offices from the beginning in 1758 to near the end of 1818,7 with a few variations: Paris through the third drawing in July 1818, Brussels/Lille through the first drawing in January 1819, Lyon through the second drawing in December 1818, Strasbourg through the third drawing in September 1818, and Bordeaux through the second drawing in November 1818.

These limits are not stated in the document, but my calculations in a modern reconstruction from published sources lead to this conclusion. My guess is that the table was constructed by Duvillard himself from available publications of the drawings; the care, accuracy, and internal consistency are characteristic of an extremely careful worker. The table is highly unusual for the time, in that it gives a three-way classification of the data, with all marginal totals. Duvillard does not settle for overall counts for the ninety numbers; he gives the counts by the place of each drawing (treating Paris as two locations, before and after the Revolution), and by where each number occurred (drawn in first, second, . . . or fifth place). His was a three-dimensional table of counted data with all one- and two-dimensional marginal totals. Since Duvillard had to break the table into two panels to fit on two

Was the Loterie Fair?  89

large pages, he also gave separate totals for the numbers 1 through 45 and 46 through 90, as well as the combined totals for all. All of this was done by hand, and the table was hand-lettered with careful legibility. Including row and column totals but not including row and column labels, it made a 94×42 spreadsheet. From Menut’s data I was able to check all of Duvillard’s figures, and with trivial exceptions the agreement was perfect. There were a very few errors in marginal totals, which is hardly surprising. Duvillard’s table was constructed with extreme care from data that agreed with Menut’s tables. The manuscript of Duvillard’s table was not accompanied by an analysis, but to a statistical eye attuned to random variation there is nothing here to point to a problem. The conclusion agrees with what I have found using a more complete data set with modern tests. This manuscript testifies to the careful oversight by the Loterie administration. The Loterie was fair as would be expected with the state acting in its self-interest: any systematic deviation would have been exploited, and would have harmed the state more than the bettors. Adam’s Smith’s invisible hand reached even the Loterie of France.



Eleven



Dreams and Astrology: The Bettors and the Loterie

A l o t t e r y t i c k e t i s a c on t r ac t between a buyer and a seller. The seller’s side is well determined, at least in the case of the Loterie, with a catalogue of available bets and prices, rules for delivery, and promises for payouts. The buyer— the gambling public— behaves in a less clearly determined manner. Consumer choice can be fickle and, especially from this distance in time, hard to track. Major complaints may lead to a record in court papers, and the surviving tickets, all of them losing bets, are not to be taken as a representative sample, and will not easily reveal the whims and strategies of a broad and varied clientele. Surviving tickets will oversample the long-odds bets. What bets were being made? For how much, and on what numbers? There is no known record of what bets were made. For example, did the public bet on simple extraits, or did they instead go for large (if rare) payouts with the terne or quaterne? (See table 11.1.) We will see that the most vocal critics of the Loterie invariably point to the very poor expected payouts for the quaterne (15 percent of the amount bet) and the quine (2 percent of the amount bet), claiming that the Loterie was a huge tax on stupidity. But did the public pay that tax? Evidently the public was not as oblivious to the “tax” as the critics thought. Table 9.2 in chapter 9 reports the income and expenditure for the Loterie from 1797 to 1828, and over that period the Loterie

Dreams and Astrology  91

T abl e 1 1 . 1 . The expected percentage return on a unit bet for each of the seven possible bets in the French Loterie, based on the payouts in force from 1777 on. The quine was not offered before 1776 or after 1804. Bet

Expected return

Extrait simple

15/18, or 83%

Extrait déterminé

70/90, or 78%

Ambe simple

270/400.5, or 67%

Ambe déterminé

5100/8010, or 64%

Terne Quaterne Quine

5500/11748, or 47% 75000/511038, or 15% 1000000/43949268, or 2%

paid out in prizes an average of 72 percent of the amount bet. If every bet had been placed on an extrait, the average percentage of return would be 83 percent; if every bet had been placed on quaternes, the return would be 15 percent. If only those two kinds of bets were allowed, the average return of 72 percent would only be achieved if the fractions of bets on them were 0.84 on extraits and 0.16 on quaternes. The weighted average is (0.84)(0.83) + (0.16)(0.15) = 0.72, or 72 percent. The bettors must have mixed their bets, but how? One common bet found in surviving tickets would have the bettor choose four numbers from 1 to 90 and make all possible simple bets: four extraits, six ambes, four ternes, and one quaterne, a total of fifteen separate bets. If each of those bets was for one unit, the expected winnings would be 62 percent of the total bet: {4(0.83) + 6(0.67) + 4(0.47) + 1(0.15)}/15 = 0.62, or 62 percent. Another common bet would choose three numbers and bet on all possible bets: the three extraits, the three ambes, and the one terne, a total of seven bets. If each of the seven bets was for one unit, the expected winnings would be {3(0.83)+3(0.67)+0.47}/7 = 0.71, or 71 percent of the total bet. The actual bets were often more complicated. Figure 9.1(b) in this

92  C h a p t e r E l e v e n

book shows one version of such a ticket, but with five sols bet on each extrait, two sols on each ambe, and three sols bet on the terne (twenty sols were then equal to a franc). With that choice of bets, the expected winnings would be {3(5)(0.83) + 3(2)(0.67) + 3(0.47)}/{3(5) + 3(2) + 3} = 0.745, or 75 percent of the total bet. Of course, it is not possible to determine the actual mix of bets, but the aggregate of all bets must have been a mixture much like these examples, although not always in those proportions on the same ticket. Evidently, the bettors were sufficiently savvy to focus on the simple bets, with concentration on the extraits, while placing some bets with a chance of a larger return. The realized tax on stupidity was much lower than the critics stated, and was in fact much lower than in most modern lotto games, where a 50 percent “tax” is common. In chapter 13 we will take a closer look at these bets. Let us turn to a different aspect of consumer choice, one that was clearly a major focus of the bettors: Which numbers should they bet? However much they bet, whatever mixture of their bets, success would be determined by the numbers chosen. And, as Casanova had insisted at the beginning, that choice was unconstrained. Determining from this distance in time which numbers they chose is a challenge. There seem to be two ways to approach this problem. The first is to examine surviving tickets. The difficulty is that these are small in number, and the route to their survival is idiosyncratic. I have acquired some tickets in scrapbooks and a few others on eBay, less than two dozen in total, and I have visited museum collections including the J. J. Johnson collection of ephemera in the Bodleian Library at Oxford. In each case the tickets originated from a few individual collections, where the purchaser saved their losing tickets as keepsakes. Even though they are too few to give a real sense of the frequency of number choice, they nonetheless can shed surprising light on the strategies employed. That study will be the subject of chapter 12. The second approach is to draw a scientifically randomized sample from the totality of all bets placed over a period of time, a sample large enough to give a broad look at the distribution of choices. That may seem an impossible goal at this late date, but in fact it has already

Dreams and Astrology  93

been done, albeit inadvertently. Let us accept that the drawings were each a true random sample of the ninety numbers in the wheel of fortune. The data pass all tests, and the only anomaly, the rash of winners in Lyon in 1820, was, if not a simple fluke, the result of a fraud that did not compromise the randomness of the selections. Remarkably, the randomness of the draw means that the winning tickets can be considered as a rigorously randomized sample of the numbers bet— remarkable since truly random sampling was not introduced until at least the 1880s, if not the twentieth century. Menut’s book, the same book that gave us the list of all winning numbers analyzed in chapter 10, also included a list of all the big winners from the reinstitution of the Loterie in 1797 until the end of 1833: 1 quine, 327 winning quaternes, and 3 large ternes. For each such win Menut gave the date of the drawing, the numbers that won, the amount won, the city where the drawing took place, and the number and location of the bureau where the bet was placed. These are, then, a true random sample of the bets placed— at least the bets on quaternes. This is a somewhat subtle point. As we shall see, the bettors did not spread their bets uniformly among the numbers available for wagering. But even if their choice was nonrandom, each possible bet on a quaterne had the same probability of being sampled: one chance in 511,038. All bets which are capable of yielding a win on a quaterne were equally likely to be included. Of course these were not the only bets placed, but since the available evidence (some surviving losing tickets) suggests that many people who bet on extraits and ambes also included a quaterne in their list of bets, we may hope to learn from this sample about the approach the entire gambling public took in choosing numbers. Popular numbers will then each have the same chance of being selected as the unpopular numbers, but they will be better represented in the sample in proportion to their popularity. The sample size (327 quaternes, 331 large wins) may be regarded as random, but that will not affect our inferences. We start with the easiest question: What numbers were the bettors choosing? Experience with modern lotto (where typically six

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numbers are chosen, from 1 to 49 or 52) would lead us to expect a markedly nonuniform distribution of choices. Studies by Kadell and Ylvisaker and by Henze and Riedwyl report data from modern lotteries showing a strong preference for low numbers (for example, birth dates and other anniversaries) and simple arithmetic progressions.1 Even in modern lotto games where more than 50 percent of the bettors’ choices were made as “quick picks” (random selections by computer, an option not available in the Loterie), nearly 1 percent of the choices were one of seven such progressions, with one of the most popular being 1, 2, 3, 4, 5, 6. Other modern patterns would not have been available in the Loterie. Some modern choices are indicated by marking a printed ticket with all the possible choices in a rectangular array. Henze and Riedwyl found that bettors would often mark numbers in a geometric pattern, such as a cross. Menut’s collection of n = 331 winning bets (including the quine and the three large ternes) is too small to reveal much about patterned choices. Consider simple arithmetic progressions such as 1, 2, 3, 4, 5; or 16, 18, 20, 22, 24; or 20, 25, 30, 35, 40. There are only 968 such progressions involving the integers from 1 to 90, and in 331 independent drawings the chance that even a single one is to be found is only 0.0073— a very rare occurrence, as the bettors in the modern lotto games no doubt discovered. But we do get reliable information on the frequency of single numbers. Figure 11.1 shows the distribution for the sample. As expected, there is a preference for smaller numbers, and also for what the French referred to as jumeaux (the twin pairs 11, 22, 33, etc.), and also for 90, as well as the sets 76/67 and 63/36. We get some limited information from this sample on the joint distribution of numbers selected in a single bet from the clumping of winners: for example, on the frequency of occurrence of two bettors on the same quaterne in a single drawing. With the distribution of numbers shown in figure 11.1, if the bettors were choosing single numbers independently from that distribution, the chance of two betting the same quaterne in a single drawing would be very small, but much more likely to occur when the drawing produced popular numbers. In fact there were thirty-two instances of multiple quaterne

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F ig ur e 1 1 . 1 . The frequency distribution of single numbers in Menut’s list of 331 large winning bets.

winners: twenty-six where two people won, five with three winners, and twenty quaterne winners in a single drawing on one day in March 1802, earning the twenty holders in cities from Paris to Toulouse a total of 277,244 francs! Figure 11.2 shows four examples of multiple big winners at the same drawing in 1824. In three of these involving only quaternes, the multiple winners bet on the same four numbers: (2, 64, 67, 77), (11, 44, 56, 90), and (4, 7, 8, 11). The other quaterne that won (1, 3, 6, 16) was in the same drawing as the large terne (1, 3, 5). Two of these groups were rich in jumeaux, consecutive numbers; the other two were concentrated on small numbers, as on dates. Inspection of the multiple wins in other years shows a tendency for them to occur in drawings with jumeaux or reverse pairs: (63, 36), (67, 76), etc. Several partially arithmetic sequences appear (e.g., 6, 24, 36, 48, and 5, 25, 35, 65). The

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Figur e 1 1.2. Menut’s list of the sixteen large winners in 1824: three cases of multiple quaternes (two in Lyon in February, three in Paris in September, two in Strasbourg in November). The large terne in Lyon in October was in the same drawing as a winning quaterne.

drawing in 1802 that produced twenty winning quaternes was 67, 76, 11, 88, 63, with a reversed pair and two jumeaux. While all the winners included 76 and 88 in their choices, all three of the remaining possible choices— (11, 63), (11, 67), and (63, 67)— were represented, ten, eight, and two times respectively. Ironically, the bettors themselves were also studying the winning numbers given in Menut, and they would have seen the abundance of “popular” numbers, from which they may have drawn the false conclusion that these numbers were more likely to win than other numbers. Just as in life, popularity can lead to more popularity, followed by disappointment. Clearly these bettors were not choosing their numbers at random. Where did the chosen numbers come from? Some (e.g., the jumeaux) came from the same lack of imagination that makes 1, 2, 3, 4, 5, 6 popular today. During one short period in the history of the Loterie, the bureaus offered an equivalent to today’s “quick picks,” in which a computer chooses numbers for the bettor at random. This was in the late 1780s, when some Loterie offices had preprinted sheets with choices already made and ready for sale, thereby eliminating indeci-

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sion as well as helping the Loterie balance its portfolio of numbers. This could lead the bettors away from piling up on a very few favorite numbers, like the jumeaux, where a large number of people might simultaneously get quaternes, such as happened in March 1802. This also saved the Loterie the trouble of going through the several-stage procedure to produce the final ticket. Evidence of this survives in a sheet from about 1790 that was recycled after the Revolution, when an “avis” (notice) was printed on its back. This sheet— including the back, labeled as “page 2”— can be viewed online in the collections of the Newberry Library.2 The examples on that sheet resemble the tickets in figure 12.2; they have the printed Blanquet signature but lack the seal, and have a blank space where the date of the drawing would have been given; each is priced at the same total sum of eighty sous. Lacroix discussed this practice in his 1816 textbook. But very few losing tickets that were printed rather than handwritten have survived from the period after the Revolution. As Lacroix concurred, this suggests that the bettors much preferred to choose their own numbers, and that the Loterie was not overly concerned with the exposure to large wins.3

Books of Advice for Bettors There clearly was a business opportunity. A large group of bettors were looking for a way to improve their chances to win, and there was a large and active group of publishers in France,4 many not afraid of offending church or state by invoking Cabalistic methodology, the mystical manipulation of numbers with practices going back to Pythagoras. But how should the publishers proceed? It would have been easy to present a list of numbers claimed to have a better chance than others, as advice for eager bettors. Several publishers did do that, but it would be hard to do convincingly. If those numbers really were more likely to win, why is that not shown in the historical record, which was easily available to all? And if those numbers really were better choices, sales would plummet once they became widely known. Ideally, the publishers’ advice would allow the owner to personalize

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the choice, to choose numbers that were only useful for the book’s owner. And several publishers rose to the challenge.

Dreams What could be more personal than a dream, pointing to a particular choice? If you were to dream of the number 74, surely that would be a sign to inspire action. But what if you were to dream about cheese? Then you would need guidance, and several publishers grandly volunteered their services to that end. I have located three such works: The Book of Dreams (1786). (Livre de rêves, ou L’Onéiroscopie, application des songes aux numéros de la Loterie Royale de France, tirée de la cabale italienne et de la sympathie des nombres.)5 The True Cagliostro (1803). (Le vrai Cagliostro, ou le Régulateur des Actionnaires de la Loterie Nationale de France et autres loteries, composées de 90 numéros; augumenté de nouvelles Cabales faites par Cagliostro pour les tirages de Bruxelles et de Paris.)6 The Golden Key (1816). (La clé d’or, ou L’art de gagner a la loterie, suivant les Calculs cabalistique et mathématiques de Cagliostro et de Cornélius Agrippa, et d’après les Manuscrits de ces célèbres Savans, contenant la composition et décomposition des 90 numéros de la Loterie, nombre de cabales immanquables et de combinaisons sympathiques pour placer avantageusement son argent; suivi de l’explication des rêves, et leur rapport avec chaque no.)7

All three of these books were published anonymously, though the latter two invoked the name of Cagliostro, referring to the Comte de Cagliostro— true name Joseph Balsamo— a famous gambler, adventurer, and swindler who lived from 1743 to 1795 (figure 11.3). He played a minor role in the scandalous Affair of the Necklace, in which a swindler attempted to sell a fabulous necklace to Marie Antoinette, but was apparently innocent of direct involvement. (He was played by the actor Christopher Walken in a movie about that affair.) All three of these books presented long lists of potential dreams and associated

Fig ur e 1 1.3. Cagliostro advises his clients. From Cagliostro 1803.

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numbers; by a rough count, they listed 2,500, 3,300, and 850 dreams respectively. The Book of Dreams and the Golden Key were published in very small formats, pages measuring five by three inches, suitable to be carried in pocket or purse for ready consultation; evidently they both were published in earlier editions. These works gave differing advice on nearly all dreams. If you dreamed about cheese (fromage), the Book of Dreams suggested the number 73 and the Golden Key offered the three numbers 21, 25, and 82, while Cagliostro gave different numbers depending upon the type of cheese: for Brie, 30, 41, and 45; for Gruyère, 7 and 32; for Parmesan, 28, 58, and 68; and for Rochefort, 16. As in gastronomy, there was no consensus on cheese. All of the books included dreams related to common foods such as ham, oysters, cake, or wine, but with very few exceptions they suggested different numbers and offered interestingly different subcategories. The Book of Dreams gave advice for only red wine (15) and white wine (32). Cagliostro allowed a dream of wine generally (21), but also the variations “white” (52, 70), “good” (24, 56), “in the cellar” (3), and “bad” (17, 67). The Golden Key waxed more eloquent, with references to dreaming of wine as a flow of blood (8, 42, 90); drinking without wine as a sign of infirmity (75, 83, 86); drinking white wine as a sign of pleasure (4, 23); and drinking the dregs of wine, also as a sign of infirmity (36, 53, 70). On the other hand, a dream of “homicide” required no qualification and only a single number was offered for it, different in each book. There were some signs of changes over time: dreaming of the king before the Revolution received the number 1; by 1803 he was downgraded to 76; and after the 1815 restoration of the monarchy such a dream received the numbers 4, 14, and 18. Napoleon did not make any of the lists under any name. Oddly, dreams of electricity or mathematics were only included in the 1787 book, and a dream of stupidity only in 1803. Pissing on the wall was apparently undreamed of prior to 1816. The Book of Dreams included both adultery (5) and affairs (81); Cagliostro did not include adultery, but did cover affairs (16, 38); and the Golden Key only recognized adultery (83, 17). Apparently no one at all dreamed of statistics in those days.

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Fig ur e 1 1.4. The frequency counts for the numbers 1 to 90 for 2,665 numbers recommended in the Dream Book. Data from Anonymous 1786. A chi-square test strongly suggests that the numbers are not randomly selected from 1 to 90 (adjusted chi-square of 122.3 on 85 df ).

Where did these books get their numbers? Figure 11.4 shows the frequency distribution for 2,665 recommended numbers in the Book of Dreams. These numbers do not pass a test of random selection, but are closer to being random than the numbers chosen by bettors (see figure 11.1). The association of dreams with numbers, particularly with the numbers of the Loterie, is an odd enterprise. The order of the ninety numbers has nothing to do with the outcome of bets. Any set of ninety symbols, even the names of ninety landowners in Genoa, would do as well, except for the difficulty of recalling them. That the numbers 81 and 82 are next to each other has no role in the Loterie; yet two dreams may be different but close together, and the creator of the Book of Dreams realized that. Pepper and pepperbox are close, and

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Figur e 1 1.5. Twenty-seven recommended numbers from a small regional 1800 book of dreams published in Caen, France— a book otherwise uninterested in the Loterie. From Anonymous c. 1800.

were assigned the numbers 82 and 81; apple and apple tree were 56 and 57. But what about a black cat and a white cat? They are alike, and yet somewhat opposite. They were made 12 and 21. White horses were 37; black horses were 73. This sort of assignment was made throughout the book, with the range moving with different categories to cover the span from 1 to 90 reasonably well. Not so with the Golden Key: the author of the list of numbers was color-blind for cats, and not at all interested in pepperboxes or apple trees. “Cagliostro” had no interest in cats of any type, and usually took it as a point of pride in assigning two or more widely separated numbers to the same dream, as if to say that the magnitude really was unimportant, as indeed was true. There must have been a huge number of smaller sources of advice based on dreams. One small twelve-page pamphlet dedicated to interpreting dreams did not associate numbers with them, but still felt it prudent to end with a suggestion— namely, that the twenty-seven numbers ending in the digits 3, 7, or 9 were most likely to be fortunate (figure 11.5).

Astrology Another way to personalize a suggested list of numbers would be to follow the practice of astrologers and key the advice to the bettor’s

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astrological sign, using complicated methods to adapt the choice of number according to the position of the planets at the time of the bet. Many books followed that practice at least partly. Menut did so with lists of “happy” and “unhappy” numbers, complicating the procedure (and making it harder for pirate publications to copy) by giving different lists according to season, or in association with women’s names, flowers, weather, and so forth. Various almanacs were published which allowed the bettor to choose numbers congenial to the season, the sign of the zodiac, or the name of a loved one. After the French Revolution, the influence of the Catholic Church on life in France was greatly diminished, and previously proscribed reading found new readership in public markets. The Golden Key included a section to this end. With an invocation of the system of Pythagoras, and various number wheels and number pentagons, the book was small; but it offered a seemingly endless variety of happy and bad numbers, all in the range from 1 to 90. The days of the months were so classified, as were the positions of the planets. Which numbers were “sympathetic” to which other numbers? The book had the answers. A complicated algorithm showed by an example how, by taking account of the phase of the moon, the day of the year, happy numbers for the city, and other factors, the reader might be led to three numbers (54, 45, 59), numbers that for the drawing on 1 December 1787 would have yielded a profit of 19,065 francs!8 If only the book had been published before the drawing. In 1800 a book was published in Italy that was devoted entirely to an astrological approach. The author, Fortunato Greppi, constructed eighty-four tables, giving some suggested numbers for each of twelve signs of the zodiac in seven different configurations of planets and other astronomical objects. Figure 11.6 gives one of the tables for Virgo.9 These tables would permit a professional astrologer to take the positions of the planets, moon, and sun and relate them to the celestial sphere. The “mansions” in the right column are twenty-eight well-defined positions in the monthly travel of the moon, and the planets would visit different “houses” and “angles” defined in terms of coordinates of the solar system. Clearly, using these tables would

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F ig ur e 1 1 . 6 . An astrological table from 1800, adapted to suggest lottery numbers based on dates and the positions of heavenly bodies. From Greppi 1800.

require more expertise than anyone but an astrologer could bring to the task, especially since Greppi offered only one unhelpful sentence of instruction.

Betting on Names An article in the Moniteur Universel of 28 December 1797 suggests another source for betting numbers. It explains that an unidentified

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citizen bet a total of 1,008 francs on all twenty-eight ambes (twentyfour francs each) and all fifty-six ternes (six francs each) corresponding to numbers associated with the name Bonaparte, with A equal to 1, B equal to 2, and so on. That citizen won on one terne and three ambes, for a net gain of 51,432 francs, a considerable fortune. If he had bet the same amount on each of the fifty-six ternes and twentyeight ambes, that would have been about eight francs each, or a total of 672 francs bet. If he also bet on each of the seventy quaternes, that would cost another 560 francs, giving a total spent of 1,232 francs. The article concludes with a short notice that General Bonaparte was received the day before as a member of l’Institut de France, thus raising a suspicion that a press agent may have been involved. There are eight distinct “Bonaparte” numbers (2, 14, 13, 1, 15, 17, 19, 5), and the question arises: Did others imitate the lucky citizen? In a random selection, these numbers should occur about 9 percent of the time, but of course they are also among the favored small numbers; so it is not surprising that over the years 1798 to 1814 they occurred in about 11 percent of the winning draws. Interestingly, in the years after the Battle of Waterloo this increased to 14 percent. Choosing numbers from names in this manner was clearly quite limiting (to the numbers 1 to 26), and an entrepreneur named J. B. Marseille, who billed himself as a “mathematicien,” responded with an extremely complicated cryptological scheme that could yield no end of sets of numbers based upon the same name (figure 11.7).10 Marseille offered more personalized service than the others. He included forms in which the owner of a copy of the book could do the same sort of calculations with the owner’s name. This service both helped the book’s owner— in the case of my copy, one Ambrose Ghilini (figure 11.8)— and reduced the book’s resale value. Incidentally, Ambroise Ghilini (1757– 1833) served Napoleon at one point, but his biographical record does not refer to any Loterie winnings.

Bet Choices in Literature Lotteries played a large role in society, and this was acknowledged by poets and novelists. A small book on the Genoese lottery, published

F ig ur e 1 1 . 7. An example of how a sample name can be employed to choose many different numbers on which to bet. From Marseille 1807.

F ig ur e 1 1 . 8 . The same method as seen in figure 11.7, here employed using the name of the book’s owner:, one Ambrose Ghilini, who served Napoleon. From Marseille 1807.

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in Italy in 1700 by Giuseppe Stampa, is an example. Stampa combined a long poem with some elementary instruction on combinatorics. David Bellhouse summarizes the poem in these words: “There are three characters: a poet, a gambler and a cabalist. Throughout the poem the gambler complains that he listened to the advice of the cabalist on how to bet and that the advice was bad. The gambler goes on to say that if he had not listened to the cabalist and had played his own numbers [1, 57 and 88], then he would have been a rich man.”11 The theme of disappointment in the advice of cabalists was also invoked in a 1787 French novel by Pierre-Jean-Baptiste Nougaret, La fole de Paris; ou, les extravagances de l’amour et de la crédulité (The Parisian Fool; or, the Extravagances of Love and Credulity).12 The narrator is the lover of Clélie, the “fool” of the title; in one chapter he laments that his love led him to accept her claim that there was a science to choosing numbers in the Loterie, and that over time this had cost him fifty Louis (1,200 francs). Clélie would note the number of the first coach they encountered, and manipulate it to produce five numbers to bet. They lost. She would consult with people she regarded as “innocents”— a child, a beggar, or a lunatic in an asylum— and ask them for numbers, which she would manipulate in similar ways. These too lost. Despite all this and much more, the novel ended on a happy note, as the narrator managed to bring his love to her senses. In 1801. Jean-Antoine Lebrun-Tossa, a French journalist, took advantage of the newly expanded interest in the equally expanded Loterie by translating a forty-four-year-old Italian novel and updating its title to Le Terne à La Loterie; ou, Les Aventures d’une Jeune Dame (The Terne at the Loterie; or, The Adventures of a Young Woman).13 The 1757 original was by Pietro Chiari, titled La Giuocatrice di Lotto o sia Memorie di Madama Tolot (The Lotto Player; or, The Memoirs of Madama Tolot).14 Lebrun-Tossa’s translation was faithful, even to the preservation of the young woman’s name, “Tolot”— an anagram of “lotto” that required a footnote for French readers. The lottery did play a role in the plot; it provided hopes of riches, hopes alternately rewarded and dashed, to the delight or surprise of the young woman.

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One example is sufficient to illustrate the sophistication of this device. In the novel, the young woman dreams of the numbers 7, 22, and 57. The numbers drawn are 72, 2, 12, 18, and 60; she loses. But she has dreamed correctly; she has simply misinterpreted the message: the number “57” says she should bet on 5 + 7 = 12! And that she should split the 22 to 2 and 2, and then combine the first 2 with 7, thus producing 72! And that, having gone ten days without seeing her lover, she should add 10 to the other 2, thus producing 12! And so forth. In places, the book reads like a satire of methods in the Golden Key. The plot is driven by romance, family conspiracies, a stolen inheritance, and, to a lesser degree, lottery winnings and losses. All ends happily; Tolot even reconciles with her duplicitous mother-in-law, and the astrologer who has furnished some winning numbers quits that job and returns to the practice of medicine. The highlight of the book is its lovely frontispiece (see figure 3.1 in chapter 3). In the autumn of 1804 the German dramatist August von Kotzebue visited Paris and offered a friend in Britain an account that shows what the bettors’ choice of numbers was like at what was literally the “street level.” Behold in one place a wheel of fortune made of glass; are you not surprised? Here extremes meet; one of the most enlightened nations of Europe seems likewise to be the most superstitious. At the corners of every street you find cunning people, who in every possible manner allure passengers to announce to them, infallibly, what numbers will be prizes in the next drawing of the numerous French lotteries; and such a prophet has always a crowded circle about him. This dirty wheel of fortune has a hole at the top; the ragged fellow who stands behind it has made a kind of an instrument of the backbone of a goose, which he applies to the hole with great gravity, and almost without moving his lips imitates the speaking of Punch, which sounds exactly as if some little demon were sitting in the wheel and addressing the auditors. If the curious draw near, the goose’s bone suddenly jumps off the hole, and the ghostly voice invites the bystanders, whose hands are already in motion, under the most splendid assurances of drawing the numbers which are to be the prizes. Two sous is the usual price of all such never-failing prophecies.

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Kotzebue went on to describe five more examples of street vendors: all with the same goal, to supply numbers for the Loterie, but each with a different pitch. And before he moved on to other parts of the Parisian scene, he added: “Don’t mind the woman who absolutely wants to force upon you a ticket of the national lottery.— ‘Seventyfive thousand livres to be gained for a trifle,’ she incessantly cries, as if she had been taught by a Brunswick lottery-office keeper; but only follows you to the corner of the street.”15 The demand for tickets had inspired a vibrant market for advice.



Twelve



The Number 45 and the Maturity of Chances

The v i e w of t he be t t or s ’ be hav ior described in the previous chapter is based on published sources: a scientifically randomized survey selected from a large group of bettors, and a look at some of the advice published and presumably read by another group of bettors. We will now do what might be termed an archeological investigation, taking a look at some of the very limited physical evidence from surviving tickets. On the basis of the number of winning quaternes after the Loterie was reestablished in late 1797, I estimate that the number of tickets sold over those thirty-eight years that included a bet on a quaterne probably exceeded 170 million. It is not much of a stretch to suppose the total sales, including those in the thirty-five years before the Revolution, could have reached 250 million— a quarter of a billion tickets. Where are those tickets now? All winning tickets presumably were turned in and then destroyed. The great majority of tickets sold did not win their bets, and would have been largely discarded. A few survive in museums, a few found their way into collectors’ scrapbooks, and others that remain may still be in French attics. The scarcity of the surviving tickets has one benefit; they can be subjected to close scrutiny to a degree that would be impossible if their number was huge. On the other hand, they are far from a random sample, and while they can shed surprising light on behavior, caution is essential in taking general messages from them. That said, surviving tickets can reveal surprises.

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Fig ur e 12.1. Three Loterie tickets for the number 45 from 1786: June (second drawings), July (first drawing), and September (first drawing). The receipt shown in figure 8.2, chapter 8, for the first drawing in June 1776 was also a part of this series.

Figures 8.1 and 12.1 show four different tickets for four different drawings in 1786. Three of the tickets are from a scrapbook in my possession; amazingly, the fourth was found on Ebay. All four tickets are alike in that each was purchased at bureau 116 in Paris, and each was for the same unusual bet, though the amounts wagered do differ. The bets were unusual in that each was for but a single number, “No. 45 Jeanne,” and for an extrait. But the extrait was not an extrait simple; rather, it was a set of five bets of equal amounts, each on an extrait déterminé. Taken together, they covered all the ways 45 could be drawn: (45, x, x, x, x), (x, 45, x, x, x), (x, x, 45, x, x), (x, x, x, 45, x), and (x, x, x, x, 45).

This was an odd bet, one might think. In a certain sense, betting on 45 for all extraits déterminés is the same as betting on 45 for an extrait simple, except that the payouts will differ because the official

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odds were not consistent. The simple bet was more advantageous than the other. Betting five francs— one franc on each position as a déterminé— would return seventy francs if in fact the number 45 was drawn in any position, while a bet of five francs on an extrait simple would return fifteen times the amount bet, or seventy-five francs. The bettor seems to have been purposely taking a less attractive bet than one otherwise available. And why choose to bet on the particular number 45? Was it simply the bettor’s lucky number? Was the bettor’s name “Jeanne”? Or was the number picked in accordance with some strange “principle of the average” since 45 is half of 90, there being 90 balls in the wheel of fortune? How can we possibly answer these questions at this distant point in time? Surprisingly, there is a simple answer to all of these questions; a speculation, but one that seems overwhelmingly likely. To engage more betting interest, both the Loterie and the commercial publishers kept a public record of the “ages” of each of the 90 numbers: how many drawings had occurred since the last appearance of the number. An attentive bettor would have known late in 1785 that the number 45 had been drawn most recently in January 1783, and that by the end of 1785 it was the only number of that advanced an “age.” At the end of January 1786, the number 45 had not been drawn in more than three years! During those three years there had been seventy-two drawings. For as long as people have been gambling, they have had a visceral belief in what has come to be called the “law of the maturity of chances.” The “law” states that after a long period in which a die has avoided one side or a lottery has avoided one number, that side or number becomes more likely to appear. Probabilists dispute this, but they do so only by essentially assuming it away— in supposing that the die or lottery has no memory. Gamblers in France in the 1780s would have held a different view. My speculation is that at some point in early 1786, our bettor— let’s call her “Jeanne”— decided it was time to take advantage of this opportunity. More specifically, starting with the second drawing of February 1786, Jeanne began to engage a planned sequence of bets. Look again at the tickets, and at table 12.1. The bets increase with each

The Number Forty-Five  113

T a b l e 1 2 . 1 . A speculative accounting for the unknown bettor “Jeanne” on the number 45. Drawings with asterisks are those for which the ticket survives (see figure 12.1). The table gives the amount bet on each of the five positions, the total amount bet at each drawing, and the total bets and payouts over time. There were two drawings each month. The second drawing of September 1786 resulted in the numbers 42, 22, 75, 47, 45, and that last draw would have returned seventy times the bet of fifteen francs on that position. 1786 drawing

Amount per bet

Total bet

Feb (2nd)

1

5

5

0

Mar (1st)

2

10

15

0

Mar (2nd)

3

15

30

0

Apr (1st)

4

20

50

0

Apr (2nd)

5

25

75

0

May (1st)

6

30

105

0

May (2nd)

7

35

140

0

June (1st)*

8

40

180

0

Jun (2nd)*

9

45

225

0

July (1st)*

10

50

275

0

July (2nd

11

55

330

0

Aug (1st)

12

60

390

0

Aug (2nd

13

65

455

0

Sept (1st)*

14

70

525

0

Sept (2nd)

15

75

600

1,050

Cumulative bet

Payout

drawing. If Jeanne started with a bet of one franc on each of the five positions, and increased that bet by one franc with each drawing, she would be betting eight francs on each position in the first drawing in June, nine francs in the second drawing in June, ten francs in the first drawing in July, and fourteen francs in the first drawing in September, just as on all of the four tickets that survive. And in all of these drawings, the bets were lost. After the first drawing in September, Jeanne would have been out a total of 525 francs. However, in the

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second drawing of September 1786, the fifth and final number drawn was in fact 45! If we speculate further that Jeanne continued betting one more time, venturing a total of 5 × 15 = 75 francs, she would then have won 15 × 70 = 1,050 francs, giving her a net profit of 450 francs for the eight-month adventure. Of course, that speculative winning ticket would not have survived. There remains the question of why Jeanne spread the bet over extraits détermines rather than placing it on one extrait simple. A bet of seventy-five francs on the number 45 as an extrait simple would have returned fifteen times that bet, or 1,125 francs. Seventy-five francs more than the same amount bet would earn with her scheme. The answer may be that she had in mind the maximum bet, which had been set at twenty-four francs in 1757. To bet at the higher level on an extrait simple was then not permitted. It would have been possible for Jeanne to stay under those rules with an extrait simple through the first drawing in April, but we have no tickets from that period. By increasing her bets with each loss, Jeanne was moving toward an old gambling strategy called a martingale. In the purest form of the strategy, a gambler would double the bet after each loss; if the first bet was for a single unit, this would guarantee the gambler a unit profit when he eventually won, assuming his fortune was not entirely consumed in the process.1 Many gamblers discovered that doubling each time could, with a run of losses, lead pretty quickly to an unhappy end. Jeanne’s version of the martingale did not guarantee a gain, but it did preserve her bankroll longer. If she had been doubling each time, her fourteenth bet would have lost 40,960 francs, and her fifteenth bet— of 80,920 francs— would have won only a net gain over accumulated losses of five francs. Jeanne’s unit bet was five francs, with a payoff of 70/5 = 14 per unit, since even if she won, four of her bets still lost. Starting with a unit bet, her actual scheme would guarantee a gain so long as her run of losses did not exceed twice the payoff on a unit bet. In the case of an extrait déterminé, a gain would be guaranteed so long as the run of losses did not reach 2 × 14 = 28 trials. By August 1786 there had been eighty-eight drawings in which 45 had not been the winning number. How long would Jeanne have

The Number Forty-Five  115

Fig ur e 12.2. Two tickets for the first Loterie drawing in February 1782.

persevered after fourteen losses? Did she indeed bet that fifteenth time? We will never know. The pattern we find with Jeanne’s tickets suggests looking at other surviving tickets to see if the bets tend to have been placed on “aged” numbers. The evidence gathered by checking a half-dozen other tickets from 1770 to 1790 is mixed, and is well represented by two tickets from the same scrapbook (figure 12.2). These two were for the same drawing (the first of two drawings in February 1782) at the same bureau in Paris, no. 624. Since the tickets have always traveled together over the intervening years, it is tempting to assume they had the same original owner; yet they show different strategies. The first ticket, which included five selected numbers, seems indifferent to “age”: all five numbers had been drawn in the previous

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twenty-five drawings, two of them twice. Many other numbers of much greater age were available, should that have been a consideration. The second ticket, with three numbers selected, is different: none of the three numbers chosen had appeared in the past thirty drawings. To be sure, those numbers were not the only choices that fit that description; fifteen other numbers, some older than the ones bet on, were also in that same group. If the three numbers on the ticket had been randomly selected from the ninety available, the chance that all three would be in the group of fifteen was only about one in 255. This suggests that age was an issue for these choices, even if not the sole issue. We cannot be sure that all the information available in making those calculations was easily available in February 1782. The compilations of ages that were available were not always up to date, but at least the gap between the present drawing and the previous occurrence of the numbers just drawn was readily available (figure 12.3). Figure 12.3 shows how one publication created an update with each drawing. There was also a blank page where the gambler could continue the updating. The publication included summary tables of the frequency of occurrence of numbers by first digit and by last digit, and a table of the time series of gaps for each of the ninety numbers over the history of the Loterie. In the J. J. Johnson collection of lottery ephemera in Oxford’s Bodleian Library, there are three tickets from 1786 to 1787 that are similar to Jeanne’s tickets. They are bets only on the single number 80 as an extrait simple, with different sums of money bet each time. The first two tickets are from the same period in which Jeanne was betting: the second drawing in April 1786 (betting twenty sous on the number 80), and the second drawing in May 1786 (betting forty sous on 80). The third ticket was from a year later, the second drawing in April 1787 (betting thirty-six francs on 80). The number 80 was associated with the name Hypolite (after the daughter of Ares in Greek mythology), and we will so designate the purchaser. All three tickets were purchased at the same bureau no. 624, the same bureau that sold the 1782 tickets shown in figure 12.2.

Fig ur e 12.3. This commercial publication in 1778 gave several types of information. The page shown gives, for each of a sequence of drawings (145 to 198), the five numbers drawn; and for each of them, the gap between this drawing and the previous occurrence of the number, measured in drawings. If the previous occurrence was the most recent drawing, an asterisk appears.

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Hypolite evidently had studied a table of gaps or of ages, perhaps like the one in figure 12.2, and she may then have known that 80 had most recently come up in the first drawing of October 1784, thirtyseven drawings or nineteen months before April 1786. Of course, in April 1786 the oldest number was 45, Jeanne’s number, but Hypolite may have had less complete information than Jeanne. While 80 was not the oldest number available at that time, it was one of sixteen that had not occurred since October 1784. To reach a bet of thirty-six francs in April 1787, Hypolite’s bets must have increased much more rapidly after May 1786 than Jeanne’s, but slower than a doubling martingale. Where Hypolite bet thirtysix francs in April 1787’s second drawing, Jeanne would have bet only 12.5 francs; a doubling martingale bet would have been 625 francs. If Hypolite persisted, she would have not had the same happy end as Jeanne: the number 80 was not to appear again until August 1788. In apparent support of the maturity of chances, Jeanne’s older number, 45, appeared earlier, in September 1786. Progressive betting after a loss was certainly a known approach at the time. In 1773 an author who used the name Gräff (German for “count”— a multilingual pun?) wrote an entire book in French presenting an extraordinarily complex approach, extending the idea to various combinations of bets.2 The author was honest and did not guarantee success; and the time any purchaser put into studying the method, presented through more than two hundred pages of tables, would be time not spent in gambling, and hence a bonus. The law of the maturity of chances has probably been ingrained in gamblers since dice were introduced in Mesopotamia about five thousand years ago. In modern times, theoreticians have not treated it kindly. There were some early exceptions. Cardano, writing in the 1560s, seemed to endorse it, reasoning that the frequency of outcomes would stabilize sooner rather than later,3 and d’Alembert raised doubts in the 1760s about formal probability methods, doubts that permitted maturing chances.4 Generally, though, the idea was and is treated with scorn, and the maturity of chances is often relegated to lists of gamblers’ fallacies or popular myths. Bayesian proba-

The Number Forty-Five  119

bilists would go even further, showing that if the chance of occurrence is unknown, a longer gap since the last previous occurrence makes it less likely for it to recur in the next trial. And these modern arguments are all correct, given the assumptions made— namely, that the game (Loterie, dice, or roulette wheel) has no memory of the past at each instance; the conclusion is built into this assumption. But is that conclusion necessarily true? Is the consistent faith of at least some gamblers in the law of maturity of chances a vain hope, or might it have some basis in fact?

Parisot The principal exception to this general scorn among mathematicians for the “law” is Sébastien Antoine Parisot (1761?– 1812?). The question marks on his dates underscore his exceptional status: while few other mathematicians criticized his work, the vast majority simply ignored him, and even basic biographical details are hard to find. Parisot published two books with a principal focus on the “law,” both in Paris at a time when new interest in probability was excited by Laplace’s 1795– 96 lectures at the École normale, which would eventually be expanded to his Essai philosophique sur les probabilités in 1814. Parisot’s first book, L’Art de conjecturer à la Loterie, appeared in 1801 and was directed explicitly at the Loterie.5 It was in small octavo format, 168 pages, each eight by five inches. Parisot’s second book was published in 1810 and was expanded in every dimension. It was in large quarto format (pages 10.5 × 8 in.) and now addressed all games of chance, including the Loterie, in a span of 654 pages, with two folding plates. This second book had a grand title: Traité du calcul conjectural, ou l’art de raisonner sur les choses future et inconnues.6 Parisot’s approach to the maturity of chances is complicated by his treatment of many varied situations and by his style of exposition, which is challenging even today. His 1801 subtitle brags that it is “founded on calculation and confirmed by experience.” Surprisingly, it lives up to that claim and, despite critical comments to the contrary, is mathematically correct. Parisot considered all manner of Loterie

120  C h a p t e r T w e l v e

bets, but for clarity, we describe only the simplest: the extrait simple. Clearly the question to be answered involved the gaps between successive occurrences of the same number in a drawing. Suppose the number 45 had just occurred as one of the five numbers drawn most recently in the Loterie. What was the chance that the same number would not occur in the next five drawings? That is, what was the chance that the gap before the next occurrence of 45 would be at least five drawings? Under the generally accepted rules of the theory of probability, the probability of losing on a single extrait in one drawing is 17/18, and in five drawings that chance is (17/18)5 = 0.751. At what point, Parisot asked, did this chance drop below 1/2? He found that (17/18)12 = 0.504, and (17/18)13 = 0.476. That is, the median of the distribution of gap lengths was about twelve; the chance of a second appearance occurring before or at the thirteenth drawing was better than 50 percent. In 1810, Parisot went further, demonstrating via the law of large numbers that the mean gap length in the long run would be 18 and the upper quartile would be about 24, since (17/18)24 = 0.254. To sum up, so far we have a median of 12, a mean of 18, and an upper quartile of 24. These numbers can already give a gambler some guidance on the rarity of large gaps. But it leaves out all the extreme cases. How rare are very small or very large gaps? For this, Parisot looked at “experience.” He had data on a total of 1,820 drawings following the reestablishment of the Loterie in 1797, from the five cities that held drawings. Of the ninety numbers in the wheel of fortune, the one that had occurred the most was 32, which occurred 130 times (suggesting an average gap of 1820/130 = 14). The number that occurred the least was 8, which occurred 78 times (suggesting an average gap of 23.3). Thus, he has a minimum average gap of 14 and a maximum average gap of 23. Since both of these gaps fell between the overall median and upper quartile, he took this as confirmation that the interval (12, 24) between these two values might be considered when betting on gaps.7 Speaking from the standpoint of a just-completed drawing, Parisot could say with full mathematical competence that the chance a particular number that had just occurred (say, 45) would occur in the next

The Number Forty-Five  121

twelve drawings was about 50 percent, and in the next twenty-four drawings 75 percent. Why would mathematicians criticize this statement? For the simple reason that, while the statement was true even in light of their own calculations, it was not the answer to the question most gamblers would ask, and was susceptible to misinterpretation. Think about Jeanne again, who in early February 1786 had begun to bet on the number 45 on the strength of it having not appeared at all since early January 1783, a full seventy-five drawings earlier. She surely believed that 45 was due for an earlier-than-usual appearance. What would a mathematician have told her— including Parisot, if he had taken a dispassionate view of the situation? All mathematical models would say that there was no memory in the Loterie, and that at the start of February 1786 the chance that 45 would appear in the next twelve drawings was about 50 percent, the same as if 45 had just occurred in January 1786. Presumably that answer would not have inspired Jeanne to begin betting, even though, as luck would have it, the number 45 actually occurred fourteen drawings later, very close to the stated median. Apparently unknown to Parisot, Laplace had posed a much more difficult problem in 1772: Suppose that with today’s drawing we start observing how many of the numbers have not appeared, and continue the series of drawings until we have seen all ninety numbers. What is the chance that it will require more than n drawings to achieve that goal? And what must n be in order that the chance is at least 1/2? In 1772, Laplace found a formula for the first of these questions, but it was too difficult for him to evaluate for the necessary numbers of trials. In 1786 he returned to the question and answered it for a simpler problem. In 1812 in his Théorie analytique des probabilités he answered the last question for the Loterie:8 Only after eighty-six drawings would the chance that all ninety have appeared reach 1/2! The gap that Jeanne experienced, 75 + 14 = 89, was very close to what one should expect the longest gap would be, on average (see appendix 3, regarding Laplace’s result). Parisot’s numbers correctly answered a general question. But for most gamblers the real question was different, a conditional one:

122  C h a p t e r T w e l v e

What was the situation for their number, which had not appeared in some time? Parisot’s answer, had he given it, would have been the same: that with no memory, the past was irrelevant. But he did not give that or any other answer. Instead, he referred to the general situation, in which for millennia gamblers had felt intuitively that a long avoidance of the number 45 had to indicate they were near the end, and that 45 was “due.” To be fair, Parisot did not state this, even if he also did not contradict it. What did he suggest, if he did not suggest searching through past records to find the oldest numbers, as Jeanne had done? Here is one example Parisot did offer in 1801. Pick in advance twelve numbers, without regard to past records (for his example, he picked the numbers 1 through 12). Then embark upon a plan: at each drawing, buy a ticket with each of those numbers specified an extrait simple. Bet one unit on each of the twelve, for a total of twelve units on the ticket. If at least one of the twelve numbers wins, do exactly the same at the next drawing. If none of the twelve numbers wins, increase your bet to five units on each number, for a total of sixty units on the ticket. Parisot illustrated this example with data from the earliest drawings in 1758 (table 12.2). Parisot was interested not in the choice of the twelve numbers, but rather in the betting patterns he presented, which were examples of types of martingales. For him, the interesting point was the interaction between the distribution of gaps and the betting system. In doing this, he expanded the notion of a martingale beyond the older definition.9 The common use of the term had simply been that when you are losing, you should bet the amount needed to bring you to profitability— a use that led to the doubling of bets after a loss. Parisot may have been the first mathematician to give the martingale a more mathematical definition that included but was not limited to doubling: “Definition. In gaming terms, by martingale we mean a series of bets made according to some progression. This progression is usually geometric.”10 As in the case of his numerical example, Parisot would, with each loss, multiply the bet by five, and with each win go back to the initial

The Number Forty-Five  123

T abl e 1 2 . 2 . To illustrate what he meant by a productive martingale— in which he would bet on the numbers 1 through 12 each time, and if he lost would increase his bet fivefold— Parisot gave these sample data. The bets that won are in bold italics. From Parisot, 1801, 222, with one typo corrected: 37 for 7 in the fourth column under the heading “numbers drawn.” Drawing

Numbers drawn

Paid

Received

1

83

4

51

27

5

12

30

2

45

87

50

47

6

12

15

3

15

38

54

11

29

12

15

4

37

19

50

88

10

12

15

5

31

71

81

50

27

12

0

6

53

10

84

22

45

60

75

7

84

16

87

37

1

12

15

8

90

39

44

89

45

12

0

9

15

5

21

6

8

60

225

Totals 204

390

Total won: 390 – 204 = 186

bet level. This used the progression 12, 60, 300, 1,500, 7,500, 37,500, et cetera. He called this a “productive martingale.” He recognized that the bets grew rapidly with losses, which is why he bet on twelve numbers: the chance that none of them would win in a single drawing was 0.48. He also offered a “compensative martingale” that only tried to control losses; it would only raise the bet by a factor of five after two consecutive losses, a progression of 12, 12, 60, 300, 1,500, 7,500, et cetera. He acknowledged that a string of losses could have a very bad effect: A martingale of this species would soon have devoured an immense sum. . . . quite commonly there happen 2, 3, or 4 drawings before the appearance of one of the first 12 numbers, and we can even go through one of those unfortunate periods like that of 1 July to 30 October 1779, which

124  C h a p t e r T w e l v e

included 7 consecutive drawings during which the first 12 numbers remained hidden. . . . We see there that fortune can be extremely ungrateful, betraying you even when the odds [against 7 drawings without even one of the 12 numbers] are 127 to 1 in your favor.11

Parisot invoked “maturity,” a term he used only occasionally; and he probably shared most gamblers’ belief in maturity, which helped him see empirical support in the limited data he had. But he did not make the mistake usually associated with that term. He was engaged in shaping bets that would not defy the laws of probability, but would permit a gambler to enjoy a good chance of a significant profit, at the risk of a small chance of a large loss. In 1810 he asked, How can we determine the periodic return of all possible numbers in the Loterie, or find the number of draws that must be passed between two appearances of the same number, in order to attack it with advantage and at its highest point of maturity?12

In answer, he gave standard calculations indicating the rarity of large gaps, and wrote: We do not pretend to lay down, in this table, the exact limits between which the ramifications of chance are enclosed; because we know well, and experience proves it enough, that an Extrait is often delayed for more than 24 drawings; . . . the limits we set are purely mathematical; there are others much safer; they are those which result from long observation, and which one can call statistical limits: one finds them in the publications solely devoted to the history of the phases of the Loterie.”

For Parisot, the mathematics was reliable, but only unconditionally (no “personalized medicine” here). And in both books he gave disclaimers, such as: Are there methods of playing such that, by obeying them, it is proved by calculation, and confirmed by experience, that one must win infallibly?

The Number Forty-Five  125

First of all, common sense suffices to warn that there must be no infallible secret; that the interpretation of dreams, that the cabalistic, mystical, sympathetic numbers, in a word, that all the nonsense with which one deludes the multitude, are nothing but traps for gullibility. The secret of winning, if it could exist, or rather the art of playing well, must necessarily flow from the principles set out in this work. It is therefore by basing ourselves on these very principles, and applying them to experience, that we will learn if we can attack the Loterie with some advantage.13

Better to build hope on sound but limited mathematics and weak but suggestive observations, than on dream books and mysticism. Follow Parisot and you will either win or have the comfort of loss despite a scientific approach. There remains the possibility that the law of maturity of chances could have actually held for the Loterie. The drawings at the same location would have used the same physical equipment for many drawings, and subtle changes in the numbers could have occurred over time. One can imagine scenarios that would have led to inequality in the chances of different numbers. The operators of the wheel could have fallen into a pattern in which the balls were introduced to the wheel in a progressive change of order, affecting future randomization, as happened in the US draft lottery of 1970. Or there could have been a physical difference in the balls due to wear. Some one of these could have built memory into the system. Or undetected cheating could have produced maturity of chances. No doubt modern equipment, if operated honestly and competently, now renders the question moot, at least over shorter spans of time, but the same may not have been true in the period of the Loterie. None of the tests that have been tried show such an effect, but the data available would only detect a strong effect that held consistently.

A True Case of the Maturity of Chances In 1795, one entrepreneurial gambler discovered a true case of the maturity of chances in a lottery in England, and he wrote and sold

126  C h a p t e r T w e l v e

a pamphlet sharing his discovery, titled Calculations and Facts Relative to Lottery Insurances.14 It was not about a Genoese lottery, but the argument was convincing and even the most hardened skeptic would have to agree. The anonymous author of Calculations and Facts considered an English blanks lottery in which 40,000 sequentially numbered tickets would be sold as chances for a much smaller number of prizes (say, 10,000) of different sizes. There would then be a series of drawings of 1,000 tickets a day over forty days. Each of the 1,000 draws would involve picking a numbered ticket from the 40,000 such tickets in a large drum, and then pairing it with one slip picked from another drum containing 30,000 blanks and 10,000 slips with the designated prizes (as in figure 4.1, chapter 4). The author had a keen sense of what equiprobability in the draws meant, and why it was doubtful in the lottery: To the end that the chances should be in every case alike, it would be necessary that the chance, for instance, of the Numbers 1 and 2 lying together, or near to each other, in the Wheel, should not be greater than that of the Numbers 1 and 40,000 (supposing the latter to be the greatest Number). But when it is considered that the Numbers are cut in slips, each belonging to a particular thousand, into the Wheel, and that the billets, when thus cut in, press each other into a mass, reciprocally binding each other with their edges, it is impossible, in spite of every motion in the Wheel to agitate and disconnect them, that the kind of order in which they were first placed can be entirely disconnected.15

The author then gave practical instructions on how to track the draws and learn when a range of neglected numbers could likely be encountered and “suddenly pull itself up” to a degree that it would present an advantage to the gambler. You might think that, once the tickets were sold, there was no possibility of taking advantage of any noticed pattern; but that would underestimate the ingenuity of eighteenth-century gamblers. A secondary illegal market grew up, selling what they called “insurance.”

The Number Forty-Five  127

The idea was that for a moderate premium you could “insure” the number of a ticket you had purchased. But it was not necessary to own a ticket to insure it, and gamblers were soon doing the equivalent of buying a ticket number, or even a range of legitimate ticket numbers, and doing so even after the weeks-long drawing had commenced. The anonymous author says that the premium was 4.5 to 8 percent of the nominal ticket price at reputable offices, and up to 17 percent at a less reputable office. According to an anonymous 1771 critique of English lotteries, “The Insurance of Tickets is much more practiced as a Game [i.e., without owning the ticket], than a Security.”16 The author of this critique went on to warn the reader that most forms of lottery insurance were illegal, including all without the ticket in hand— a vain attempt to protect a state monopoly. Of course, the reader had to trust that the office that sold the insurance could still be found after the drawing. The available bets varied but, for example, one could buy “insurance” that a particular number would be chosen on a particular range of days, or that a particular range of numbers would be so chosen, or that a particular number would win a prize (figure 12.4). At some times one could “insure” a ticket for a few hours without ownership; this was called “riding the horse.”17 Thus, a gambler watching the drawing carefully might become convinced after the thirtieth day of drawing that ticket no. 12,742 (which had not yet surfaced) was due, and could bet that it would appear on days 31 through 34, with or without a prize. The 1795 author of Calculations and Facts gave guidance for how to recognize favorable situations, taking particular advantage of the way groups of numbers tended to stick together. Among those who became involved with insuring lottery numbers was John Law. While he was in the Netherlands in 1712, before he settled in France and produced his ill-fated currency reform, he advertised lottery insurance for a Dutch blanks lottery. His advertisement made the following offer: For a particular lottery with 30,000 numbered tickets, you could give him a list of ten different numbers and 100 guilders. If none of those numbers won even a small prize (3,800 prizes were offered), he would pay you 300 guilders. It was not

128  C h a p t e r T w e l v e

Fig ur e 12.4. An advertisement for an English lottery insurer.

required that you actually owned the tickets you “insured.” John Law was already aware in 1712 that one could create value without having a guaranteed backing. We know this because a Dutch mathematician and actuary, Nicholas Struyck, mentioned it in a 1716 book on probability, attributing it to “Jan Law.” Struyck correctly calculated the expected value of this purchase as giving Law 22 guilders on average per contract, so that the purchaser could on average expect to win back only 78 percent of his 100 guilders. Struyck thought that a poor investment, and did not seem to realize that buying the tickets at the lottery was much worse, offering only about a 50 percent return on a purchase.18 It is possible that the maturity of chances was helpful in the blanks or class lotteries in England or the Netherlands, although even there the difficulties would have been challenging to even the most dedicated gambler. In the Loterie in France, where the drawing was done in one day, it would not be useful. But even so, these examples are an indication of how carefully the results of lotteries were watched, and

The Number Forty-Five  129

how bettors might notice the slightest deviation from simple chance and try to profit from it. The archeological evidence of the surviving tickets suggests that the law of maturity of chances attracted considerable attention.

The Law of Maturity of Chances in Balzac In England in the 1700s, there were plays (The Lottery and The Gamester) that made reference to different ways of wagering, and there were many morality pamphlets sold for a penny, telling harrowing stories of ruin after engagement with the lottery, usually ending in death: “He was executed according to his sentence; and would to God that this history might prove a warning to all, against trying their fortune in the LOTTERY!”19 At a later period in France, there were many more references in novels to lotteries of different types, as discussed in the previous chapter. A lottery is an ideal tool for a novelist; it can change the direction of an individual’s life in a moment, in a way that would otherwise be hard to explain. For a while after the 1836 end of the Loterie, it remained in public memory. Writing in 1842, Honoré de Balzac used it effectively in the novel Un Ménage de garçon (translated to English at various times under the titles The Bachelor’s Establishment, The Two Brothers, or The Black Sheep). Balzac could assume his readers would recognize terms like “terne” and aspects of the Loterie administration, including dates and places of the drawings, with only the slightest hints.20 In the novel he memorably characterized an old bettor, Madame Descoings: “This stubborn player never missed a drawing: she was faithful to her Terne, which had yet to be drawn. This Terne was going to be twenty-one years old, he was reaching his majority.” After perhaps 3,600 drawings over twenty years without her three terne numbers all appearing in the same drawing, you might think that only the law of maturity of chances would save her, but a limited amount of data suggests that the success rate of lottery tickets in novels is much higher than in mathematics and real life. More precisely, one of two outcomes tends to occur: either (1) the bettor wins at the

Fig ur e 12.5. Title page of Jules Verne’s Un billet de Loterie: Le numéro 9672 (1886).

The Number Forty-Five  131

crucial moment and the book ends happily, or (2) the bettor entrusts someone to purchase the ticket with those numbers but the trust is misplaced; no ticket is purchased, but the numbers win anyway, and the bettor dies of a broken heart. Unfortunately for Madame Descoings, her story was in the second category. By 1886, society’s memory of the details of the Loterie had faded, and novelists could no longer use it to drive a plot forward. Jules Verne’s novel Un Billet de Loterie, 9672 was based on a lottery of a very different type, more in the spirit of a raffle (figure 12.5).21 A fiancé is missing, but has left a note and a lottery ticket bearing the number 009672 in a bottle. Much happens, but it suffices to say that the outcome of this story, unlike that of Madame Descoings, was in the first category.



Thirteen



How Much Did They Bet, and Where?

T h e pe r iod af t e r t h e R e vol u t ion was mostly a time of growth and prosperity for the Loterie, with three drawings a month in each of five French cities. At the peak, more than a thousand agencies sold tickets for the Loterie, 150 in Paris alone. These agencies were active over a long period in about seventy-five cities in France, as well as in Geneva and Neuchatel in Switzerland, and a few places in Italy and Germany. During the peak of the Napoleonic empire, there were even more: in 1804 a sixth Loterie location was established in Turin, Italy, that accepted bets from Paris, and by 1812 about fifty agencies in the Netherlands were taking bets for French drawings. The geographic spread is easy to describe; further details are more difficult to discover. Menut’s list of winners covered the post-Revolution Loterie through 1833, thirty-six of its thirty-eight years of post-Revolution operation. The list opens a door to statistical inference. A part of Menut’s list (figure 13.1) shows the eight successful bets on quaternes in the year 1827. For example, the first line says an agency in Paris (bureau no. 89, which was at no. 2, Boulevard des Italiens) sold a winning quaterne for the third drawing in February that year in Paris, with a payoff of 4,850 francs. The fifth line reports that an agency in Marseille (bureau no. 958) sold a winning quaterne for the first Paris drawing in April, with a payoff of 9,700 francs. Five of the wins in

How Much Did They Bet, and Where?  133

Fig ur e 13.1. The eight winners of bets on quaternes in 1827. From Menut 1834.

1827 were at drawings in Paris; one was in Lille, another in Lyon, and the last was in Bordeaux. For the moment, let us omit the three large ternes and the single quine included in Menut’s list, and consider only the 327 winning quaterne bets. Since the probability of any single bet on a quaterne winning is 1/511,038, we can estimate the number of bets (winners and losers) by multiplying the number of winning quaternes by 511,038, thus giving us 327 × 511,038. This estimates that a stunning 167,109,426 quaternes were bet over thirty-six years of the Loterie. These years had a total of 5,916 drawings; this would amount to an estimated average of 28,247 quaterne bets per drawing. Of course, this is only an estimate (plus or minus 25,000 for the total number of quaterne bets, and plus or minus 5 for the average number of bets per drawing). The number of separate tickets sold with a quaterne bet could be slightly less, according to how many people bet on more than one quaterne per ticket. One question is, just how did the bettors split their bets between quaternes and the other bets? It would be reasonable to guess that bets on the quaterne were probably only a small fraction of all bets placed. Two observations support this and suggest that the bets on the extraits through the ternes occupied the public much more than did quaternes. There are few surviving losing tickets that include bets on quaternes in the various collections I have seen, but that could be partly due to the tendency for bettors to follow the pattern seen in figure 13.2, with the quaterne as only one bet out of fifteen made on that ticket, any one of which would have made this a winning ticket

134  C h a p t e r T h i r t e e n

Fig ur e 13.2. A losing ticket from the first Paris drawing, January 1827. The bettor chose four numbers (33, 37, 61, and 87) and bet a total of 8.20 francs on the fifteen possible different bets. On this day, the numbers 66, 81, 19, 39, and 26 were drawn.

if successful, and thus returned to the Loterie to claim the prize. This would happen a bit more than one time in five. Of course, this is not a random sample; but in the case of this example, the other bets outnumbered the quaternes by fourteen to one, and many more tickets did not include even a single quaterne. If we take the fourteen-to one ratio as general, it would suggest that the total number of bets (not tickets) over thirty-six years was on the order of 15 × 167,109,426, or about 2.5 billion bets! There is one other source of information. With this large an operation, it was necessary for the central office to publish regular instruction manuals for the agencies, to ensure that proper uniform procedures were followed in well over a thousand offices. A collection of these manuals covering the period 1797 to 1815 contains much of interest, including sample tickets illustrating challenges such as the agencies would be expected to face in pricing tickets.1 Figure 13.3 (top) shows a ticket on which five numbers were specified and twenty-five bets were made: 0.90 franc on each of ten ambes, one franc on each of ten ternes, and 0.20 franc on each of five quaternes, for a total bet of twenty francs. The minimum allowable bets were 0.10 franc per ambe and 0.05 franc per terne or quaterne, with an upper limit of twelve francs for quaternes. The second ticket (bottom) illustrates a different challenge. It represents a bettor who wants

How Much Did They Bet, and Where?  135

F i g u r e 1 3 . 3 . Two sample tickets from official agency manuals for 1804 (t o p ) and 1815 (bot tom ). From Administration de la Loterie 1797, 1800, 1804, 1815.

to bet in only two categories: on all possible ambes composed entirely of numbers ending in the digit 9 (like 19 and 59; there are thirty-six such pairs from 1 to 90), and on all possible quaternes composed entirely of numbers ending in a 9 (there are 126 of these). Each ambe would carry a 0.20 franc bet (one step up from the minimum) and each quaterne 0.05 franc, the minimum then allowed, for a total bet of 13.50 francs. Each manual included five samples, and three of those did not involve quaternes at all, thus suggesting that a large part of the Loterie’s customers preferred the bets with shorter odds, and payoffs closer to the actual chances. These were, of course, sample tickets constructed to instruct agents on a range of possibilities. How often bettors tried such bets, if ever, is unknown. Since the winnings were in principle a constant multiple of the

136  C h a p t e r T h i r t e e n

T abl e 1 3 . 1 . A reconstruction of the amounts bet in a random sample of 327 quaterne bets in the Loterie over the years 1797 to 1833. Amount bet ( francs)

Number of bets

0.05 to .20

198 (61%)

0.21 to .50

68 (21%)

0.51 to 1.00

38 (12%)

1.01 to 6.00

22 (7%)

10.00 Total

1 (.3%) 327

amount bet, we can work backwards from Menut’s data to infer the amounts that actually were bet. I say “in principle” since the sums reported are only approximately consistent with this supposition; apparently, varying agency fees were extracted from the winners. Table 13.1 shows the frequency distribution of payoffs, and indicates that, with few exceptions, the wagers on quaternes were quite small— usually only five to twenty centimes. More than 90 percent of them were one franc or less. Clearly, most gamblers bet only small stakes, but some were willing to risk enormous sums. Who were they, and where were they? Tables 13.2 and 13.3 break down the bets by the cities in which the Loterie was drawn, and table 13.4 lists all of the 331 big winners over this period, by locations of the drawings and by locations of the agencies where the bets were placed. Recall that Parisian bettors could purchase tickets for a drawing at any Loterie, while others had a choice of betting on the drawings in Paris or on the Loterie in their own section of France. The overwhelming impression from these data is that Paris was the center of activity: 200 of the 331 winners (or 60 percent) were drawn in Paris; 143 of 331 (or 43 percent) were paid to bets placed in Parisian bureaus. Now, in 1836 the population of France was recorded by census as 33,333,019 (exclusive of Corsica), of whom 909,126 were in Paris and 1,106,891 in the Department of the Seine. By those num-

How Much Did They Bet, and Where?  137

T a b l e 1 3 . 2 . Estimated quaterne wins in French Loteries 1797– 1833, by cities in which they were drawn; and 1836 city populations, in thousands. Quarterne wins

Percentage of wins

Population

Percentage of population

Paris

197

60%

909

70%

Lyon

47

14%

151

12%

Strasbourg

32

10%

58

4%

Bordeaux

25

8%

99

8%

Brussels/Lille*

26

8%

88*

7%

327

100%

Loterie

Total

1,305

100%

*The Brussels Loterie moved to Lille in 1814; the population figure given here is the average of the two cities’ 1836 populations in thousands.

Tabl e 13.3. Estimated Loterie quaterne bets 1797– 1833 by city drawn, and relative to number of drawings Drawings

Estimated number of bets

Bets per drawing

Paris

1,259

100,674,486

79,964

Lyon

1,171

24,018,786

20,511

Strasbourg

1,161

16,864,254

14,085

Bordeaux

1,156

12,775,950

11,052

Brussels to 1814

475

6,132,456

12,910

Lille from 1814

694

7,154,532

10,309

5,916

167,109,426

28,247

Loterie

Total

bers, Paris was greatly overrepresented in the Loterie. But France at that time was predominantly agricultural, and (again by the 1836 census) only 2,427,992 of its citizens resided in towns with populations of 25,000 or more; and 37 percent of these people were in Paris. Viewed this way, the bets on the Loterie were not that far from being uniformly distributed across the urban population of France at that time. The list in table 13.4 shows that the bets did come from all over,

T abl e 1 3 . 4 . Where the Loterie bets were placed: 331 large winners from 1797 to 1833, listed by the agency office where each bet was placed and the Loterie location where each was drawn. From Menut 1834. Agency

Paris Lyon Bordeaux

Strasbourg

Brussels

Lille

Total

Paris

86

18

10

17

7

5

143

Lyon

9

9

0

0

0

0

18

Bordeaux

5

0

11

0

0

0

16

Strasbourg

3

0

0

8

0

0

11

Brussels

0

0

0

0

0

0

0

Lille

1

0

0

0

0

0

1

Grenoble

5

4

0

0

0

0

9

Marseille

7

4

0

0

0

0

11

Metz

2

0

0

4

0

0

6

Caen

5

0

0

0

0

0

5

Rouen

5

0

0

0

0

0

5

Dijon

4

0

0

0

0

0

4

Bayonne

3

0

1

0

0

0

4

Toulouse

3

0

0

0

0

0

3

Amiens

3

0

0

0

0

1

4

Nantes

2

0

0

0

0

1

3

Perpignan

2

1

0

0

0

0

3

Toulon

2

1

0

0

0

0

3

Versailles

2

0

0

0

0

1

3

Orléans

0

0

0

0

1

2

3

L’Orient

1

0

0

0

1

0

2

Besançon

1

1

0

0

0

0

2

Boulogne

2

0

0

0

0

1

3

Le Hâvre

1

0

0

0

0

1

2

Montpellier

1

1

0

0

0

0

2

Rochefort

1

0

1

0

0

0

2

Les Andelys

2

0

0

0

0

0

2

How Much Did They Bet, and Where?  139

T abl e 13. 4 . (cont.) Agency

Paris Lyon Bordeaux

Strasbourg

Brussels

Lille

Total

Coutances

2

0

0

0

0

0

2

Nîmes

2

0

0

0

0

0

2

Hagenau

0

0

0

2

0

0

2

Bêziers

0

2

0

0

0

0

2

Châlons s. S.

0

2

0

0

0

0

2

Agencies with one quaterne winner at a Loterie: Lot. de Paris: Angers, Anvers (Antwerp), Beauvais, Blois, Calais, Colmar, Compiègne, Crefeld (Krefeld), Dourdan, Eveux, Ferney, Gand (Ghent), Genève, Laon, Mantes, Meaux, Melun, Mülhausen, Namur, Nancy, Nemours, Neuchâtel, Nogent-le-Rotrou, Pau, Perigueux, Provins, Quimper, Rennes, Rhodes (Rodez), Rouffach, Salins, Sedan, Thionville, Tours, Valance, Vannes, Vitry, Weissembourg. Lot. de Lyon: Arles, Le Puy, Bagnois, Turin. Lot. de Bordeaux: Poitiers, La Rochelle. Lot. de Strasbourg: Landau, Mayence (Mainz). Lot. de Bruxelles: Chartres, Le Mans, St. Denis. Lot. de Lille: St. Lô, St. Omer.

from small towns as well as large. Presumably the descendants of Fermat were betting in Toulouse, and the relatives of Laplace and Quetelet were betting in Caen and in Gand. With small bets spread broadly over the urban population, the appeal of the Loterie seems to have been universal. The spread was not uniform. The map at the upper left figure 13.4 shows the eighty-two cities in France where at least one large winner won over this period. There is a greater concentration along the northern coast, in the area around Paris, near the German border, and along the Mediterranean coast. These were the affluent parts of France. To get an idea of what these parts of France were like at that time, consider the three contemporary thematic maps also shown in figures 13.4. The map at upper right in the figure was prepared by Charles Dupin in 1827 to show levels of education in France. The districts were shaded according to the number of students per capita, with lighter shading indicating more education. In the map seen at lower left, Adolphe d’Angeville exhibited in 1836 the concentration of industry in the different departments; darker is more. And in the map at lower right, A. M. Guerry in 1833 showed the level of donations to the poor, an indication of disposable income. Altogether we see that

140  C h a p t e r T h i r t e e n

F ig ur e s 1 3 . 4 . (t op l e f t ) A map of France on which each of eighty-two cities where a large winning ticket was sold is marked with a penny. One additional city, Antwerp, is off the map. (top r ig ht ) Map from Dupin 1827, showing numbers of students per capita (lighter is higher). (b o t t om l e f t ) Map from d’Angeville 1836, showing levels of industry (darker is higher). (b o t t om r ig h t ) Map from Guerry 1833, showing levels of donations to the poor (darker is higher). Source for d’Angeville 1836: Bibliothèque nationale de France.

the Loterie was most popular in the richer, more educated, and more industrial areas, and much less popular in the agricultural heartland. The largest recorded payoff, 689,620 francs, was awarded to a bettor in Marseille who bet the quaterne for the drawing held on 9 August 1810 at Lyon. Presumably the bet was for ten francs, an extraordinary sum for such long odds, and some agency fees account for the dis-

How Much Did They Bet, and Where?  141

T abl e 13. 5 . The 189 large wins in cities with regional Loteries, 1797– 1833. Calculated on the basis of data from Menut 1834. (a) Number of large wins, cross-classified by the location of each Loterie drawing and the location of the agency where each bet was placed. The regional Loterie moved from Brussels to Lille in 1814.

Loteries

Agencies Paris

Lyon

Strasbourg

Bordeaux

Brussels

Lille

Total

Paris

86

9

3

5

0

1

104

Lyon

18

9

0

0

0

0

27

Strasbourg

17

0

8

0

0

0

25

Bordeaux

10

0

0

11

0

0

21

Brussels/

12

0

0

0

0

0

12

143

18

11

16

0

1

189

Lille Total

(b) Total amount won, cross-classified on the same basis as in (a).

Loteries

Agencies Paris

Lyon

Strasbourg

Bordeaux

Brussels

Lille

Total

Paris

2,426,945

200,880

69,254

107,598

0

5,012

2,809,689

Lyon

1,174,645

156,306

0

0

0

0

1,330,951

Strasbourg

450,970

0

214,582

0

0

0

665,552

Bordeaux

164,790

0

0

217,882

0

0

382,672

Brux/Lille

219,971

0

0

0

0

0

219,971

4,437,321

357,186

283,836

325,480

0

5,012

5,408,835

Total

crepancy. Menut reports that nearly as large a sum (683,000 francs) was won on 9 October 1824 by a Parisian, also betting on the drawing at Lyon. However, that bet was on a terne at more favorable odds, and if the reported win is not a typographical error, the sum wagered would have been 125 francs. This is such a huge amount to risk at those odds that it would raise the suspicion of fraud. Table 13.5 gives a slightly different view of a portion of these data, dealing with bets made in the cities with regional Loteries. In particular, it displays the trade between cities. The bets predominantly orig-

142  C h a p t e r T h i r t e e n

inated in Paris. Three-quarters of these were made for Parisian drawings, but a quarter were directed to the regions. On a per-capita basis, the distribution of originating bets among the cities was fairly even, with the population of Lyon being slightly less interested in betting, although this difference does not pass the screen of a significance test. The sizes of the bets— as measured by the average size of the winnings— are fairly uniform as well, with one exception. Bets placed in Paris on the Lyon Loterie yielded a payoff of 1,174,645 francs, nearly half as much as was paid for bets placed in Paris on the Paris Loterie. This is largely due to the 683,000 francs won on the earlier mentioned terne, which was bet in Paris and won on 9 October 1824. But, coupled with the earlier noted phenomenon of a cluster of four quaternes in Lyon won by Parisian bettors in seven months, this provokes a suspicion that something was afoot. After all, is it not surprising that a bettor would wager 125 francs (on the order of 2,500 euros today) on a 11,748-to-1 bet to be drawn in another city? Of course, people have always done surprising things. Aside from the unusual success of Paris bets in Lyon, there is one other phenomenon that stands out here. Paris bettors were by far the most active: 75 percent of the big wins came from agencies in Paris. And bettors in that city won 82 percent of the awards from these wins. Sixty percent of the Paris bets were placed on the Paris Loterie; the remainder were spread fairly evenly over the Loteries in four other cities. Strasbourg had about half the population of those other cities, but its Loterie did as well or better than the others did. Presumably the Strasbourg Loterie benefited from border crossers from Germany.



Fourteen



Muskets, Fine-Tuned Risk, and Voltaire

A l r e a d y b y 1 7 0 0 t h e J e s u i t s c h o l a r Claude-François Ménestrier, writing in Lyon, could list five categories of lottery, none of which included the species that principally occupies this book. He listed 1. state and political lotteries, which he associated with England; 2. commercial lotteries, used to sell or allocate buildings, real estate, or merchandise, which he associated with the Netherlands; 3. lotteries as games, simply for entertainment with small nonmonetary prizes; 4. lotteries by princes or lords to distribute gifts by chance to random selections of people; and 5. charity lotteries, to raise money for the poor or for churches.1

These were not small categories. A listing by Ewen of substantial British state and private lotteries starts with Queen Elizabeth I in 1567– 69 and includes well over 200 entries before 1826, including 126 British state lotteries between 1769 and 1826. A book by Florange devotes 570 pages to French lotteries before 1873 that involved bonds and annuities. Recent histories by Legay and Bernard paint broader pictures of considerable activity throughout the period. Archives and collections of lottery ephemera show, through large numbers of los-

144  C h a p t e r F o u r t e e n

ing tickets and advertising bills, that a considerable amount of societal energy was dedicated to these enterprises.2 All of these other lotteries differ from the French Loterie and its relatives in one important way. The Loterie was designed as a series; the others were each a separate offering, even though there could be substantial similarity between successive examples. As Ewen states, “The officials seem to have been at great pains to vary details in the most wanton fashion, so that hardly ever were two schemes alike, yet throughout the entire run of State Lotteries (1664– 1826) the laborious method of drawing remained the same.”3 In contrast, for the Loterie— aside from some minor changes in payoffs in the 1770s, the brief experiment with primes gratuites in 1776, and a few early changes in bets offered— each drawing from the 1750s to the 1830s was the same in all important respects. Because the other lotteries tended to be started anew at each offering, the temptation to reinvent and redesign was irresistible, and was not resisted. Human ingenuity rose to the occasion, as evidenced by the numerous unsuccessful proposals that accompany losing tickets in archives, or even in print.

Hautefeuille One dramatic example came from the pen of the early French economist Jean de Hautefeuille (1647– 1724).4 He made several unsuccessful proposals, but one in particular catches attention for its audacity. In a short pamphlet in 1713, he proposed a new way to draw a lottery “in the presence of all the people of Paris.” This method was not limited to any specific type of lottery; it could be used in any case where one or more numbers were to be drawn at random to determine the winner. He would construct a “round table of whitened fir, eight or nine feet in diameter, divided by black lines in ten equal parts, from the circumference to the center. Between each of these divisions will be painted, in very large characters, the ten primitive numbers, 1 2 3 4 5 6 7 8 9 0, not in this natural order, but between the big and the small, as we see in the figure which will produce a greater effect of chance.” He provided a picture (figure 14.1)

Muskets, Fine-Tuned Risk, and Voltaire  145

F ig ur e 1 4 . 1 . From Hautefeuille 1713. The author proposed placing a large spinning disk on a boat on the Seine and selecting numbers by musket fire at the disk; the figure shows the results where four fired balls would give the locations N, D, C, M, or “1713.” Source: Bibliothèque nationale de France.

Hautefuille would mount this large disc on a spindle of iron, bent in the middle to allow the disc to be viewed from the side, and he would secure the spindle in a frame on a small boat in the Seine, anchored between the Pont Royale and the Pont Neuf. Ropes would be attached to the disc in such a way that two men pulling them could set the disc to rotating with great speed. A second, larger boat with sufficient adornments to comfortably seat the lieutenant general of police and other observers, would be anchored thirty or forty paces downstream. On the bow of the large boat, facing upstream toward

146  C h a p t e r F o u r t e e n

the disc, would be a musket loaded with one ball and aimed at the disc. At a designated time a child would fire a rocket, and a shooter would then discharge the musket, hitting the rapidly spinning disc. The boatmen in the small boat would then move that boat to bring the disc close to the lieutenant general of police, who would certify what viewers from shore would have already seen: which of the ten numbers was hit— say, 7. The process would then be repeated, giving, say, 4. The winning number would then be 47. If a four-digit number were needed, the process would require four steps. Hautefeuille described an ornate ceremony, with the lieutenant general arriving on horseback to be ushered to the large boat by trumpets, drums, and oboes. He acknowledged that while one disc was enough, the process would be sped up if two or more were used. In doubtful cases the lieutenant general’s decision would rule. There is no record that the procedure was ever tried.

The British First Classis Lottery of 1711 In summer 1711, a new type of “blanks” lottery was introduced in London. It was referred to as a classis lottery, and it bore some similarities to earlier lotteries run in the Netherlands since the 1560s, but was structured on a different principle. It was informally described as “the adventure of two millions” since the aggregate prize value exceeded £2,600,000. Before the drawing, 20,000 tickets were sold, each for £100, a very large sum at that time. Two or more people could share a ticket; fractional ownership was permitted. The prizes would be dispensed as annuities over sixteen years, and all tickets were guaranteed an annuity of at least £110, so in a sense this was a fancy way to get the public to loan money to the government to pay war debts. All 20,000 tickets were combined in one wheel (see figure 4.1 in chapter 4 for an example of the type of wheel). The prizes (there were no “blanks”) were divided among five different wheels or “classis.” The drawing was to start with the first classis and continue through the fifth classis over about two weeks, the number and size of prizes increasing with each classis. The details can be found in Ewen’s book,5 but for our purposes a summary will do. Each draw would consist of

Muskets, Fine-Tuned Risk, and Voltaire  147

T abl e 1 4 . 1 . Prize allocation among classis at the 1711 British lottery (Ewen 1932, 137) First classis (1,330 tickets)

Third classis (4,000 tickets)

Fourth classis (5,340 tickets)

Fifth classis (6,660 tickets)

1 at £3,000

1 at £4,000

1 at £5,000

1 at £20,000

50 at £200

1 at £2,000

1 at £3,000

1 at £4,000

1 at £5,000

1,279 at £110

1 at £1,000

1 at £2,000

1 at £3,000

1 at £4,000

1 at £500

1 at £1,000

1 at £2,000

1 at £3,000

4 at £400

1 at £500

1 at £1,000

1 at £2,000

5 at £300

4 at £400

1 at £500

1 at £1,000

100 at £200

5 at £300

4 at £400

1 at £500

2,557 at £115

150 at £200

5 at £300

4 at £400

3,836 at £120

200 at £200

5 at £300

5,125 at £125

250 at £200

1 at £1,000

Second classis (2,670 tickets)

6,394 at £130

one ticket being taken from the main wheel with the twenty thousand tickets, and it would be matched with one ticket taken from that day’s classis, both without replacement. The draw would go on until that classis wheel was empty, and then the drawing would move on to the next classis. It took more than a day to draw an entire classis. The largest prize was an annuity for £20,000 to be paid out over thirtytwo years, awarded with the fifth classis. The other prizes would be spread around; for example, the third classis would comprise 4,000 tickets drawn, and the five largest prize values were £4,000, £3,000, £2,000, £1,000, and £500 (table 14.1). In addition, there were four prizes at £400, five prizes at £200, and 150 prizes at £200. All of the remaining 3,836 tickets in that classis would be rewarded annuities of £120. Figure 14.2 gives the results of the first day of drawing the third classis. The awards list shows two deviations from the description I have given. One is that the first and last draws within each classis were

Fig ur e 1 4.2. Awards of premiums at the classis drawing on 4 August 1711. Note the bonuses given for the first drawn ticket in the day, and the draws immediately before and after the large prize winners.

Muskets, Fine-Tuned Risk, and Voltaire  149

awarded £500 automatically as a premium. In figure 14.2 the first drawn ticket, no. 14,417, was awarded £500 on that basis, but because the classis was not completed on August 4, there was no last drawn bonus that day. Also, on that day, three of the four largest awards for that classis were drawn: £3,000 for ticket no. 9,539, £2,000 for ticket no. 19,398, and £4,000 for ticket no. 7,718. And these take us to the second deviation, which is the reason why I selected this particular lottery for presentation. The design of a lottery should maximize its attractiveness to the bettor, subject to whatever constraints are imposed on the needs for a profit. Mostly that involves setting the prizes at a level where there is a tradeoff between the number of prizes awarded and the size of the largest prize. For the 1711 classis lottery this meant offering everybody a prize that was at least the size of the £100 bet, and not emphasizing that, since the payment of prizes would take place over many years, the lowest prizes would be net losses if the interest rate was taken into consideration. The largest prize here, £20,000, was meant to dazzle, as Casanova would later say; but it would make only one of 20,000 ticketholders extremely happy. The 19,999 others would suffer various forms of disappointment. Psychologists now know that not all disappointments are equal, but this lottery was unusual in recognizing that in the design. Bettors who have “just missed” a large prize will feel more aggrieved than those who were not even close. This lottery built into the payouts a special allowance for that. If your number was drawn immediately before or immediately after one of the larger prizes (£1,000 or larger, here), you would receive a special premium: a sum equal to a year’s interest for the prize you had just missed, 6 percent at the time of the drawing.

The Tomkins Picture Lottery Let us now consider a private lottery with an entirely different design and an unusual angle that provides a contrasting approach: the Tomkins Picture Lottery of 1821. P. W. Tomkins (1759– 1840) was an

150  C h a p t e r F o u r t e e n

accomplished engraver who wished to hold a lottery to dispose of his large collection of paintings, watercolors, and prints, in order to obtain funds to cover his large debts. To accomplish this, he got the House of Commons to approve his holding a lottery to sell the collection. The prizes consisted of 16,550 lots, each consisting of multiple pictures, some in “elegant frames” and some in volumes that were “superbly bound.” The grand prize was valued at £7,500; the total value placed on all lots was £152,225. The 14,816 least valued lots of the 16,550 were valued between £4 and £9. A total of 16,550 × 2 = 33,100 tickets were to be sold at three guineas each (one guinea was equal to one pound and one shilling, or £1.05). That would bring a total of £104,265 to the seller, presumably enough to clear his debts. All prizes were on view at the lottery office. The unusual design feature that makes this lottery worth mentioning here was that there would be two sets of tickets, each set numbered from 1 to 16,550. The sets were distinguished by one set of tickets being printed in black ink and the other set printed in red. The prizes were to go entirely to only one of these sets; the winning set would only be determined with the last drawn number. If the last drawn ticket had an even number, all the red tickets would be winners; if an odd number, all the black tickets would win. This allowed the lottery bill to use the sales pitch, “The purchaser of a red ticket and a black ticket is sure to gain a prize.” Perhaps this would double the sales, unless the bettor realized that purchasing one of each color would also guarantee a losing ticket as well. These two lotteries offer two contrasts: one between the two of them, the other between these two and the Loterie. The 1711 classis lottery presented a novel wrinkle designed to reveal the results slowly, one ticket at a time, with hopes rising and falling as the draws went on. But it would diminish the disappointment for at least a few otherwise losing bettors: the 38 out of 20,000 who preceded or followed a big winner. The Tomkins Picture Lottery was designed to guarantee that all 33,100 tickets remained in play to the very last draw, when fully half of them would be immediately disappointed, relegated to

Figur e 1 4.3. Advertisement for the Tomkins Picture Lottery in 1821, with the “perfectly novel scheme” that awards all prizes to half the tickets, with that half decided only by the last drawing.

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the “no prize” category at the moment when the last ticket revealed the winning color. The contrast with the Loterie is more dramatic. As Casanova had emphasized, in the Loterie the bettor was much more in control. A bet on a quaterne could be insulated from the “near miss” phenomenon by the simple device of betting on the same numbers for all possible ternes, ambes, and extraits, and that could be done all on the same ticket! Indeed, it appeared that most quaterne bettors did something like that, to judge by the surviving tickets. How did the two design innovations work in the marketplace? The 1711 classis design or slight variations did appear in a few other places on the continent. Ewen says it may even have been used as early as 1446 in Bruges, but he gives no details or citation.6 Inspection of three large scrapbooks of tickets shows it was used in the Loterie de la Charité de Lyon in 1712, in several lotteries in the Netherlands over the period from 1731 to 1742, and in a Hamburg lottery in 1796. But from Ewen’s detailed descriptions, the 1711 appearance in London was the only such use in Britain before the period from 1819 to 1826, when it was added to several of the state lotteries as the date of the suppression of British state lotteries approached, and the lotteries were working feverishly to extract as much profit as they could before the axe fell on 18 October 1826. Apparently, the gain in sales was in most instances judged to not be worth the trouble. The Tomkins design suffered a different fate, and the plan may never have been used again. On 27 June 1817 the House of Commons gave Tomkins permission to hold the lottery. Sales of tickets began before March 1819, and the lottery drawing was initially promised to take place before December 1820. Sales must have been slow, because in December 1820 Tomkins asked the House for a delay and it was granted, to no later than the end of July 1821. But on 26 June 1819 a fire at the print shop destroyed some of the prizes, and Tomkins delayed the sale, apparently holding it on 24 July 1821. Ewen seemed unsure that it made that date, and there seems to have been no subsequent use of the scheme.7 A design that conceivably left half the participants tasting sudden and bitter defeat would not be destined for popularity. In any event, Tomkins was able to carry on in his business for several more years.

Muskets, Fine-Tuned Risk, and Voltaire  153

Voltaire and the Lottery The ubiquitous presence of unexecuted proposals for lottery innovations in archives and collections testifies that many, perhaps most new ideas were not pursued. It is also true that not all ideas that were pursued succeeded, and some of the failures were spectacular. One such failure involved the poet-philosopher François-Marie Arouet (1694– 1778), better known by his pen name, Voltaire. About 1715, the French government had taken on debt to improve and maintain Paris’s large city hall, the Hôtel de Ville. The debt was in the form of perpetual annuities sold to the public, mostly to the upper classes, and these provided monthly payments to the holders. By the late 1720s the maintenance of the debt— those payments— had become a burden, and the finance minister at the time, Michel Robert Le Peletier des Forts (or DesForts) developed a plan to get the holders of the annuities to turn them in for what he considered to be a reasonable reimbursement. With the financial support of the king, they would hold a lottery, where the holders of debt would tender their annuities to the government for reimbursement, and as an incentive each would get a ticket in a lottery to be held monthly, where the king would guarantee prizes to a total of at least 500,000 francs each month. In 1728, at a dinner party hosted by a Madame Dufay, Voltaire and Charles Marie de La Condamine (1701– 74) discussed the plan for the lottery and discovered a flaw that opened a door for a sure profit. The plan rested on two key conditions of the lottery. First, the Act of 19 October 1728, which Le Peletier des Forts wrote creating the lottery, gave one lottery ticket at no cost to each submission for reimbursement, regardless of the size of the tendered annuity: A onefranc annuity and a 1,000-franc annuity would each receive equal chances in the lottery. Second, the very large guarantee of 500,000 francs bore no relationship to the amounts tendered. If Voltaire and La Condamine could subdivide an existing contract to one-franc components and form a group they called “le Société” to acquire nearly all the lottery tickets each month, they would be guaranteed a huge prize for a small investment. Since the lottery was only offered

154  C h a p t e r F o u r t e e n

to annuity holders, the number of tickets was not so large that this was impossible. And they actually managed to pull it off, with many individual contracts of one franc each. The best account of the episode, by Jacques Donvez, uncovered most but not all of the details.8 Evidently Voltaire reached an agreement with some of the notaries charged with dispensing the lottery tickets. Donvez writes that the notaries “had to keep registers in which they wrote down the number of the ticket, the names or words or identification that each annuitant will want to choose, the amount to which he was contributing and the price of the ticket. The note [in the Archive] explains verbatim that ‘he [Voltaire] snatched all the Registers without filling them.’ So he bought the blank tickets from the notaries, then left to write in any names. The notaries obviously had an understanding with the philosopher.”9 The lottery began in January 1729, with monthly drawings. The Voltaire– La Condamine plan began only with the February drawing, where thirteen one-franc annuities in La Condamine’s name returned 13,000 francs in lottery winnings. The coconspirators presence grew in the next few months, and after the 9 April drawing Voltaire moved back to Paris and they found their pace. From then on, the list of winners never failed to include names of members of le Société. Some of the names were those of real people, acquaintances of Voltaire who may not have known they were members; and many other names were fictional creations. After the first few months the Société members apparently won nearly everything; in October 1729 the total awarded in the drawing was 1,040,000 francs, all but 36,000 of which went to le Société. The last drawing was 9 June 1730, but no data after February 1730 is available. Still, Donvez found no evidence that le Société ceased its activity earlier than the last drawing. Their gains must have been several millions of francs. When the finance minister learned what was going on, he declared that Voltaire and La Condamine’s plan was contrary to his intention, and he attempted to refuse payment. Voltaire declared that this was unjust, and on appeal, the Council of State ruled in his favor, declaring that the plan may well have not been as des Forts intended, but it

Muskets, Fine-Tuned Risk, and Voltaire  155

was as the act had been written. Le Peletier des Forts was removed as finance minister on 19 March 1730, apparently for reasons unrelated to the lottery. In one memoir, Voltaire described him as “a Tartuffe”— a reference to the lead character in Moliere’s 1664 comedy The Imposter, a wicked hypocrite. Voltaire and La Condamine were well-to-do before this and quite wealthy afterwards. La Condamine had a distinguished career as an explorer, geographer, and mathematician. In the period 1735– 44 he explored in South America, helping to measure a meridian arc near Quito in what was then Peru and mapping the Amazon.



Fifteen



The Loterie in Textbooks and Manuals

The Loterie and Mathematical Education As long as there have been mathematics textbooks, authors have incorporated methods with current practical applications to attract use by teachers and to engage students. The teachers have often been more appreciative of this than the students. For example, Euclid’s proposition 3 on simple proportionality was the basis of pro rata pricing in commercial arithmetic, and in the eighteenth and nineteenth centuries this became a cornerstone of elementary mathematics under the name “the rule of three”: If a/b = c /d and any three of these four quantities were known, the fourth could be found readily.1 For years, students in introductory mathematics courses across Europe were tortured by problems involving the rule of three, where the nondecimal units (feet and inches; pounds, shillings, and pence; hours, minutes, seconds) could turn a simple calculation into a nightmare. Problems involving the Loterie played a similar role. To teachers wishing to engage young minds, the Loterie was an attractive opportunity, and it seems every textbook from the 1780s onward seized upon it. Basically, they all had the same treatment, using combinatorial counting to cover the simple evaluation of the chances of winning the various bets. There were marked differences nonetheless. Let us look at three of the earliest and most important examples of books published during the years of the Loterie with educational intent: a 1783 book by Charles-François de Bicquilley; a book by Nicolas de Condorcet, written in the late 1780s and pub-

The Loterie in Textbooks and Manuals  157

Fig ur e 15.1. Bicquilley’s comparisons of the Loterie payoffs for the extrait, ambe, terne, quaterne, and quine, to what the payouts would be if each were a bet with no advantage to the Loterie. Bicquilley 1783, 117.

lished posthumously in 1805; and Silvestre François Lacroix’s 1816 text. Condorcet’s book was a draft for an educational program he had planned before the Revolution, working with Lacroix.2 All of these books included basic calculation of the probabilities of winning for all of the Loterie bets, as a simple application of combinatorial calculation (see appendix 1). Condorcet had written a Discours when he was working on his program in the 1780s, where he began a description of his plan this way: “Games of chance and lotteries come first; this is the simplest case, since the number of possible combinations is known and finite, and the probability of each can be determined.”3 All three of these books also reported the contrast between these probabilities and the odds at which the Loterie paid off the winning bets. For example, Bicquilley found the probabilities as fractions (e.g., the probability of winning a terne as 1/11,748), and then stated it as one chance in 11,748— meaning that if a successful unit bet returned 11,748 it would be a fair bet, contrasted with the Loterie’s payoff for a successful bet on a terne of one unit bet being paid 5,200. He showed in a table how these differences (e.g., 11,748 – 5,200 = 6,548) increased as the chances decreased. Bicquilley then gave a short statement that could be considered an apology for the Loterie’s profits: “I believe it is unnecessary to remind the Reader here that the fairness of which we have just spoken is purely numerical, & in no way prejudices the legitimate fairness which can be found in the advantages to the Lottery in view of the freedom that the bettors have in choosing according to their interests, the essential expenses of its administration, as well as the useful

158  C h a p t e r F i f t e e n

& laudable purposes to which these profits are put to use.”4 Bicquilley agreed with Adam Smith in noting that an agency would require a profit to offer such a service,5 at least to the level of expenses, and he foresaw the dubious claim that state lotteries use today, often in the form of the statement, “All proceeds go to support education”— a claim that is never audited, and which is meaningless when the liquidity of the funds is recognized. Bicquilley apologized; Condorcet and Lacroix indicted. In the Discours whose first sentence is quoted above, Condorcet immediately went on to say that the only usefulness for the application to games and lotteries was “that of proving how vain are all the hopes of those who indulge in these games, those who are too often the dupes and the victims: perhaps mathematics, by demonstrating the ridiculousness of their speculations, will have more effect than a moralist in exposing the disastrous consequences.”6 In the 1805 book, Condorcet emphasized in colorful terms the hopelessness of winning the large prizes: To give an idea of these last probabilities of winning to those who are not accustomed to this kind of calculation, we will observe that the expectation of winning a Quaterne is less than the risk of a man of age fifty of dying of apoplexy in an hour. That of winning a Quine is less than the risk of seeing two out of eight people of age fifty being stricken with apoplexy in a day; or, if you will, the risk of two people of that age being stricken with apoplexy in three days. He who would bet a Quaterne for each draw, would have an equal probability for the draw or non-draw of his ticket only after 376288 draws or more than 15678 years. He who would bet a Quine would have an equal expectation of draw or non-draw only after 30103000 draws, or 1254292 . . . years. If we suppose the lottery were created six thousand years ago at the same time as the world, it would still be more than 9668 to bet against 362, or 300 against 1, that his Quine would not yet have been drawn.7

Condorcet assumed two drawings a month, which was only true from 1776 to 1800. Lacroix, Condorcet’s collaborator in the 1780s,

The Loterie in Textbooks and Manuals  159

published his own much more advanced treatment in 1816, but he echoed the earlier work with different numbers, adapted to the expanded Loterie, now with many more drawings per month. The times he gave were down a bit from Condorcet’s, but still absurdly large. To get an idea of the difficulty of winning, he wrote, just consider that by making 2 draws per month in 4 different cities, which would give 96 draws in a year, it would take more than 457,804 years for all Quines to appear, assuming that none would have been repeated during this immense interval. The appearance of all the Quaternes would require, under the same hypothesis, more than 5,323 years: one would find much more still, if . . . we were looking for the number of draws it would take to have probability 1/2 of winning a Quine or a Quaterne.8

Condorcet and Lacroix attacked a problem that is still with us: How do you explain a really remote chance in terms that an audience might understand? Their efforts were unusual at the time, and creatively ingenious. But their premise— namely, that if they could explain the futility of winning a big prize, then the bettors would desert the Loterie— was arguably wrong. The idea among mathematicians then, and sometimes today, that it is gullibility and ignorance that propels most gamblers, is dubious at best; and in the case of the Loterie, it is wrong as to the judgment of remote chances, as we saw in chapter 13. The gamblers overwhelmingly preferred the short to the long odds.

Laplace Laplace’s most widely read work was his Essai philosophique sur les probabilités.9 That work began life as a lecture he prepared for the École normale of 1795– 96, when the Revolutionary government arranged to bring the best scientists together to give presentations accessible to common citizens. In principle, there were no technicalities in the lectures, and Laplace used no formulas or mathematical notation, but there were places where his success in engaging a general au-

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dience was debatable. His reference to Euler’s number e = 2.71828 . . . as “the number whose hyperbolic logarithm is unity” was not helpful for many readers. But in general, his prose was elegant and clear. Laplace’s lecture was first published in the volumes of the Séances des écoles normales in 1795, and these were reprinted without change in 1800. From that point on, the work evolved, with Laplace adding and rewriting at every publication. Three different versions were published from 1801 to 1813, and three more versions appeared in 1814, including the lecture’s first appearance as a separate book and a second edition of the same. The book itself then went through a total of five editions from 1814 to 1825, some with major changes.10 Laplace’s treatment of the Loterie evolved in interesting ways. In the Séances of 1795 there was little mention of the Loterie, perhaps due to the lecture being given during the short period of “suppression” at that time. But the Loterie was not ignored, and its brief mention was marked by an appearance of a new argument, at least new to the discussion of the Loterie. In 1738, Daniel Bernoulli published what is now seen as a remarkable early statement of modern utility theory (figure 15.2). He observed that the value to an individual of a unit of money depended upon the individual’s fortune; this led him to what is now called the principle of diminishing marginal utility. More specifically, Bernoulli speculated that the value in utility of a unit sum gained by an individual was inversely proportional to the person’s total fortune.11 Laplace accepted Bernoulli’s observation and his principle, referring to Bernoulli as “quite ingenious,” and he used it as support for his accusation of “the immorality” of the Loterie. He considered a simple bet. Suppose you begin with a fortune of 100 francs, and bet ten francs with equal chances of winning or losing. The expected value of the outcome of the bet in ordinary terms is 100 francs, making it in classical terms a “fair” game. But the expected value in terms of utility is less than the utility of the initial 100-franc fortune, since the utility of ten gained to a fortune of 100 is less than the utility of ten lost. The pain of loss exceeds the joy of winning. If each additional franc is worth less, then a bet with “fair” expected value in terms of francs

The Loterie in Textbooks and Manuals  161

Fig ur e 15.2. This figure shows Bernoulli’s value or utility function. The vertical axis QA gives the value corresponding to the fortune (axis AR). As the fortune increases, so does the value of a unit gain in fortune, though at a decreasing rate. The curve is a logarithmic function. Bernoulli 1738, fig. 5.

will always be an unfair bet in terms of expected utility. So even a “fair” bet runs counter to the individual’s own values, and of course the Loterie is not a “fair” bet. To Laplace, it was therefore immoral. The argument is an interesting use of Bernoulli’s idea, but it is not a strong argument. A single-franc change in a fortune of 100 francs is imperceptibly smaller on the upside than on the downside; for ten francs, the difference is still not great. Only when the bets are a ma-

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jor fraction of the fortune does the difference become pronounced. And Laplace seems to have realized this. Between the 1810 version of the Essai and the 1812 version, he changed the example. Bernoulli was no longer described as “quite ingenious,” and the comparison was now given as a fortune of 100 and a bet of 50, so the change was now a substantial fraction of the fortune. Interestingly, those new numbers are exactly the same as the numbers Bernoulli gave in the example he used in 1738. This change persisted through the versions of the Essai from 1814 through 1825, but with mention of Bernoulli moved to the “Historical Note” at the end of the book, and with his “ingenuity” restored. Laplace did a few other calculations on the Loterie in his works, but the most remarkable was the one already mentioned in chapter 12, in which he calculated the median of the distribution of the number of drawings needed to achieve the appearance at least once of every one of the ninety numbers. Starting in 1772, he kept coming back to the problem until he finally succeeded in 1812: to a very good approximation, the answer lay between eighty-five and eighty-six. He included this result in the 1814 Essai, as a warning for how long a bettor had to wait before he could have even weak confidence in his favored numbers appearing. The calculation Laplace presented was a tour de force involving difference equations and delicate asymptotic approximations. Though others, including Lacroix,12 reported Laplace’s result, no one bettered his analysis. There was, however, one person who thought he had done so. In 1834 a M. Gauthier d’Hauteserve published a small elementary treatise on probability in which he claimed to get Laplace’s same result, eighty-five, by simple arithmetic.13 He constructed the table in figure 15.3 as follows. He supposed that the drawings proceeded in a nonrandom manner until only one of the ninety numbers had failed to appear. The nonrandom manner was based on the fact that with each drawing, the expected count of new numbers to appear would be exactly 17/18 times the count of numbers that had not yet appeared. So, start with ninety undrawn numbers at the first drawing; the expected count of the remaining undrawn numbers after that drawing

The Loterie in Textbooks and Manuals  163

(and in this first case it is the exact count) would be 90 × 17/18 = 85. The expected count remaining after the second draw would then be 85 × 17/18 = 80.278 to three places (yes, keep the fraction). Continue multiplying by 17/18 with each drawing, and that would give the numbers in column 2 of the table, except for what must be due to rounding errors. Column 1 gives the drawing number; column 3 gives the count of newly drawn numbers (using fractions) removed from the list of undrawn in that drawing, and column 4 gives the cumulative count of new numbers seen so far. D’Hauteserve’s idea was to continue this process until only one number remained to be seen, for then the problem became easy! Parisot had given the answer for one number in his study of the maturity of chances, and d’Hauteserve had also derived it earlier in his book: for one number, the median of the distribution of the number of draws before that number appears is between twelve and thirteen! (See chapter 12.) His table had him getting the count of undrawn numbers down to only one number at the seventy-third drawing. Add seventy-three to Parisot’s twelve or thirteen and you get eighty-five to eighty-six, exactly Laplace’s result, with a trivial investment of time and energy. The agreement was illusory, however. The method d’Hauteserve used was not sound, and his rounding errors had led him to seventy-three by mistake. Working more accurately, the last remaining number would not be reached until between the seventy-ninth and eightieth drawings, giving a result near ninety-one or ninety-two, well above Laplace’s. Sometimes error plus accident equals success. Very little seems to be known about d’Hauteserve. Given the extent of interest in the Loterie, surprisingly few books about or even including probability were published during its run from 1757 through 1835, though I believe that all those on the Continent mention it as an example. Of course, the situation was much different for ephemeral works offering advice to bettors, a major industry in itself. But books on probability beyond those already mentioned were few and widely scattered. In France I find, in addition to the three just discussed and those by Parisot, only the Encyclopédie Méthodique (1789). In Germany there was a strange philosophical

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Fig ur e 15.3. Table from d’Hauteserve 1834, p. 74.

book by Karl Heinrich Frömmichen (1773) and a review of legal decisions by Johann Heinrich Bender (1832); in Amsterdam a pamphlet by Huyn (1788) that we will encounter in the next chapter; in Brussels an elementary text by Adolphe Quetelet (1828); in England a book by Rouse (1814); and in Vienna a short text by J. J. Littrow (1833). Mostly these just give the results of basic calculations of the chances of winning, and contrast those to the payoff odds. Perhaps the most interesting deviations from this are the books by Littrow and Quetelet. Littrow’s short book is mostly concerned with applications in astronomy, but after saying the usual things about the Loterie, he goes on to state Laplace’s result for the probability that all ninety

The Loterie in Textbooks and Manuals  165

T abl e 15 . 1. Part of d’Hauteserve’s table, recalculated to greater accuracy. 1

2

3

4

1

90.00

5.00

5.00

2

85.00

4.72

9.72

3

80.28

4.46

14.18

4

75.82

4.21

18.39

5

71.61

3.98

22.37

6

67.63

3.76

26.13

7

63.87

3.55

29.68

8

60.32

3.35

33.03

9

56.97

3.17

36.19

10

53.81

2.99

39.18

67

2.07

0.11

88.05

68

1.95

0.11

88.15

69

1.85

0.10

88.26

70

1.74

0.10

88.35

71

1.65

0.09

88.44

72

1.56

0.09

88.53

73

1.47

0.08

88.61

74

1.39

0.08

88.69

75

1.31

0.07

88.76

76

1.24

0.07

88.83

77

1.17

0.06

88.90

78

1.10

0.06

88.96

79

1.04

0.06

89.02

80

0.98

0.05

89.07

numbers have appeared after a given number of drawings and states the median number, which he gives as 85.14 He does not cite Laplace, nor does he give any explanation where the number comes from; and trying to figure this out would surely have been a challenge for any of his readers.

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Fig ur e 15.4. Quetelet’s text book tables. (top) Quetelet’s “Moral value” of the different onehundred-franc bets: the value given as 67.50 should be 67.42. Quetelet 1828, 109– 11. (bottom) Quetelet’s table of Loterie finances for five years: amount paid to the Paris bureaus, amount paid out to the winners, and amount returned to the government (part of which would go for expenses). Quetelet miscopied the means (Moyenne) for the first two columns, which should be 25,388,800 and 18,950,000. Chabrol 1823, tableau 99.

Quetelet’s extremely brief book treats the Loterie extremely briefly, but it still manages to make two nearly opposite points while offering two displays. His first displayed table (figure 15.4a) shows the nearly steady decrease in expected winnings for a 100-franc wager on the six bets then available, saying only, “One may be amused to learn the

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true value of a 100 franc ticket,” which he termed the “moral worth,” implying that the Loterie was taking an unconscionable amount of your money, the expected value of a 100-franc bet on a quaterne being a mere 14.68 francs. Having shown how much the government took, he seems to then show how little it got. Of the 126,944,000 francs in gross returns from 1816 to 1820, the treasury kept 32,194,000 francs, or 25.36 percent of the total collected. “The profits that a government derives from the institution of lotteries is a kind of tax on which it relies as confidently as on taxes of any other kind: to be convinced of this, cast your eyes on [figure 15.4b]. . . . The treasury therefore receives a little more than a quarter of the amount paid into the bureaus.”15 That the Loterie pays out nearly three-quarters of the total amount bet should have suggested to Quetelet that the clientele were giving most of their attention to bets on the more “moral” options with shorter odds.

The Loterie Manuals The mathematical publications reached only a limited audience of educated people. There was one other set of publications that reached an even smaller audience, but one that was closely targeted and intensely interested; their livelihoods depended upon close reading of every detail. I refer to the sets of instruction manuals that were published exclusively for the “intelligent machines” Casanova had stated would be required: Loterie employees. There were three components to these publications. One was simply the relevant official acts that set the legal boundaries, a routine republication of facts well known but easily forgotten, to ensure that everyone knew the rules. The second component was manuals for the agencies, aimed primarily at the supervisors but also at the sales force. The third was a smaller set of manuals for the inspectors, who traveled the country and checked up on the far-flung agencies to ensure that the supervisors were keeping a very close watch at the local level. These manuals survive in few copies; they were not for sale and never in wide circulation. But they were kept by the inspectors and

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agents, and the comments that follow are based upon what is probably a complete set of the post-Revolution manuals.16 To read in these manuals is to feel a pervasive sense of paranoia, with every entry aimed at some unobserved but real worry. The acts go into detail about possible counterfeit tickets and sales of illegal foreign lottery tickets, both of these actions violating the agency’s trust. The manuals for agencies warn of care in destroying unsold tickets, double- and triple-entry bookkeeping of records on sold tickets, the numbers bet on, and the packaging of materials to go to and from Loterie headquarters. The manual for inspectors was sixty-two pages long, covering what to check for in the agency logbooks and registers, what to report, how to detail the correspondence (with templates for common reports), and instructions in how to look for signs of debt or slackness and how to dismiss agents when necessary. The manuals for agents (receveurs, receivers of bets) were the most detailed, and the one in 1815 ran to more than a hundred pages. Their emphasis was on multiple cross-checkable records of every sale, even mails to and from the Paris office. For present purposes, though, we will focus on what could be called mathematical instruction. The agents were not expected to be mathematicians, but they were in a very sensitive position where mathematical errors could be extremely costly, especially if they became embedded in routine practice. These were types of instruction that would never be found in schoolbooks, and every such instance signaled that a real problem was being addressed. The first and quite possibly most important lesson for the agents was in writing numerals. The digits had to be uniform and unambiguous, a lesson the Loterie had surely learned the hard way. Before the Revolution, the Loterie had solved this by having the tickets printed, leaving only fraudulent post-drawing alteration as a worry. After the Revolution, handwritten tickets were adopted, and attention shifted to written numerals, probably as a labor- and time-saving device, perhaps reflecting a rising level of literacy. Figure 15.5 shows the idealized digits. The numbers bet on were to be written in ascending order, separated by double lines (to avoid

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F i g ur e 1 5 . 5 . The prototype for the post-Revolution rules for written digits, as published in the 1815 manuals, with two actual tickets from 1812 (61, 67, 71, 75, 77) and 1826 (33, 37, 61, 87).

confusing 3 and 7 with 37, for example) and with “./.” designating the end. Digits 1 and 0 would have both underscore and overscore; the other digits would have overscores, excepting 6, which would have an underscore. In practice there was some variation, but the methods for distinguishing 0 and 1, as well as the double lines and some version of the end mark, were strongly adhered to, suggesting that these were enforced. The main concern, to judge by the first post-Revolution Instruction à L’Usage des Receveurs in 1800, is surprising. It involves what must have been the least commonly encountered bet offered, the ambe déterminé, in which two numbers were specified as well as the positions among the five draws where they would occur. It quickly becomes clear that what worried the Loterie administration was not so much the particular bet, problematic as that could be if not carefully described, but the combination of several bets together in one instruction. We have already encountered one example, in Jeanne’s 1786 bets on five extraits déterminés at once on the number 45 in each of the five possible positions. That bet was clear, and seemed to cause no problem. With the ambe déterminé, the situation became more complex. Notation for the five different positions was required; the Loterie used the letters P, S, T, Q , C, for première (first), seconde (second), troisième (third), quatrième (fourth), and cinquième (fifth). For example, a bet on the ambe déterminé with the number 37 in the second position and 45 in the third position could be written as 37 S, 45 T.

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F ig ur e 15 . 6 . A sample ticket for all 216 bets with only the eight jumeaux, from the Loterie instruction manual of 1800.

The main complications and worries came with multiple bets. Suppose a bettor wished to bet one franc on every ambe simple that included the number 25 and where the other number ended in the digit 7. Unambiguous notation was one problem (“ending in 7” was described as la finale sept ; I’ll use “x7”), but the crucial other problem was counting: each ambe would be a separate bet, and the price would be the number of bets multipled by the amount bet on each. If the agent undercounted, the Loterie would lose. The particular bet on the number 25 paired with x7 was not difficult to price; there are nine numbers among 1 to 90 that end in 7, and so nine pairs were involved. Since it was an ambe simple, where order did not matter, the total bet was nine francs. The bets could get quite complicated, even with simple descriptions. A bet on all the ambes and all the quaternes where every one of the two or four numbers included ends in the digit 9 is a bet on 36 ambes or 126 quaternes. Woe to the agent who counts and charges for only some of these. And what if a bettor, such as Balzac’s Madame Descoings (see chapter 12), falls in love with the eight “jumeaux” 11, 22, . . . , 88, and regularly bets on all of the extraits, ambes, ternes, quaternes, and quines involving only the eight jumeaux at the minimum levels? That amounts to 218 separate bets, conceivably at every drawing. The manual helpfully supplies a sample ticket (figure 15.6). Figure 15.7 shows a sample ticket that combines the jumeaux and the ambe déterminé, betting that any one of 20, 30, 40 would be the

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F ig ur e 15 . 7. A sample ticket for all 3 × 8 = 24 bets on ambes déterminé, with the numbers 20, 30, or 40 drawn second, and one of the jumeaux drawn fourth. From the Loterie instruction manual of 1815.

second number drawn, and that one of the eight jumeaux will be the fourth number drawn. If bets like these were undercounted routinely, the loss to the Loterie could be enormous in the aggregate. The agents of the Loterie must have been among the best motivated students of combinatorial mathematics in France!

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Sixteen

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The Suppression of the Loterie in 1836

The Lot er ie was f inal ly and per manent ly suppressed on 1 January 1836. The French word suppression seems like a strange way to describe the closure of a nearly eighty-year-old enterprise that had been created by the government and had enjoyed a long and successful life. We are used to hearing of the suppression of an insurrection, an uprising, or a revolution; in each case this is an act of failure. Or we speak of the suppression of a youthful ambition or a boisterous child, usually as an act that may have seemed necessary, but which still cut short creative efforts at a real cost to an individual or society. Or we may refer to the suppression of an urge, perhaps an addiction, perhaps an expression of affection— in either case with real consequences for the good or otherwise, which, through the avoidance of destructive behavior or through missed opportunities, changed the future and may even have led to lasting regret: Robert Frost’s road not taken. None of these usages seems appropriate for the Loterie. The 1836 closure was not the first suppression of the Loterie; that was in 1776, when suppression was simply a legal way to change the Loterie from a support system for the École militaire to a source of funds for the central budget. The second suppression of the Loterie, in 1793, was a recognition that in the midst of the Terror it was simply impossible to carry on. But the 1836 suppression was different. It really

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was final; no similar lottery on the same principle would be established in France until well into the twentieth century. And it was not a capricious act of the moment; rather, it was the result of a number of different moral, cultural, and economic forces coming together at a particular time in history.

Religion and Addiction Gambling, be it lotteries, dice or card games, or casino games such as roulette, has never received a very favorable press. It has been damned by the Church, and called a seedbed for crime by civic groups. Most of the early objections to gambling were moral, and the earliest of these were religious. Even when the first full book on this topic was published in 1619 by Thomas Gataker, it was in fact a reaction to earlier, less substantial work. Gataker was a distinguished preacher, and he worked on his book while he was employed from 1601 to 1611 by Lincoln’s Inn in London, one of the four societies of barristers that comprise the London bar. The office of preacher at Lincoln’s Inn was an office of the Church of England, and Gataker’s appointment to that post preceded that in 1616 of the poet John Donne. Gataker’s book, Of the Nature and Use of Lots: A Treatise Historicall and Theological,1 was a response to a sixteen-page tract by James Balmford published in 1593, which held that drawing lots, or any other use of chance such as dice or cards, was only permissible in the eyes of God for deciding controversies, and impermissible for sport. Balmford’s idea was that the omnipotent God always decided the result, and to appeal to the word of God on frivolous matters or for personal gain was sinful. If Dice be wholly evil, because they wholly depend upon chance, then Tables and Cards must needes be somewhat evil, because they somewhat depend upon chance. . . . In the Scriptures . . . Lots were used but only in serious matters. . . . The proper end of a Lot is to end a controversy: and therefore for your better instruction [by] advancing of the name of God, [and] is to be used religiously, . . . and not to be used in sport: as wee are

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not to pray or sweare in sport. . . . Man must not pervert [Lots] to play, and by playing to get away another man’s money.2

Gataker’s 360-page book was much more than a simple reply to Balmford. It was a closely argued and densely footnoted (often in Greek or Latin) treatment of the broad subject. To Balmford’s main argument, Gataker’s answer was that just because God could, and on some occasions did, decide such chance events, it did not follow that he always gave attention to them: “It followeth not; If sometimes extraordinarily, then ever.”3 At length, and with close attention to scripture, Gataker argued that the drawing of a lot or the roll of a die was a neutral event, neither evil nor good; it was the human use or abuse of the act that could be evil. He was sometimes interpreted as giving a defense of gambling, which was not his purpose. He also took issue with Balmford on his claim that the use of lots to resolve controversies was proper. “For a Lot, as we have seene, is casuall: and to put a necessarie act to a casuall event, cannot be without sinne, since it maketh that casual and contingent which Gods law maketh necessarie.”4 Gataker specifically disapproved of the drawing of lots as a substantive role in an election, such as the election of a pope. Apparently he would have disapproved of the Genoese system in that same era of choosing five leaders at random; but he might have allowed the use of the same system as a lottery, as long as the stakes remained low. Balmford published a 147-page reply in 1623; Gataker answered in an expanded second edition in 1627. At about the same time, the bishop of Geneva, and later Saint, François de Sales wrote Introduction à la Vie Dévote (Introduction to a Devout Life); the final version was published in 1619. De Sales addressed gambling succinctly in chapter 32 of part 3, “Forbidden Games.” He wrote, “The games of dice, cards and the like, where the winning depends principally on chance, are not only dangerous recreations, like dances, but they are also absolutely bad and blameworthy games in their very nature. This is why they are forbidden by law, both civil and ecclesiastical.” To De Sales, it was offensive that such games often favor one who neither has skill nor has exerted effort.

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Even if the game is undertaken with mutual consent and does not harm others, it is contrary to reason: “Gain must be the reward of effort, but it becomes the reward of chance, which deserves no reward, since that does not reflect on the winner at all.” Finally, he argued, “There is no joy in these games without winning; and is not this joy guilty, since it supposes the loss and the displeasure of others? Such pleasure is undoubtedly unworthy.” Balmford would have recognized a kindred spirit. De Sales’s book was not intended as a philosophical work, but for three centuries this would serve as a basic textbook in Catholic behavior for young people, and for that reason alone it was extremely influential, at least among moral philosophers.5 Neither Balmford nor de Sales had much effect upon practical matters; gambling remained both frequently illegal and widely condoned. Nearly a century later, the same philosophical debate was reignited in dueling publications from the Netherlands by French and French expatriate philosophers. Even the arguments remained much the same. The debate included the philosophers Jean Barbeyrac (in 1709) and Jean La Placette (in 1714) playing the role of Gataker, and a Protestant minister, Pierre de Joncourt, as Balmford. The argument was carried out at a higher philosophical level than in Gataker, with classical and biblical citations in evidence on both sides, and arguments based in part on ideas of natural law. There was no reference to Gataker in the main texts, although Barbeyrac did recruit Gataker as a supporting voice in other writing.6 Even so, these early tracts were aimed at gambling in general and not at lotteries in particular. Lotteries went unnoted by Gataker and were mentioned only in passing in the later French debate— and then with a note of ambivalence, since among the most common lotteries of that era were those intended to raise money for the Church. A different argument against gambling— namely, that it tempts people to addiction and thus to ruin— also has a long history. Probably the earliest book to present that case was by the early Belgian doctor Pascasius Justus (or Pâquier Joostens). In 1561, Justus published a small tract (figure 16.1) that may have been the first to rec-

F ig ur e 1 6 . 1 . The title page from the third edition of Pascasius Justus’s Alea. The book was published in a very small format. This title page is 2 × 3.5 inches.

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ognize compulsive gambling as not just simple greed, but rather a self-destructive behavior that the gambler himself might recognize was counter to his best interests— and also as a lost sense of reality, a form of melancholy, similar to alcoholism, that could be treated with some sort of early psychotherapy.7 Justus was himself a former compulsive gambler, and in initiating this type of critique he also started what was to be a long series of similar matches of author to subject. In the 1770s, Jean Dusaulx, also a former compulsive gambler, added an eloquent voice in his books La fureur du jeu (1775) and De la passion du jeu (1779). Neither of these works made more than a passing reference to lotteries, and none at all to our Loterie. These authors seem to have learned their unfortunate lessons in casino-type games, and had not been swayed by a passion for lotteries.

Injustice and Immorality A third species of objection arose in the last half of the eighteenth century that was often specifically targeted at lotteries. The essence of the issue was that the odds were poor, and a lottery was a bad bet. In his 1776 Wealth of Nations, Adam Smith put it this way, referring to the blanks lotteries that were common in England at that time: That the chance of gain is naturally over-valued, we may learn from the universal success of lotteries. The world neither ever saw, nor ever will see, a perfectly fair lottery; because the undertaker could make nothing by it. . . . The vain hope of gaining some of the great prizes is the sole cause of this demand. The soberest people scarce look upon it as a folly to pay a small sum for the chance of gaining ten or twenty thousand pounds; though they know that even that small sum is perhaps twenty or thirty per cent. more than the chance is worth. . . . There is not, however, a more certain proposition in mathematicks than that the more tickets you adventure upon, the more likely you are to be a loser. Adventure upon all the tickets in the lottery, and you lose for certain; and the greater the number of your tickets the nearer you approach to this certainty.8

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Smith’s assessment of lotteries was objective, not accusatory. He even included an explanation why it could not be otherwise: No such operation could take place without the promise of profit by the operator. His was a rational argument for why it might not be sensible to buy even one ticket. At about the same time, the French naturalist Georges-Louis Leclerc, the Comte de Buffon, went Smith one better by introducing a form of a utility calculation into the discussion. Buffon’s argument was in a famous essay on moral arithmetic, as part of his great thirty-six-volume Histoire Naturalle, in a supplementary volume published in 1777. That publication date would place it a year after Smith’s Wealth of Nations, but the essay was written at least a decade earlier. In it, Buffon introduced the notion of moral certainty: Some events are so unlikely that we must plan our lives by assuming they will not happen: for practical calculation, they should be assigned the probability of zero. An example he gave considered a man of age fifty-six (quite possibly Buffon’s age when he wrote this), who arose one morning and sought to plan his day. He was in good health; what was the chance that he would die before the day was over? From mortality tables Buffon found that such a probability for an average man would be about one in ten thousand, and thus for a healthy man it would be even less. The proper action for such a man was to plan the day completely neglecting the chance of death; he was morally certain to survive the day.9 An event of probability no greater than one in ten thousand, Buffon concluded, can be assumed to be morally certain not to happen. By the time Buffon got to lotteries, thirty-five pages later in his essay, he had relaxed the limit to one in one thousand. The display of hope is the lure of all deceptive gamblers. The great art of the lottery operator is to offer large sums with very small probabilities, a hope swollen by the force of greed. These deceivers further magnify the attraction of their prize by offering it for a very small amount of money paid, so small anyone can deprive themselves of it for a hope which, although much smaller, seems to grow with the grandeur of the total sum. We are not told that when the probability is below a thou-

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sandth [un millième], hope becomes zero, however great the sum promised, since everything, however great it may be, is reduced to nothing as soon as it is necessarily multiplied by nothing, as here the large sum of money is multiplied by the zero probability, and, as is generally true, any number multiplied by zero is always zero.10

If Adam Smith had adopted Buffon’s view, that the expected utility of a large prize awarded with a miniscule probability was zero, he would have seen that even an entry into a fair lottery was irrational. It was not long before a form of this argument— namely, the claim that the payoff structure of the Loterie was designed to deceive and to take advantage of the uninformed— was employed as an attack upon the Loterie. Condorcet and Lacroix employed it, as we saw in chapter 15, but in two of the earliest published such cases it came with disguised motives. In 1788, Pierre Nicholas Huyn published a tract that for the most part gave the rules of a series of casino games and calculation of the probability of success in each. Huyn was an entrepreneur (see Shafer 2022) who had considerable ownership stakes in casinos, and he was not content simply to advertise them in this way; he also added a five-page attack on the competition, the Loterie. After briefly describing the rules and the chances of winning for each Loterie bet, he delivered a diatribe against that competing operation: There is no more pernicious game than the Lotto. . . . A merchant who would not risk a bet at a public [casino] game for fear of compromising himself, ruins himself betting the lotto in the silence of the office: he is like the poor man who, under a fatal and illusory bait, loses the fruit of his labor; the servant after losing his wages steals from his master to attempt to regain them. . . . The insidious practice of publishing the collection of winning numbers offers the most dangerous bait to which fools are prone. They publish with emphasis payouts of 70000 for one; but they are careful not to tell the extent of the difficulty of obtaining it.11

Indeed, the gazette Mercure de France routinely published the results of the drawings of the Loterie, and the building where the Lo-

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terie was at that time drawn had once housed the East India Company. This combination inspired a witty anecdote that was circulating at the time, and which Huyn ended by quoting: the claim that there was an inscription on the door of that building reading, In this place where Colbert enriched France, Mercure sells expensive hope to fools.12

Huyn’s self-interested attack made much of the fact that the odds at the Loterie got progressively worse as the prize became larger, while at casinos all the odds were much closer to even. He did not point out that at the casino, the rapid repetitive play could remove all of a bettor’s stake in short order at one sitting; inflated odds were not necessary for that to happen. At the Loterie— with only two drawings a month at that time, a much more stately pace— a payoff schedule underpaying for large prizes was deemed necessary precisely because the Loterie could not count on repetitive play mitigating the chance of huge loss.

Talleyrand Within a year of Huyn’s publication another tract appeared, also couched in moral terms but with even more complicated hidden motives. It was written by one of the most famous figures in the French Revolution and the succeeding Napoleonic period: CharlesMaurice de Talleyrand-Périgord, usually referred to simply as Talleyrand, but writing in this instance as the bishop of Autun. Talleyrand (1754– 1838) was born to aristocracy; he became a priest in 1779, then the agent-general representing the Catholic Church to the Crown, and finally a bishop in 1789. He played several key parts in the Revolution, and later served as foreign minister successively to Napoleon and to Louis XVIII. He remained the indispensable man through five regime changes— a feat perhaps unmatched in European history. Talleyrand’s tract is usually described as the text of a speech he may

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have given to the National Assembly on 12 December 1789. It was an unabashed polemic against lotteries, and was specifically targeted at the Loterie royale. A good portion of it could have been inspired by a reading of Huyn, but it was better developed and offered a few nuances that had been missed by or been uncomfortable to Huyn. The Loterie, Talleyrand said, was both unjust and immoral. It was unjust because the payoff odds were so far from fair, and became even worse the more attractive the prize sought. It was immoral because it played on the hopes of the gullible lower classes, bringing misery and hardship to those who could least afford it. He departed from Huyn in also castigating casino games as being no different, arguing that if by some miracle a Loterie were to offer fair payoffs and give up all expected profit, it would become just, but would still be immoral, still luring people with false hope to the belief that they could win a huge prize with minute chance. In that, Talleyrand might have been channeling Buffon. And while he dwelled on the harm to the lower classes, he acknowledged that the deceptive practices also took a toll on the professional classes, and indeed on all classes, tempting even responsible and able people to ruin and disgrace. Talleyrand’s critique was unusual in giving specific attention to the deceptive notion of the maturity of chances. He wrote, It is almost a necessary result of the structure of this Loterie that one who first risks only small bets, soon finds himself drawn into larger bets, victim of a foolish yet ordinary illusion. He attaches himself all the more to a choice of number, the longer it has been fatal to him; he even sees himself as obliged to make new sacrifices, so as not to lose the fruit of the old. Consequently, he will choose again and again the same numbers, in the deep conviction that they will finally give in to his perseverance, and that, by the antiquity of their most recent appearance, they acquire higher priority every day to reappear before the others. It is as if in such a game, the future could somehow depend on the past; that the same numbers, mixed at random, are constrained in their movements by the preceding draws, and a number that has not appeared for a certain number of draws, will now appear more easily than any other particular number,

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and will offer itself in the next draw to the indifferent hand of the child who will make the draw.13

Talleyrand both followed and preceded others who have made this case without offering any data in support of the claim of vast misfortune. His casual mentions of suicides, bankruptcies, and overloaded hospitals may have resonated with his audience, but no data at that time could have tied such events to the Loterie specifically. “Has a man killed himself ? Question the people: it is the Lottery that cost the life, they will tell you most often.” He even considered the Loterie a cause of the Revolution: “Among the causes of this revolution, the Loterie, no doubt, must occupy one of the first ranks.”14 Talleyrand did not shrink from confronting the financial loss that would be incurred if the Loterie were suppressed. He reported that the Loterie had been making twelve million francs a year, with three million going to expenses and so contributing nine million francs to the budget. He considered that spending a quarter of the receipts on salaries, locations, and material was an “enormous” cost. He estimated that about three million francs more were spent by French citizens on illegal foreign lotteries (with payoffs even worse than the Loterie), so that a total of about fifteen million francs was spent each year by the people on this unproductive enterprise, a total waste in his view. He argued that with the Loterie suppressed, the foreign sales would be more visible and would soon disappear. The French nation would eliminate fifteen million francs of wasteful spending, and that money instead would go to productive uses and benefit the economy to make up for most or all of the lost nine million francs of income. “Who will dare to think that nine million, though real, but coming from such a corrupt source, can redeem so many misfortunes before the eyes of the whole Nation?”15 What was Talleyrand’s motive in calling for suppression? His arguments echoed those of other clergy over many years, and it may seem strange even to raise any question of motive, since Talleyrand was the representative of the Church at the highest level of government. But he was not typical in behavior. He seldom rose to speak in the Na-

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tional Assembly, but he had done so on 10 October 1789, with farreaching consequences. He had then calmly addressed the dire economic straits that faced the government, and suggested that it could all be met by nationalizing the enormous amount of property held by the Church, including roughly 10 percent of all real estate in France. It was a bombshell to the clergy, but he had prepared well, with plans to compensate the priests generously and thus forestall a rebellion at that level. When the Count de Mirabeau, a powerful orator, made the official motion to accomplish this on 13 October, Talleyrand had prepared the ground so well that the deed was accomplished within three weeks. The Church was stripped of vast economic and political power, but the Revolution was saved financially, France was changed forever, and all this was precipitated by that single act of the Church’s agent in the Assembly. Talleyrand had to resign his bishopric within two years (the pope was not pleased). The completion of this redistribution took more than a decade and was not smooth in all details. The clergy were reclassified as employees of the state, and many resisted for several years. The economic change required lands to be sold and titles to be conferred, and in the process a whole new class of entrepreneurs was created. But the change was permanent. When the great Cathedral of Notre Dame burned in April 2019, it was national property, not Church property. Let us then consider Talleyrand’s motive regarding the Loterie. First, the timing: It is true that he apparently advanced his Loterie proposal as a motion on 12 December 1789, just two months after his earlier proposal on Church property was adopted. But while the motion on Church property was rushed through with careful handling, the Loterie proposal was never again heard from. Indeed, it was not even published in the official newspaper, the Moniteur Universel, but was relegated directly to the Assembly archives, and only printed as a “motion” in the 1870s.16 What is more, it was not even new in December 1789: the full text of the tract had already appeared six months earlier as a separate publication, “Par M. L’Évéque D’Autun,” by July of that year. It was reviewed in the Mercure de France on 4 July 1789, with many long passages in quotation and the whole argument co-

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gently summarized. It attracted sufficient attention that rejoinders appeared: Already in 1789 a writer named Jabbe-Minoblant published a seventy-two-page answer with two columns showing Talleyrand in one and a reply justifying the Loterie in the other.17 If Talleyrand had really held suppression of the Loterie as a priority, he could have advanced his motion well before December, but it would have had no chance of passing. It would have been impossible to cut off such a source of revenue in July 1789. Talleyrand’s Memoirs, written in the 1830s, reveal that the tract was a residue of earlier plans already formed during the mid-1780s. When he had been the Church’s agent-general to the Crown in 1782, he had realized that with the dire financial situation the king faced and the immense wealth of the Church, some significant concessions would have to be made. Looking back in 1830, he wrote, The abolition of the lotteries was one of my favorite ideas; I had investigated all the chances and all the consequences of that baneful institution. At the same time, I noticed that the clergy, being attacked and scoffed at by philosophers, were daily losing public regard. My object was that they should regain the esteem of the people, and for that purpose, I was anxious to hold them up to the eyes of the nation, as the protectors of strict morality. By inducing the clergy to submit to some pecuniary sacrifices in support of that principle, I should have served, not only public morals, but also the very order I had consented to join.18

Talleyrand’s plan in 1782 was to have the Church buy the Loterie from the king, then close it down, and further to promise to grant annually to the king the nine million francs in foregone proceeds as a present. The plan depended upon the Church agreeing to a substantial sacrifice. They declined. “It may be observed that my first political campaign was not very fortunate, a result which I dare to attribute to the fact that my proposals were far too radical for the men with whom I purposed to make use of them.”19 When Talleyrand made his radical suggestion regarding Church property in October 1789, he had better prepared the ground. The Church paid a huge price,

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vastly more than was requested in 1782, though that sale of the Loterie would have been insufficient to avoid the Revolution. Talleyrand’s dusting the tract off as an active motion in December 1789 was a costless attempt to throw a small bone to a much-shriveled dog that had been forced to move out of the house. Talleyrand was finished as a man of the Church, but well started on a different career. He may have been sincere in his feelings about the Loterie, but he did not again raise a public voice about it, even as he helped preside over a sequence of governments, all of which benefited from the Loterie’s contributions.

Laplace In 1817, Pierre Simon Laplace was elevated by the restored Bourbon monarch Louis XVIII to the rank of marquis. As such, he became a member of the Chambre des Pairs, an upper chamber of the French Parliament created in 1814, akin to the British House of Lords. It consisted of the “pairs,” or peers of France, as appointed or so designated by the king. Laplace had played other roles in government before, most notably after the Coup of 18 Brumaire (9 November 1799) that brought Napoleon to power, when Napoleon quickly appointed him as minister of the interior. That appointment lasted but a month, because after Napoleon had consolidated control, Laplace was replaced by Napoleon’s twenty-four-year-old brother Lucien; Laplace’s appointment as minister had been merely a placeholder (figure 16.2). When Napoleon was interviewed by General Gourgaud in his forced retirement on St. Helena, he clearly did not think Laplace had exceeded expectations in the position. He said, “Laplace was not addressing any question from its true point of view; he looked for subtlety everywhere, had only problematic ideas; he brought the spirit of the infinitesimal into administration.”20 Two days after Bastille Day in July 1819, Laplace rose in the Chambre des Pairs to speak during a meeting dedicated to the national budget. He spoke about the Loterie, specifically about its suppression. He began,

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Fig ur e 16.2. Two cover sheets signed by Laplace during his short term as minister of the interior. The first is dated 24 Brumaire, an 8 (15 November 1799), a mere six days after the Coup of 18 Brumaire, when Napoleon seized power, and before new stationery for the Consuls de la République was prepared. The second is dated 8 Frimaire, an 8 (29 November 1799), two weeks later, on new stationery. Laplace’s appointment ran from 12 November to 25 December 1799.

Gentlemen, the state of our finances permits us to decrease taxes. The law we are considering applies this decrease to direct tax contributions and to deductions from salaries. Would this plan be the most advantageous? . . . I think that we can achieve a more useful end by suppressing the tax of the Loterie.

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Fig ur e 16.2. (cont.)

Laplace argued his position on moral grounds, much as Talleyrand had, pointing to the mathematical disadvantage the Loterie gave to those who played, and arguing that this was visited to large degree upon the poor. The poor, excited by the desire for a better life and seduced by hopes whose unlikelihood it is beyond their capacity to appreciate, take to this game as if it were a necessity. They are attracted to the bets that permit

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the greatest benefit, the same bets that we see are the least favorable. . . . We applaud the orator who would turn his audience away from the Loterie, forcefully recounting the thefts, the misery, the bankruptcies, and the suicides that are its children.

Like Talleyrand, Laplace scoffed at the claim that the Loterie was a voluntary tax paid only by the willing, raising the argument that the statistical regularity in the profits of the Loterie showed the social necessity of the urge to bet: We are told further that this tax is voluntary. No doubt it is voluntary for each individual, but for the set of all individuals it is a necessity, just as their marriages, births, and all sorts of variable effects are necessary, and nearly the same each year when their number is large, just as the revenues from the lottery are as constant as is agricultural production.

Laplace ended his brief remarks seeming to concede that his was a losing cause. He noted that society comes together to support establishments that work to the betterment of the people, such as savings banks and insurance companies. But so too we must severely proscribe establishments founded upon ignorant illusions and cupidity; no benefit can compensate for the evils they produce. We must then extremely regret that the suppression of the Loterie has not been placed at the head of the agenda for reducing taxes, as a tribute to morality.21

The arguments of Talleyrand and Laplace were sufficiently similar so that it is tempting to ask whether they had in any way collaborated. Laplace had met Talleyrand in 1791, when he was asked to help provide the scientific basis for the metric system. Talleyrand had also initiated the founding of the Institut de France as a successor to the Académie des sciences, and had worked with Laplace on that as well, starting in 1791. There seems to be no reason to believe they were particularly close, but their careers followed a similar arc: both

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survived the Revolution and played early governmental roles under Napoleon, and both emerge after the Bourbon Restoration in 1814, again with different public positions. There is a small trail of correspondence between them of no great moment, the warmest being a 17 May 1813 note from Talleyrand to Laplace (then chancellor of the Senate), thanking him for a copy of the newly published fourth edition of his Exposition du Système du Monde. It is plausible that Talleyrand played at least a supportive role in Laplace’s elevation to marquis four years later. And when Laplace spoke in 1819, Talleyrand was plausibly among those in attendance— he too was a member of the Chambre des Pairs at the time. Laplace would certainly have been aware of Talleyrand’s 1789 tract, but there is no reason to believe that tract influenced Laplace’s dim view of the Loterie. That view, and particularly his opinion of the illusion of the maturity of chances, was on display in bits and pieces from Laplace’s earliest work on probability. Still, that he chose to speak at that budget meeting on that subject was probably not a matter of pure chance. We may never know if Talleyrand urged him to do so, or if Laplace simply saw an opportunity to offer a form of thanks for aid in making the transition from Napoleon’s regime to the Bourbon Restoration. Laplace had dedicated his Théorie analytique des probabilités to Napoleon in 1812, and then he had hastily issued a second edition of his book in 1814 without the dedication.

The Economics of the Loterie in the 1820s The Loterie was at its peak in profit and sales during the years of the Napoleonic empire, with agencies in the Netherlands, Germany, and Italy adding to a strong demand in France. During the years 1809 to 1813 the average annual net profit returned to the French budget was 15,500,000 francs; the peak year was 1810 with net profit of 24,200,000 francs on sales of 88,260,000 francs, of which 64 percent was paid back to winning tickets and 7,300,000 francs went to expenses (see table 9.2). Over those five years the fraction paid back to winners averaged 70 percent. For comparison, the current return to winners by

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Figur e 16.3. The spread of amounts won in large Loterie payoffs, indicative of the amount bet in each case over the years 1797 to 1834. It shows little change in dispersion over time.

the Illinois Lotto appears to be less than 50 percent, though their public bookkeeping is not clear on that. After the loss of empire in 1814 and the Bourbon Restoration, there was a large drop in gross sales and net profit (roughly 40 percent down in each category). By 1820 there had been a slight rebound, and over the following decade the finances were stable. For the years 1824 to 1828 gross sales averaged 52,900,000 francs and net profit averaged 9,900,000 francs. With the exception of the anomalous year 1814, over the entire post-Revolution period the fraction of gross sales (amount bet) that was returned to winners was always between 61 and 79 percent, rates that today’s lotto purchaser would envy. Talleyrand’s dire fear that the payout rates for the most extreme bets (quine 2.3 percent and quaterne 14.7 percent) would deceive people apparently did not apply to the average bettor. The data on large winning bets can be used to estimate the number of bets. We look at the stability over time, using the list of large payoffs (winning quaternes and three large ternes) given by Menut, who only included winners in France (figure 16.3). Outside the 1814– 15

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Figur e 16.4. The number of large Loterie bets decreased over time. The decrease was roughly steady from 1797 to 1815, but the number was relatively constant from 1816 to 1829, and then dropped considerably after the ordinance of 1829.

anomaly, the vertical spread over the period from 1797 to 1834 is similar throughout: people who bet on quaternes placed wagers whose size distribution did not tend to change over time. However, though it is not obvious from figure 16.3, the number of people placing quaterne bets did decline in time. Another display (figure 16.4) focuses on changes in willingness to bet quaternes. Here we have the number of large bets, not their size, on the vertical axis. In looking at the illustration you can see a contrast between three different time periods: up to 1815, 1816 to 1829, and 1830 and after. The average number of such bets per year is about equal in the first two of the three periods, but with this difference: Up to 1815 there is a pronounced downward trend, roughly a 50 percent drop in the number of quaternes bet in France from 1805 to 1810. Since there was in fact an increase in Loterie revenues over that period, this suggests that it was the bets placed in the empire but outside France that made up the difference. On the other hand, from 1816 to 1829 the betting tendency was fairly stable. In the third period, the years 1830 to 1833, the bets on quaternes, and in fact all bets, were markedly fewer, signifying

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a loss of business. By extrapolation from the thirty-one winning bets from 1826 to 1829, I estimate that the French bet on nearly 16,000,000 quaternes in those four years. From the fifteen winning bets in the four years from 1830 to 1833, I estimate they bet on about 7,700,000 Quaternes, a 50 percent drop from the previous period. It is to this sign of the Loterie’s impending death that we turn now.

The Blade Falls, Slowly There was no single event that led to the end of the Loterie; it was a cumulation of years of complaints by the moralists, and a strong economy that had greatly reduced any need for the benefits of the Loterie, added to the slowing demand for an old and unchanging product. Laplace died in 1827, and if there was a push from the aging Talleyrand, who was then in the ceremonial role of grand chamberlain of France, it was behind the scenes. The Bourbon Restoration was becoming increasingly unpopular in 1829 and would fall to a milder revolution in July 1830. In any case, the first administrative strike against the Loterie was an ordinance adopted on 22 February 1829. At the time, France consisted of 86 administrative departments, and the ordinance stated that the Loterie could not be extended to the eight departments where it did not currently exist, and would be permanently suppressed in twenty-eight other departments. Further, it set the minimum bet on a single ticket at two francs in the remaining fifty departments, supposedly to decrease the attraction to the poor. Most importantly, it set in motion a process that would have been hard to reverse. The Loterie administration itself was a governmental unit, and not in a good position to argue publicly against this trend. It did find a way, though, even if muted. On 14 April 1830 the Moniteur Universel, an official outlet for the government, published a long report from the Loterie. Most of the report was a detailed description of the economic data since the Revolution, and it included the table reproduced as table 9.2 in chapter 9 of this book. The report reviewed the history as described in that table and summarized above: a steady net profit

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to the state of seven to ten million francs a year, modest (and decreasing) expenses, and from 1797 through 1828, the Loterie had returned a total of 335,236,115.35 francs to the national budget. The Loterie administration’s report closed by noting the ordinance of 22 February 1829 and its effects; it constituted a remarkably direct denunciation of the government’s action, coming from an active government agency. These returns could not protect the Loterie from attacks and criticisms which attach to the very principle of this kind of tax. . . . We will confine ourselves to pointing out that it is not the Loterie which gives rise to the passion for gambling; it is a passion of all times and all places which led to the establishment of lotteries; and, not being able to destroy this fatal inclination, which has its roots in the heart of man, we have sought to regulate the game and make it productive for the State, and we have succeeded in reducing its unfortunate consequences by preventing individuals and foreigners from exploiting it with greed and bad faith.

The government, the report stated, suppressed any public notices of the sale of tickets; and forbade publication of popular opinion regarding the change in law curtailing the Loterie’s activities. Finally, by a royal decree of February 22, your Majesty suppressed the unproductive offices of twenty-eight departments, for fear of soliciting a dangerous taste which had not revealed itself; it then reduced agents’ receipts, in order to stop them from sharing with bettors to whom they could offer a gambling incentive bonus; it raised the minimum on bets, in order to make the lottery less accessible to the savings of the working class, and it thus succeeded in stopping the progress of a passion whose administration collects the tributes for the government, without ever consulting those who bring them to them.22

With the decree of 1829 curtailing growth, lowering salaries, eliminating advertising and advocacy, and forcing the closing of agencies in a third of the departments in France, the Loterie was being bled to death; the end was inevitable. The events leading up to the July

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revolution three months later would have absorbed all political attention. With a new government came new people from a different background and less interest in the past. Some insight into the decision can be gleaned from a seventy-two-page report prepared for the Chambre des Députés early in 1832, prior to its vote on a new postRevolution budget for national income. It was printed by the Revue Encyclopédique in March of that year, although the résumé carries an 8 April 1832 date; the translated title is A Critical Examination of the Tax Base.23 Its author, Émile Pereire (1800– 1875) and his brother Isaac were important French financiers, rivals of the Rothschilds, and founders of the bank Crédit Mobilier in 1852. The book was a critique of the draft budget bill, giving much attention to the reduction of a number of taxes. It dedicated only two pages to the Loterie, treating it as a tax; but those few comments are revealing. Pereire’s short opening made his view clear: After having recognized the immorality of this tax, its disastrous consequences and the insignificance of its results, the budget committee [of the Chambre des Députés] proposed to maintain it until 1836; we do not hesitate to find this deadline too long, and we believe on the contrary that it becomes every day more urgent to put an end to this odious exploitation of the credulity and the misery of the people.24

After a short and, as we will see, dubious look at the figures as they had appeared in the Moniteur Universel in 1830, Pereire assessed the situation: If we consider also that the State has lost the work of the individuals it occupied in the collection of these 335 million, work which can be evaluated at 156 million, since it has thus been unproductively remunerated, we will see that the net product was ultimately only 169 [sic ; actually 179] millions. We have seen that the sum that had been collected by the lottery in 32 years amounted to . . . an average of 55 million and a half per year. If an equal amount had been used in the purchase of five percent annuities at par, there would not be a penny of public debt in France. If the same sum

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of 55 millions had been, since 1797, paid each year for the cause of savings, it would have produced in 1828 three billion nine hundred millions.25

Pereire made a large error in describing the contribution of the Loterie. He clearly had the 1830 table before him— the one reproduced as table 9.2 in chapter 9 of this book— and he may have meant to subtract the expenses (156 million) from the gross profit (492 million), but instead he subtracted expenses from the net profit of 335 million! He was double-counting expenses, certainly not a sanctioned accounting practice, and it was quite misleading. Pereire’s final statement hints at a hidden motive. He may have been sincere in his moral repulsion from the Loterie, although his language was a bit colorful for a tax report, but he was a banker already looking for other ways the money spent on the Loterie by bettors could be used: savings and annuities, both banking products. Several of the lotteries that would be allowed to continue due to their charitable goals were based upon annuities or bonds. Pereire was not alone in feeling this way; one leader of the move to suppress the Loterie in the Chambre des Députés was Benjamin Delessert, a financier and a founder of a savings bank. Self-interest and societal interest are not necessarily in conflict, but there is a distinct hint of self-interest in Pereire’s passage and the background of the new class of politicians. Pereire’s report estimated an annual increase in income of 1,874,700 francs through savings from not having to collect receipts from a suppressed Loterie, and an annual loss of 8,000,000 francs for the loss of income from the Loterie. This was dwarfed by an annual loss of 42,300,000 francs from cutting the taxes on wine. On 21 April 1832 the next and fatal step was taken. Whether influenced by moral considerations, the bankers’ wishes to play more of a role in this domain, or simply the weight of decades of disputation and no one rising publicly to defend the Loterie, the minister of finance was instructed by the Chambre des Députés to gradually abolish the Loterie, so that it would cease to exist on 1 January 1836. The 21 April 1832 act authorized the minister of finance to start by reducing the number of drawings and the number of offices in

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preparation for the suppression; also, he could further increase the minimum bets and reimburse, where necessary, the agents whose offices were being closed. There must have been a few loose ends, perhaps agencies that tried to go on alone at a reduced rate. Descotils and Guilbert show a ticket from Lyon they say was dated 1839, but I believe that is misreading an 1835 handwritten date.26 In any event, on 21 May 1836 a final law was passed that prohibited all types of lotteries, taking great care to define the prohibition as extending to “all operations offered to the public in order to give rise to the hope for a gain which would be acquired by lot” (article 2). That act also enabled the prosecution of agents for foreign lotteries. Exception to the suppression was made for lotteries of furniture or objects of art exclusively intended for charitable purposes or the encouragement of the arts. France had, with some finality, outlawed “hope for a gain” in its most easily practiced form.

Conclusion

I n t h e y e a r s j u s t a f t e r 1 8 3 6 there were few lotteries in France, but eventually the barriers weakened. At least forty French lotteries between 1851 and 1867 were accepted as allowed exceptions to the law of 1836.1 These were overwhelmingly charitable lotteries, in support of churches, orphanages, or groups of artists; none of them were of the same type as our Loterie. In his 1837 treatise on probability, Siméon-Denis Poisson continued the tradition of earlier texts by calculating the odds for the Loterie and noting how unfavorable they were, but he also said the Loterie was now “happily suppressed by a recent law.”2 By 1843 Antoine Augustin Cournot, while still criticizing the odds, was now referring to it as “l’ancienne loterie.”3 The Loterie had been consigned to history. But what, with hindsight, can we say about this unique experiment? No other lottery of this scale, before or since, survived for such a period while consistently embracing a risk that was calculable and potentially fatal at every drawing. Casanova had promised success as long as the king gave financial backing and the winnings were paid promptly. He also warned against any attempt to control the risk, such as by trying to balance the portfolio of risks by limiting the number of bets on individual numbers. Of course, Casanova wrote his account of the lottery late in life, also with the benefit of hindsight, and he may have conveniently forgotten some bad advice he had given. But the advice he reported was certainly in character, and

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he did not spare himself in more embarrassing parts of his memoirs, so it seems reasonable to credit him with a wise appreciation of the behavior of other gamblers. The success that was evident throughout most of the nearly eighty years of the Loterie was remarkable, and that five successive governments in a revolutionary period maintained these principles more or less throughout was also remarkable. Such consistency was not to be taken for granted. Fredrick the Great in Berlin had started a lottery in 1763 with the same principles, but very soon he lost his nerve and pushed the risk off on others, with consequent problems. That lottery survived for a while but never flourished, and soon the “Klasse” lotteries took over that stage. They were like the blanks lotteries in that they carried a wholly different set of risks. Would the tickets sell sufficiently to cover the prizes? Would a secondary insurance market take away business? These risks were not calculable the way Casanova’s were. Frederick faced a calculable risk and blinked; the French accepted the risk, and over time they also came to accept the law of large numbers as protection, even though their comfort surely came more from experience than from mathematical proof. The experience of the administrators was mirrored in the experience of the public, as they developed a confidence or trust in the operation that carried their interest over the changes of regime.4 It is also true that Casanova’s second warning, not to try to micromanage bets to balance the portfolio, may not have been always strictly adhered to. At various times the Loterie administration reserved for itself the right to do just that, though if it ever tried to do so in any systematic way, there seems to be no record of the attempt. It would have been an administrative nightmare, with thousands of bets at hundreds of agencies and as many as fifteen drawings a month, and the bettors would have always been one step ahead. To enforce balance would have required an immediate method of communication between agencies at the level of the individual bet. And it would have been fruitless; it would have limited the chance of a huge loss (say, when some number in that week’s news was bet by an unusual number of people), but that would have come at the cost of a profit should that number not come up, as well as a cost of enthusiasm in the betting public.

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In the 1780s the adminstration did try one scheme to achieve a more diverse portfolio: it preprinted tickets with a purposely diverse set of numbers and offered them as an easier way to bet, with no need to consult dream books or tea leaves— a sort of eighteenth-century version of the modern “quick-picks” in which the computer chooses your numbers. Otherwise the Loterie officials bowed to necessity, and avoided intervention. Better to concentrate on maintaining the rigorous randomization in the drawing in order to allow the law of large numbers to work in their favor, and instead concentrate their efforts on protecting against fraud through post-drawing ticket sales. Over the course of the Loterie there was minor but constant adaptation and experimentation with safeguards, some of which seem eerily like methods we find in use today. In the early years, the focus was on methods of distinctive printing in the effort to prevent counterfeiting. By the later years, the Loterie administrators found that dispersion of multiple copies of agency records was more effective. The early tickets used hard-to-duplicate patterns to uniquely identify days of the drawing and the numbers chosen, and this was effective only to the extent that agents could be counted on to scrupulously study every claim in minute detail. Clever differences in pattern can work well when there are only a few different patterns to internalize, such as in distinguishing currency of different denominations. But there was such immense variety in the details of bets that the eyes of even a diligent agent would soon glaze over and miss the few real cases of fraud— cases so rare that most agents would never get the reinforcing reward of discovering a fraud. The method was sound in conception but unsound in implementation. Better to have unconfusing tickets, and to rely on agency records as the backup and final authority on any large payoff. A few small frauds may have been missed, but overall the integrity of the Loterie was safer. After the Revolution, the Loterie used handwritten tickets with multiple panels, including tabs to be detached and dispersed for filing and later cross-checking if needed. Instead of cryptic codes, the tickets emphasized unambiguous handwriting. The French did take one other step to limit risk over long periods in the Loterie’s history: they eliminated the quine as an available

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bet. This may have been more for the worry about fraud than about simple gambling risk. The probability of a successful counterfeit, perhaps an inside job duly entered in the register with the connivance of an agent, was surely higher than the one chance in 44 million for an honest win. And the demand for the quine was low. Witness the very few actual wins when the quine was allowed; only one seems to have occurred in its history within France. When the only known winning case is a dimly receding legend, it is hard for even an addicted gambler to raise a hope of winning. Over its eighty years the Loterie became a part of popular culture in France, a status signaled both by large sales and regular criticism. As noted in chapter 13, there were more than 150 million bets on the quaterne and more than 500 million bets overall in thirty-three years of post-Revolution operation. For three decades after the Revolution, the Loterie was active over all of urban France. In 1782 LouisSébastien Mercier took a shot at the Loterie in colorful terms in his Tableau de Paris, an eighteenth-century critical commentary on society, with biting wit: “Loterie royale de France: Another source of great evils, and newly opened. It is a scourge which is renewed no less than twice a month. This loterie, fatal in every possible way, is a real contagion that has come to us from Italy.”5 Ironically, it was a motion by Mercier in the Council of Five Hundred in 1795 that started the process toward reestablishing the Loterie after the Revolution. When he was confronted on the floor by his own words from 1782, he denied any contradiction, saying that he had referred to the old organization, not the principle.6 By the time of the 1836 suppression, at least some voices were nostalgic. Balzac, writing in 1842: This passion [for the Loterie], so universally condemned, has never been studied. No one saw the opium of misery there. The Loterie, the most powerful enchantress in the world, did not she create magical hopes? The spin of the roulette wheel makes players imagine a profusion of gold and pleasures but lasts only as long as a flash of lightning; the Loterie gave five days of existence to this magnificent lightning bolt. What is the so-

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cial power today that can, for 2 francs, make you happy for five days with the prospect of perhaps delivering all the joys of civilization to you? Tobacco, a tax a thousand times more immoral than gambling, destroys the body, attacks the intelligence, it stupefies a nation; the Loterie did not cause the slightest misfortune of this kind.7

Why then was the Loterie closed? It seems unsatisfactory, even if true, to say its suppression was simply a combination of complaints about the odds, the immorality of gambling, the social harm from addictive gambling, and a new governing class wanting to encourage investment. These are unsatisfactory reasons because they were weak arguments even at the time, with the possible exception of the last of them, which may not have been weak in the eyes of those in power. And they were arguments that had failed for the preceding seventy years. Still, the situation was much the same in Great Britain, where state lotteries had been suppressed in 1826, with similar arguments aired publicly, and similar conflicts of interest lurking quietly.8 In both France and Great Britain, the only state actions to suppress, prior to the final act, were directed against competitors to the state lotteries, with the goal of preserving the state monopoly. In both cases the public approval was signaled by strong demand for the lotteries, at least until near the very end; yet in both cases there were no prominent public defenders of the lotteries. The religious arguments had become muted with the weakening of the clergy after the Revolution, and should be expected to have been all the weaker with the Church enjoying the benefits of special lotteries itself. The accusations of social harm were supported by a few real cases of abuse that received public attention, showing by example that gambling could be additive and could drive people to bankruptcy or suicide. In the hands of a good writer this can become a strong rhetorical argument, but it never came with more than anecdotal evidence. For example, in the summer of 1787 a Lyonnaise pharmacist named François Tissier with a weakness for the Loterie was taken advantage of by a Loterie agent and his wife. They got him to go heavily into a martingale, and he built up debt; they then blackmailed him for

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thirty thousand francs. The matter ended up in the courts.9 But such cases of real abuse required access to levels of credit that were beyond most people, and they were unlikely to garner widespread reaction. Casino gambling, where the bets were essentially unlimited and the time between bets was minimal, was much more prone to this kind of problem. The data discussed in chapter 13 of this book are consistent with most Loterie bets being small, and there was a gap of at least a week between drawings at a single location. This would not prevent people with other motives from putting on a moral cloak as cover for other, conflicting motives. An absurd claim in the temperance press of fifty suicides the night after the drawing of a British lottery in the 1820s cannot have had wide credence, even though the report was so striking that it reached across the Atlantic.10 Another question, more interesting than that of why the Loterie was suppressed, is: Why did people gamble? Nearly a half billion tickets for the French Loterie may have been sold over eighty years; were these people all like mythical lemmings, following a crowd over a cliff to destruction? Was the success of the Loterie due to irrational behavior or deception? At the time this was posed as a moral question, blaming the attraction on deception, a common assumption still popular among opponents of lotteries. Some said then, and say now, that the lottery is a tax on stupidity. On the whole, this is a dubious assumption. Public skepticism about lotteries has always been widespread. Robinet reported in 1782 that a common proverb at that time was “Loterie, filouterie,” or lottery is trickery. Gataker had an earlier version in 1619: “In Lotery is Bouery [bowery],” or lottery is deceit— which he reported was due to the Dutch.11 Leti in 1697 satirically announced that the origin of the word “loterie” was from “lot” and “rie”— literally, in French, “The chances laugh.”12 The odds for the Loterie were widely and commonly available; that “deception” could persist for decades, despite this common knowledge, widespread skepticism, and a lack of personal experience with large payouts, is not credible. A more interesting explanation is rooted in the mathematical arguments of Daniel Bernoulli, Buffon, and Laplace. These argu-

Conclusion  203

ments about poor payouts had been voiced throughout the life of the Loterie, but only by mathematicians, not a notoriously strong lobby. Even people (such as Adam Smith) who realized that any entrepreneur must make a profit, would raise eyebrows at the change in payoffs from the extrait’s 15 for 1 (where 18 for 1 would be even) to the quaterne’s 75,000 for 1 (511,038 for 1 would be even) or the quine’s 1,000,000 for 1 (where nearly 44,000,000 for 1 would be needed for it to be a fair bet). Buffon considered such small chances as for the quaterne and quine to be zero, and thus for him even fair lotteries were irrational. Other theoreticians with a strong belief in expected value felt that long-run averages should drive individual decisions, even when the long run was thousands of lifetimes, and they could be impressed by the argument that an expected value much smaller than the price of entry was irrational. The expected values in monetary terms for a single franc bet on an extrait, ambe, terne, quaterne, or quine are, respectively, 0.83, 0.67, 0.47, 0.15, and 0.023 francs. But anyone with a sense of the risk for potential fraud in the Loterie would be less impressed. Daniel Bernoulli in 1738 had advanced the idea that the decision should be based on the expected utility of the reward, not the reward itself. He suggested a particular utility function with decreasing marginal utility, where the utility of an additional unit decreased as the reciprocal of a person’s fortune. He thought this precise relationship was just a convenience, but he and several later people did believe that the basic phenomenon of decreasing marginal utility was an evident general truth (figure 17.1). The consequences of this idea presented a troublesome problem for any attempt to model human behavior. As we have seen, Laplace recognized that with decreasing marginal utility and with expected utility as a criterion, it would follow that gambling in even a fair game was irrational: even then a gamble implies an expected utility loss. If you have 100 francs and gamble for an additional franc, the franc you might lose (your 100th) is more valuable than the franc you might win (your 101st). Thus, Laplace stated, gambling would always and everywhere be immoral. And no one rose to the challenge of making

204  C o n c l u s i o n

Fig ur e 1 7.1. Daniel Bernoulli’s utility function (“fig. 5”), exhibiting decreasing marginal utility. The axis AR may be thought of as representing wealth or income, and the curve sS gives the utility as the logarithm of the wealth. Detail from figure 15.2 in this book.

a case for gambling on a societal level. For just this reason, Alfred Marshall would later refer to gambling as “an economic blunder”; he granted that the pleasure derived from the excitement of gambling might exceed the expected loss in utility, but that would mean abandoning expected utility, and “we are then thrown back upon the induction that pleasures of gambling are in Bentham’s phrase ‘impure,’ since experience shows that they are likely to engender a restless, feverish character, unsuited for regular work as well as for the higher and more solid pleasures of life.”13 But did Marshall correctly represent what Bentham thought? Jeremy Bentham (1748– 1832) is considered the founder of utilitarian philosophy, though he did not approach it mathematically. He is best known for statements like, “The greatest happiness of the great-

Conclusion  205

est number of people is the foundation of morals and legislation”— a statement that raised the blood pressure of my late colleague William Kruskal, who correctly insisted that this was in general a mathematical impossibility, and that inevitably some tradeoffs were required. Bentham never published on lotteries, but he left one short piece that found its way into a University of Geneva archive in French translation as “Des lotteries.” The scholar who found it summarized Bentham’s argument as follows: In these pages, five objections to lotteries are listed and in turn rejected: that they are fraudulent; that they foster reckless spending; that they stimulate the gambling spirit; that they create unhappiness in those who draw blanks; and that, as an institution of finance, they are unsuitable because they are too expensive to run. Bentham denies that there is any fraud in the matter; insists that there is more temptation to overspend on ordinary, everyday sensual enjoyments than on lottery tickets; contrasts the coolness with which such a ticket is purchased with the fever which prevails at a gaming table; calculates that the pleasure of expectation, spread as it is over a considerable period, is bound to be greater than the momentary pain of disappointment; and argues that the replacement of a “burthensome” source of government income, such as a tax, by an “unburthensome” one such as a lottery, would justify any expenditure that is no more than a percentage of the yield.14

Bentham evidently took a more sanguine view of lotteries than Marshall thought. But the basic problem still remained. Well into the twentieth century, even people like Marshall, who accepted the idea that expected utility was the reasonable way to assess bets, were forced to accept the view that the notion of utility must be more complex than the simple amount of money Bernoulli had in mind, where each additional unit of money conveyed less utility to the holder. Like Marshall, they had to introduce excuses such as the “pleasure of gambling” or Bentham’s “pleasure of expectation” to account for widely observed behavior. That is, they would introduce vague considerations that went outside simple utility theory: the ex-

206  C o n c l u s i o n

Fig ur e 1 7.2. The “wiggly” utility function, from Friedman and Savage 1948, 297.

citement surrounding the idea of the prize, which was the very hope that the law of 1836 had extinguished. The simple solution came only in 1948 in an elegant article by Milton Friedman and Jimmie Savage, “The Utility Analysis of Choices Involving Risk.”15 Friedman and Savage noted that the broad assumption of decreasing marginal utility— that the utility function will be everywhere increasing at a decreasing rate— was neither necessary nor warranted when it was expected utility that was at issue. All that should be generally assumed was that the utility of wealth increased with wealth, and that individuals were “consistent” in their preferences. The consistency they required was that adopted by von Neumann and Morgenstern in their 1944 book Theory of Games and Economic Behavior: basically, that if A was preferred to B and B was preferred to C, then A was preferred to C, where A, B, and C could be either levels of wealth or gambles involving levels of wealth. Within such a framework they could encompass individuals as rational and consistent who would base actions on a “wiggly” utility function such as in figure 17.2. This curve could represent the utility of a given level of wealth

Conclusion  207

for an individual whose preferences agreed with decreasing marginal utility for lower levels of wealth and for higher levels of wealth, but whose behavior between those regimes was consistent with increasing marginal utility. One conceivable story to help understand this is that within the lower or upper regions, the basic quality of life changed slowly, but lifting oneself from lower to upper offered such a substantial gain that the person would be willing to take on increased risk to make the transition. Friedman and Savage required not that the individual thought in terms of such a specific function, but just that they behaved consistently with expectations computed on its basis, and so that predictions could then be made from it that could test the theory. They not only had solved an old and open problem, but had laid a framework for utilitarianism that would prove immensely influential in microeconomics. This leaves us with one question this history raises. Why was the plan of the Loterie not adopted in any other country to the extent it was in France? For various periods it was popular in many German states, but usually those lotteries were small and short-lived. It was never adopted in Great Britain, and aside from the period when Napoleon carried it to the limits of the French empire, it was never a big presence in the rest of Europe. And yet, at least superficially, it was far superior to the blanks lotteries or the Klasse lotteries. It did not require advanced sales, and could be drawn as frequently as was wished. The payoffs could be easily adjusted, assuming that they were not so different from a neighbor’s payoff that they would be bested by competition. The public liked the plan, and had great choice in bets. The main and probably major factor behind the lack of spread, was the type of risk that the state had to take on to gain the profits. In support of this speculation, consider the success of the different lotto games at present. These modern reincarnations are quite different from the French Loterie. One is the casino game keno, which entered the United States in the 1880s and can still be found in some places. Its name is said to have been derived from “quine,” but the connection is remote at best. The more direct descendants are the various forms of lotto that still flourish as contributors to states’ budgets.

208  C o n c l u s i o n

Typically the bettor chooses six, not five, numbers from 1 to 46 or 54 or some similar limit, but otherwise there is no choice of bets— it is as if all bets terne through quine were assumed to be everyone’s choice. Because the bets are bundled, there is no reason to call attention to payoff odds versus mathematical odds. But the most telling difference from the French Loterie is that modern methods of communication make it possible for the state to eliminate the risk of having many winners with the same numbers, by splitting the prize among winners— what we now call a parimutuel pool. This, and the fact that the pool accumulates by “rolling over” funds from previous drawings with no winning claims, also makes it impossible for gamblers to determine accurately their bet’s expected value, much less its expected utility. In most cases the expected value seems to be well under half the price of the bet. The state takes at least half the amount bet, more than twice what the French took, and has pushed the risk of multiple winners off on the bettors themselves. Security and oversight are weaker than in the French Loterie, since the potential losses are not born by the state. In June 1998 a “pick three” lottery in Arizona (where three digits 0 to 9 are chosen daily) went for an entire month with the wheel of fortune (actually a computer algorithm) missing the number 9. For that month, no one whose three-digit number included a 9 had any possibility of winning, but it was a matter of indifference for the state. And, of course, it was a bettor who noticed this, not the state.16 In both foreign and domestic policy, states both modern and ancient routinely take on risk that is often vaguely formulated and whose consequences are not fully articulated. The major risk associated with the French Loterie was precisely defined and calculable from simple basic principles. Are these risks fundamentally different, or essentially the same? There is a long tradition of considering them as fundamentally different. Ancient Hindu law drew this distinction in its regulation of gaming. There were two sorts. One was called Choperbàzee, exemplified by betting on games of dice. The other was Shemàbhee, exemplified by betting “when Persons cause Elephants to fight with Elephants, Bulls with Bulls, Cocks with Cocks, Nightin-

Conclusion  209

gales with Nightingales, or any other Animals in the same Manner.”17 The distinction was clearly between Indian dice, elongated rectangular solids with four sides which bettors universally agreed were equally likely to come up, and battles between elephants or nightingales, in which the chances were unknowable and ignorance and vain hope substituted for calculable chance. More recently, starting in 1921 Frank Knight and John Maynard Keynes made similar distinctions between economic risks using different terms— emphasizing the difference between situations in which all possible outcomes of an action were known and their chances were objectively calculable, as in games of Indian dice, and situations in which they were not. Many statisticians such as Jimmie Savage would argue that the distinction was not fundamental but just a matter of degree of information, and that the same tools of analysis with probability and expected utility were applicable in all cases. Indeed, the Hindu lawyers would have agreed. They did not permit either type of gambling, and stipulated the same penalties in both cases: the winning gambler would have to pay the magistrate half the winnings, and a gambler caught cheating would have two fingers cut off. In any event, the temptation to make the distinction is deeply rooted, and as the sociologist W. I. Thomas told us in 1928, “If men define situations as real, they are real in their consequences.”18 That the Loterie risk was transparent meant, first, that it was impossible to avoid confronting it, and second, that it would be impossible to avoid blame for accepting it should things go badly. With other less clearly articulated risks, one can substitute “possible” for “impossible” in the previous sentence. Whether the risk would be embraced or cautiously avoided in favor of some course of action that did not carry such a burden would then depend critically upon the advice the state would receive, and the confidence its ministers— whose fate could well hang in the balance— had in the advisors. In France in January 1757, the ministers needed the money and had to turn one way or the other. D’Alembert was one advisor; there is no firm record of what he said, but there is no reason to believe he would have dissented. On the other hand, Casanova’s firm and con-

210  C o n c l u s i o n

fident voice, his mastery of the needed mathematics, his convincing experience with gambling, and his emphasis on the relevance of the successful use of the statistical law of averages in insurance were exactly the assurance the ministers would have needed. Fredrick the Great had the advice of Leonard Euler, whom he trusted as one of the greatest mathematicians of the century. But while Euler was an immensely better mathematician than Casanova, he was not as expert in the statistical implications of the mathematics, and would have been a much more conservative voice. Casanova did not invent the Loterie; he was not the first to present it to the French ministers, who had other schemes to consider as well. But he was the perfect advocate to present it to the court, and to argue its virtues and advantages at the crucial time of judgment, with force of reason and mastery of the facts of the case. It was indeed Casanova’s lottery.

Acknowledgments

This r e sear c h has been c ar r ied o ut over a twenty-fiveyear period, and my debts of gratitude have mirrored the growth rate of the US national debt. Not all of my six hundred emails with Bernard Bru over that period have concerned the Loterie, but his answers to my many questions have been a master class in French and mathematical history. David Bellhouse’s knowledge of English archives has been particularly helpful; Gabriel Sabbagh’s generosity in sharing his scholarship on French economists in particular has opened new doors to me. Teresa Ging’s role in the start of the work is recognized in the text. And the number of people who have answered my questions or corrected my speculations over the years is too large to enumerate, but I cannot omit mention of Branko Aleksic, David Bevington, Pierre Crépel, Robert Darnton, Lorraine Daston, Persi Diaconis, Stewart N. Ethier, Richard W. Farebrother, Christian Genest, Marc Hallin, James Hanley, Peter McCullagh, Xiao-Li Meng, Fred Mosteller, Glenn Shafer, J. B. Shank, Noel Swerdlow, Antony Unwin, and two anonymous reviewers. It would not have been possible to complete this book over the past year— a year of pandemic in which university libraries and archives were locked down— without the eager assistance accumulated over the past forty years from many wonderful booksellers, including (but not limited to) Christoph and Detlef Auvermann, Lucinda Boyle,

212  A c k n o w l e d g m e n t s

Stéphane Clavreuil, Deborah Coltham, Julien and Françoise Comellas, Roger Gaskell, Arnoud Gerits, Jonathan Hill, Andrew Hunter, H. P. Kraus, Michael Kühn, Hugues de Latude, Nina Musinsky, Jeremy Norman, Alexandre Piffault, Barbara Scalvini, Susanne Schulz-Falster, Ian Smith, Rick Watson, Jeff Weber, Graham Weiner, and Christian Westergaard. In the earlier parts of this investigation, I was helped by librarians and archivists in Paris, Berlin, London, Oxford, and Chicago. I would single out two for special thanks: Alice Schreyer (University of Chicago, and now Newberry Library) and Julie-Anne Lambert (Bodleian Library Oxford). In the later stages I am grateful for the encouragement and advice from Karen Merikangas Darling, my editor at the University of Chicago Press, and her excellent editorial team. And last but not least, this book would not exist but for the untiring support and shared interests of my wife, Virginia, to whom the book is dedicated.



Appendix One



Probability

I n t h e F r e nc h L o t e r i e, f i v e n u m b e r s would be drawn from a “wheel of fortune” containing the numbers 1, 2, . . . to 90. The evaluation of the chance of winning is a simple problem in combinatorial probability. For any integer numbers 0 ≤ k ≤ n, the number of ways of choosing k specific objects from a set of n distinguishable objects is n k

=

n! k !( n − k )!

where n! = 1 × 2 × 3 . . . n, and by convention, 0! = 1. Then the number of ways five Loterie numbers can be drawn is 90 5

90 · 89 · 88 · 87 · 86 · · · · 2 · 1 5 · 4 · 3 · 2 · 1 · 85 · 84 · · · · 2 · 1 90 · 89 · 88 · 87 · 86 = 5·4·3·2·1

=

= 43,949,268.

The chances of the different bets winning can be found in two ways. We illustrate by finding the chance that a particular bet on a terne (three numbers specified) will win. Suppose the gambler bet on the terne 17, 27, 40. The number of ways the gambler can win is the number of possible draws of five numbers that include these three.

214  A p p e n d i x O n e

That requires drawing these three, and there is one way in which this can be done: 33 = 1. And there are 872 ways to choose the other two. So the number of ways to win is 3 3

87 2

=1 ·

87 · 86 = 3,741. 2

Then the chance of winning is 3 3

87 2 90 5

3,741 1 = . 11,748 43,949,268

=

An alternative way to compute this chance is to imagine that the draw has already been made but the result is unknown to you. In how many ways can you choose three of the five winning numbers? That 90 90 ·89 ·88 is, 53 = 5·4 = 117,480 ways you can 2 = 10. There are 3 = 3·2 choose three from ninety. The chance of winning is 5 3 90 3

=

1 10 = 11,748 117,480

which is exactly the same as before. All of the probabilities for the Loterie can be calculated in this way. If you bet on k distinct numbers, the chance of winning is k k

90 − k 5− k 90 5

=

5 k 90 k

.

This gives the chances as Bet

Chanc e

Extrait Ambe Terne Quaterne Quine

5/90 = 1/18 10/4005 = 1/400.5 1/11,748 5/2,555,190 = 1/511,038 1/43,949,268

The chance of winning a bet on an extrait déterminé is even simpler: 1 1 × 89 1/90. The chance of winning a bet on an ambe déterminé is 90 =

Probability  215

1/8,010, the chance of winning on your first number multiplied by the chance of winning on the second, given that your first was successful. If the Loterie drew six numbers instead of five, and you bet on k distinct numbers, the same logic would give the chance of winning as k k

90 − k 6− k 90 6

=

6 k 90 k

.

This gives the chances for a Loterie where six are drawn as Bet

Chanc e

Extrait Ambe Terne Quaterne Quine

1/15 1/267 1/5874 1/170,346 1/7,324,878



Appendix Two



Laplace’s Lottery Theorem

I n one of h i s e ar l i e st pape r s, read to the Paris Académie des sciences on 2 February 1772 and published in 1774, Laplace gave as an example a theorem regarding the Loterie, somewhat generalized. It was not formally designated as a theorem, even though there were “theorems” in the paper, but was described as the fourth of five “problems” he had solved. The statement was: A Lottery being composed n numerals of which p are chosen at each drawing, we ask the probability that after x drawings all of the numbers have appeared.

His one-page solution was elegant and correct, exhibiting his facility with the recurrence relationships that had been the main focus of the paper. The following exposition catches the spirit of his solution without needing that facility. We will, as Laplace did, seek the probability that after x drawings, not all of the n numbers have appeared (the stated problem’s solution would then be 1 minus this n probability). Let p be the number of ways you can choose p distinguishable objects from a set of n; that is, the binomial coefficient n! (p!(n – p)!) . This is the number of possible outcomes in one drawing, so the total number of possible outcomes in x independent drawings n x would be p . Laplace then sought to determine the number, S,

Laplace’s Lottery Theorem  217

of these cases that would not exhibit all n numbers over the x drawings. A simple example illustrates his approach. Let us look at how the calculations would go with n = 3, and let A, B, and C be the sets of outcomes that, respectively, do not exhibit 1, 2, and 3. Of course these sets are not mutually exclusive. In this example we require the number of outcomes of A, B, and C all together; that is, A ∪ B ∪ C, eliminating double-counting. Denoting “number of cases” by #, the general equation for what we seek is #(A ∪ B ∪ C) = #A + #B + #C – #(A ∩ B) – #(A ∩ C) – #(B ∩ C) + #(A ∩ B ∩ C),

or, since the sets A, B, and C are identical in size, as are their intersections, #(A ∪ B ∪ C) = 3 × #A – 3 × #(A ∩ B) + 1 × #(A ∩ B ∩ C).

Recognizing that the coefficients 3, 3, and 1 are found as binomial coefficients with n = 3 (e.g., the second “3” is the number of two set intersections chosen from the n sets), this is =

n n n #A − #(A ∩ B) + #(A ∩ B ∩ C ). 1 2 3 x

x

Now, #A = n −p 1 , #(A ∩ B ) = n −p 2 , #(A ∩ B ∩ C ) = the solution can be written thus: #( A ∪ B ∪ C ) =

n 1

n− 1 p

x

−

n 2

n− 2 p

x

+

n 1

n− 3 x , p

so

n− 3 p

x

.

This leads directly to Laplace’s solution, by generalization to arbitrary n: #S = Σ(−1)k+1

n k

n− k p

x

, the sum being from k = 1 to n − p.

218  A p p e n d i x T w o

The probability that the n possible choices have not all been exposed x by the xth drawing is then #S/ np . Laplace stated that to apply this to the Loterie, you would set n = 90 and p = 5. But he did no calculation, and he evidently believed that his solution, while elegant and correct, was for practical purposes useless, in that it defied actual calculation in the one situation of practical interest, the Loterie. Writing in a referee’s report on a paper read to the Académie des sciences by Legendre in 1781, Laplace said, Legendre rightly observes that although the problem is solved analytically, it is not however numerically, when the number of numbers is not small, as in the Loterie of France. The formula is then made up of a large number of alternately positive and negative terms which are excessively large, while the final result is below unity. The numerical calculation of this formula then becomes impracticable and the use of tables of logarithms which one could help oneself becomes illusory, it is therefore much to be desired that we have a general method of approximation, by means of which we can easily obtain the numerical value of this kind of expression, when the numbers of which they are functions are considerable. (Bru and Crépel, 1994, 158)

Laplace— and, apparently, Legendre— seem to have concluded from the formula that actual calculation was not feasible. They had reason to worry; for example, here are the first twelve of the eightyfive terms for the series when x = 40: 9.14743138 – 40.293069 113.869804 – 232.08754 363.615995 – 455.77773 469.728121 – 405.97106 298.635431 – 189.13386 104.068105 – 50.112795

Laplace’s Lottery Theorem  219

This series will eventually sum to 0.999988, thus confirming that it is nearly certain that not all numbers will appear in forty drawings. But there is no sign of convergence, and in the precomputer era these calculations would not be attempted. The probability of doing them all correctly to the required number of places was, then, vanishingly small. Legendre’s paper, which was not published and may not still exist, may have reawakened Laplace’s interest in the problem. As part of a memoir read to the Académie in 1783, Laplace managed to get a reasonable and computable approximation to the distribution for a simpler case, when n = 10,000 and p = 1. For that case he found that the median of the distribution of the number of drawings needed to reveal all ten thousand numbers was 95,767.41 (Laplace 1786, 334– 38). But the case of the actual Loterie apparently still eluded him. Nearly twenty years later, when he was preparing his great treatise Théorie analytique des probabilités, he returned to the subject and finally managed to develop a way to get a good approximation for n = 90 and p = 5. For that case, he found the median number of drawings to reveal all ninety numbers to be 85.53 (Laplace 1812, 191– 201). Laplace’s own computations were entirely in terms of these elegant approximations. Apparently Laplace did not realize that for a larger x (say, 60 or 70), the situation changes dramatically. When x is sufficiently large that the chance of there being more than a few numbers that have not occurred is small, only the first few terms are needed, and these would have been easy to compute even in the 1700s. In 1785 one of Leonhard Euler’s last articles was published; it was on exactly this problem. Euler had read the memoir to the Berlin Academy on 8 October 1781. By that time his blindness had become quite severe; he died in September 1783 and may not have had a chance to revise the article for publication. He did not cite Laplace, but that cannot be taken as an indication that he had not seen Laplace 1774; the art of citation was practiced differently then. Euler started with the problem Laplace had presented in 1772; he gave the same solution, expressed in different notation. It was here that Euler introduced his compact notation for nk . And then he did something

220  A p p e n d i x T w o

Laplace probably had not attempted: for n = 90 and p = 5, he computed the chance that after x = 100 drawings all numbers had not been exposed, and from this he found the chance that all had been exposed to be 0.7419. (Actually, Euler erred in calculating one logarithm; the correct answer is 0.7410.) He then went on to find the answer for x = 200 drawings (0.999), as well as the chances that all but one or all but two of the numbers had occurred after one hundred drawings. Euler did this simply by direct calculation, using logarithms. For in these cases the series converges very quickly, despite appearances that this might not be the case. Here are the first ten partial sums for the calculation for x = 100: 0.29640405 0.25574361 0.25917979 0.25897888 0.25898753 0.25898725 0.25898726 0.25898726 0.25898726 0.25898726

A very good approximation can be found with only three or four terms! In the 1790s Jean Trembley, a minor Swiss mathematician at the Berlin Academy, wrote a long paper on the same subject, with many citations to both Laplace and Euler. Trembley reproduced Euler’s calculation, getting the correct answer, and also found the median of the distribution as between 85 and 86, just as Laplace would later show in 1812. All of this was accomplished with a few logarithms (Trembley 1799, 76– 77). Here are the first ten partial sums for these values of x:

Laplace’s Lottery Theorem  221

85

86

0.69861498 0.47048274 0.51684438 0.51025782 0.51095452 0.51089746 0.51090117 0.51090098 0.51090098 0.51090098

0.65980304 0.45644921 0.49542724 0.49020791 0.49072789 0.49068781 0.49069026 0.49069014 0.49069015 0.49069015

Laplace’s result for the median 85–86 was quoted in some texts (e.g., Littrow 1833, pp. 54– 55), for possible guidance to believers in the law of the maturity of chances. Jahn (1839, pp. 43– 54) discussed both Laplace and Euler, and recalculated Euler’s results. For a modT abl e A . 1 . The probability p that all 90 numbers are revealed in k drawings; 40 ≤ k ≤ 200. k

p

40

0.0000

45

k

P

k

p

95

0.6702

150

0.9831

0.0003

100

0.7410

155

0.9873

50

0.0028

105

0.7988

160

0.9904

55

0.0136

110

0.8449

165

0.9928

60

0.0425

115

0.8812

170

0.9946

65

0.0971

120

0.9095

175

0.9959

70

0.1774

125

0.9312

180

0.9969

75

0.2765

130

0.9479

185

0.9977

80

0.3837

135

0.9606

190

0.9983

85

0.4891

140

0.9703

195

0.9987

90

0.5859

145

0.9776

200

0.9990

222  A p p e n d i x T w o

Fig ur e A.1. The curve describes the probability distribution of the number of drawings (k) of the Loterie required to reveal each of the numbers 1 to 90 at least once. The median of the distribution would be about 85 or 86.

ern treatment of Laplace’s theorem, see Kolchin et al. 1978, chapter 7, especially formula 2, p. 212. Kolchin refers to Markov for this formula, and indeed Markov treated this subject in his chapter 4. See Markov 1912, 101– 8. Markov cited Euler 1785, as well as an article by Lagrange which cited Laplace 1774.

Notes

Introduction 1. 2. 3. 4. 5. 6. 7. 8. 9.

Buchan 2018; Murphy 1997; Wennerlind 2011. Hamilton 1936; Murphy 1997; Spang 2015. Ewen 1932, chapter 2; Brown 1999. Menut 1834. École militaire 1762, 1782. Stigler 2014. Hacking 1975. Hacking 1990. Daston 1988;, Legay 2014.

Chapter 1 1. 2. 3. 4. 5.

Wilson 1941; Zweig 1928. Casanova 2013– 18. All of the dialogue has been translated from Casanova 2013– 18, tome 2. Swijtink 1986; Daston 1988. D’Alembert 1761, p. 7; 1768, p. 86.

Chapter 2 1. 2. 3. 4.

Bellhouse 1991; Felloni 2005; Assereto 2013. Felloni 2005, p. 76. Harnage 1662. Charpentier 1920; Kynnersley 1885.

Chapter 4 1. Ashton 1893, pp. 79– 85; Farebrother 1999. 2. Harnage 1662. 3. Casanova 1757.

224  N o t e s 4. 5. 6. 7.

Casanova 1757, pp. xi– xii. Schuchard 2011. Zollinger 2006. Legay 2014, pp. 72– 73, gives a map of their locations.

Chapter 5 1. On the Internet at Fond Turgot 1190– 1926, 745AP/48. 2. In an edict of 20 Juillet 1777. 3. In an edict of 20 Juillet 1777.

Chapter 7 1. 2. 3. 4. 5. 6. 7. 8.

Bernard 1994; Garvia 2007. May 1771. Harnage 1662. Bleackley 1923, p. 242. Ashton 1893. Tuer 1882. Thomas 1976, p. 56. See http://www.stat.uchicago.edu /~stigler/AnonymousLettercirca1768-V2 .pdf, http:// www.stat.uchicago.edu /~stigler/LetterSupportingEnglishLoterie-French.pdf.

Chapter 8 1. Laulan 1951. 2. Wennerlind 2011, p. 144.

Chapter 9 1. Leonnet 1963, pp. 41– 45.

Chapter 10 1. 2. 3. 4. 5. 6. 7.

Fienberg 1971; Mateu et al. 2004. Menut 1834. McCullagh and Nelder 1989, p. 191– 92; Joe 1993. Joe 1993. Thuillier 1997. Duvillard 1806. Duvillard’s MS Table from January 1819 is in BnF Arsenal N. a. f. 20587.

Chapter 11 1. Kadell and Ylvisaker 1991; Henze and Riedwyl 1998. 2. https://archive.org/details/case _oversize _frc _27593. 3. Lacroix 1816, p. 109.

Notes  225 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15.

Darnton 2018. Anonymous 1786. Cagliostro 1803. Anonymous 1816. Anonymous 1816, p. 27. Greppi 1800. Marseille 1807. Bellhouse 1991; Stampa 1700, especially pp. 19– 21. Nougaret 1787. Lebrun-Tossa 1801. Chiari 1757. Kotzebue 1804, pp. 754– 55, 763.

Chapter 12 1. Shafer 2022. 2. Gräff 1773. 3. Stigler 2014. 4. Swijtink 1986. 5. Parisot 1801. 6. Parisot 1810. 7. Parisot 1810, p. 114– 15. 8. Laplace 1812, p. 200– 201. 9. Shafer 2022. 10. Parisot 1801, p. 105. 11. Parisot 1801, p. 123. 12. Parisot 1801, p. 309. 13. Parisot 1801, p. 121. 14. Anonymous 1795. 15. Anonymous 1795, pp. 2– 3. 16. Anonymous 1771, p. 33. 17. Ewen 1932, chapter 8; Anonymous 1771, p. 36. 18. Struyck 1716, p. 90; Buchan 2018, p. 94; Stigler 2021. 19. Anonymous c. 1796. 20. Balzac 1842. 21. Verne 1886.

Chapter 13 1. Administration de la Loterie 1797, 1800, 1804, 1815.

Chapter 14 1. 2. 3. 4.

Ménestrier 1700, pp. 17– 19. Ewen 1932; Florange 1928; Legay 2014; Bernard 1994. Ewen 1932, p. 232. Felix and Sabbagh 2021.

226  N o t e s 5. 6. 7. 8. 9.

Ewen 1932, pp. 136– 38. Ewen 1932, p. 233. Ewen 1932, p. 296. Donvez 1949, pp. 37– 55. Donvez 1949, p. 46.

Chapter 15 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.

Stigler 2012, 2016. Bicquilley 1783; Bru and Crépel 1994, ch. 5; Lacroix 1816. Bru and Crépel 1994, pp. 602– 3. Bicquilley 1783, p. 118. See chapter 16. Bru and Crépel 1994, pp. 602– 3. Condorcet 1805, p.133. Lacroix 1816, p. 108. Laplace 1814. See Bru 1986 for even earlier use of some of these ideas by Laplace, and Stigler 2005 for a full account of the publishing history. Bernoulli 1738. Lacroix 1816, pp. 83– 89, 281– 84. D’Hauteserve 1834. Littrow 1833, pp. 54– 55. Quetelet 1828, pp. 111– 12. Administration de la Loterie 1797, 1800, 1804, 1815.

Chapter 16 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17.

Gataker 1619. Balmford 1593. Gataker 1619, p. 158. Gataker 1619, p. 97. Such as Thiers 1686. Dunkley 1985 describes the skirmishes in detail. Van Houdt 2008; Justus 1642. Smith 1776, vol. 1, p. 132. Buffon 1777, p. 57. Buffon 1777, pp. 92– 93. Huyn 1788, pp. 49– 51. Huyn 1788, p. 51; Mettra 1787, p. 310. Talleyrand 1789, pp. 22– 23. Talleyrand 1789, p. 33. Talleyrand 1789, pp. 44– 45. Mavidal 1878, pp. 548– 54. Jabbe-Minoblant 1789.

Notes  227 18. Talleyrand 1891, pp. 39– 40. 19. Talleyrand 1891, p. 40. 20. Gourgaud 1823, p. 90. 21. Laplace 1912. 22. Moniteur Universel, 14 April 1830. 23. Pereire 1832. 24. Pereire 1832, p. 65. 25. Pereire 1832, p. 66. 26. Descotils and Guilbert 1993, p. 33.

Conclusion 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18.

Avenel 1887, pp. 29– 32. Poisson 1837, pp. 68– 70. Cournot 1843, p. 105. For the role of trust in nineteenth-century government administration, see Porter 1995. Mercier 1782, 1783. Leonnet 1963, p. 42. Balzac 1842. Raven 1991; Zollinger 2016, ch. 5. Bret 1787. Gordon 1833, p. 10. Robinet 1782, p. 183; Gataker 1619, p. 27. Leti 1697, vol. 2, p. 280. Marshall 1891, p. 742, math note IX. Stark 1954, referring to p. 536 Dumont Papers, Geneva MS box LI, pp 54– 67. Friedman and Savage 1948. Kaigh 2001. Halhed 1776, pp. 287– 88. Sills and Merton, 1991, p. 229.

References

The most ancient wor ks on t he history of lotteries have been cited in the text. Ashton 1893 is still useful in regard to English lotteries. Among modern sources for European lotteries in general, the best more recent works are parts of Daston 1988; Legay 2014; Zollinger 2006, 2016; Leonnet 1963; Bernard 1994; Descotils and Guilbert 1993; and Schädler 2012— the latter three being wonderfully illustrated as well as excellent in their historical accounts. The Loterie was featured in illustrated exhibits by the Musée Carnavalet (1936) and Loterie Nationale (1936). Casanova’s memoirs (1997 in an English translation, definitively in French 2013– 18) are the best and really the only major source for information about his life. There are many biographies, but few add much other than a proper note of skeptical fact checking, which Casanova usually survives very well. There is a fine recent biography by Kelly (2008) and a collection of studies edited by Stefanovska (2021). Zweig 1928 and Wilson 1947 remain unmatched as general assessments. There are a number of gorgeously illustrated exhibition catalogues that include excellent historical commentary, including Prévost and Thomas 2011 and Ilchman 2017. Henry 1882 gives detailed references regarding Casanova’s understanding of mathematics. Administration de la Loterie.(1797, 1800, 1804, 1815. Loterie Nationale: Instruction à l’usage des receveurs, sur les obligations qu’ils ont à remplir (Brumaire, an 6); Instruction à l’usage des receveurs de la Loterie Nationale (Brumaire, an 9); Instruction aux receveurs de la Loterie Imperiale (Vendémiaire, an 13); Instruction aux inspecteurs de la Loterie Imperiale de France (Vendémiaire, an 13); Instruction aux receveurs de la Loterie Royale (1815). Paris: Imprimerie de la République, Imprimerie de la République, Imprimerie impériale, Imprimerie royale. Anonymous. 1771. The Lottery display’d; or, The Adventurer’s Guide; shewing the Origin, Nature, and Management of the State Lottery: The Errors and Losses incident to the Drawing, Registering and Examining; The Method of guarding against their pernicious Effects; and of recovering Prizes, hitherto sunk through imperfect Intelligence, or Loss of Tickets. Also, The Nature of Insuring Tickets, with Rules for estimating the Premium, at any Period of the Drawing. London: A. Caldwell. ———. 1786. Livre de rêves; ou, L’Onéiroscopie, application des songes aux numéros de la Loterie Royale de France, tirée de la cabale italienne et de la sympathie des nombres. Nouvelle édition. Paris, Desnos & chez les receveurs de loterie.

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Index

agencies, and Loterie bets, 3– 4, 67, 78, 132, 134, 141, 193 Aldous, David, 85 Aleksic, Branko, 211 Altona (lottery), 56– 57, 85– 87 ambe (defined), 24 ambe déterminé (defined), 46 Ashton, John, 58, 223n1 (chap. 4) Assereto, Giovanni, 223n1 (chap. 2) assignats, 2 astrology, 102– 4 Autun, bishop of. See Talleyrand-Périgord, Charles-Maurice de Avenel, Henri, 227n1 (conclusion) Balmford, James, 173– 75 Balzac, Honoré de, 129, 170, 200– 201 Barbeyrac, Jean, 175 Bellhouse, David, 107, 223n1 (chap. 2) Bender, Johann Heinrich, 164 Bentham, Jeremy, 204– 5 Bernard, Bruno, 143, 224n1 (chap. 7), 225n2 (chap. 14), 229 Bernoulli, Daniel, 160– 62, 202– 4 Bernoulli, Jacob, 7 bets: limits on, 27, 35, 37, 39, 48– 49, 134; location of, 137– 42; number of, 133– 34 Bicquilley, Charles-François de, 156– 58

birthday problem, 83– 84 Blanquet, Antoine, 42– 45, 97 Bleackley, Horace, 224n4 (chap. 7) bookkeeping, 4, 168, 190 Bordeaux (loterie), 67, 70, 81, 84, 88, 133, 137– 41 Boucher, François, 40 Bret (Avocat), 227n9 Brown, Eric, 223n3 (introduction) Bru, Bernard, 226nn2– 3 (chap. 15), 226n6 (chap. 15), 226n10 (chap. 15) Brumaire, Coup de, 185– 87 Brussels (loterie), 55, 56– 57, 70, 84, 87– 88, 137– 41 Buchan, James, 223n1 (introduction), 225n18 (chap. 12) Buffon, Georges-Louis Leclerc, Comte de, 178– 79, 181, 202– 3 cabalism, 10, 78, 97– 98, 107, 125 Cagliostro ( Joseph Balsamo), 98– 100, 102 Calzabigi, Giovanni, 13– 17, 22, 40– 41 Calzabigi, Ranieri, 15– 17, 22, 37, 40– 41 Casanova, Giacomo, 9– 17, 22, 26– 29, 37, 40– 41, 58– 61, 67, 71, 92, 152, 197– 98, 209– 10, 223n3 (chap. 4), 224n4 (chap. 4); memoirs of, 9, 11, 17, 26, 198, 229 Catholic Church, 97, 103, 143, 173, 175, 180– 85, 197, 201

240  I n d e x centimes, 8, 68 chance of winning at Loterie, 47, 91, 158– 59. See also Loterie: expected and realized return Charpentier, Léon, 223n4 (chap. 2) Chiari, Pietro, 107 chi-square test, 81– 84, 87, 101 Chistyakov, Vladimir, 222 coded patterns, 65– 66, 199 Colbert, Jean-Baptiste, 180 Condorcet, Nicolas de, 156– 59, 179 Copenhagen (lottery), 56– 57, 85– 87 Cournot, Antoine Augustin, 197 Crépel, Pierre, 226nn2– 3 (chap. 15), 226n6 (chap. 15) d’Alembert, Jean, 16, 17, 118, 209 Darnton, Robert, 225n4 (chap. 11) Daston, Lorraine, 223n9 (introduction), 223n4 (chap. 1), 229 debt collection, 71, 74– 76, 168 Delessert, Benjamin, 195 De Moivre, Abraham, 7 denier, 48, 52 Descoings, Madame, 129, 131, 170 Descotils, Gérard, 196, 229 d’Hauteserve, Gauthier, 162– 65 dice, 7, 118– 19, 173– 74, 208– 9 digits, written, 168– 69 Donne, John, 173 Donvez, Jacques, 154 dowries as prizes, 64– 65 draft lottery (US), 78– 80, 125 drawings by date, 70 dreams, 20, 98– 103, 108, 125 Dunkley, John, 226n6 (chap. 16) Dusaulx, Jean, 177 Duverney. See Pâris-Duverney, Joseph de Duvillard, Emmanuel-Etienne, 87– 89 École militaire, 11– 12, 16, 23, 42– 43, 71, 172, 223n5 (introduction); loterie, 32– 34, 41– 43, 84 École normale, 119, 159– 60 Elizabeth I (queen), 3, 143 Encyclopedie, 17, 163

Euclid, 156 Euler, Leonard, 160, 210, 219– 22 Ewen, C. L’Estrange, 143– 44, 146– 47, 152, 223n3 (introduction), 225n17 (chap. 12) extrait (defined), 24 extrait déterminé (defined), 46 fable, 76 Farebrother, Richard William, 223n1 (chap. 4) Felix, Joel, 225n4 (chap. 14) Felloni, Giuseppe, 19 Fermat, Pierre, 7, 139 Fienberg, Stephen E., 80, 224n1 (chap. 10) Florange, Charles, 143, 225n2 (chap. 14) fortune, wheel of, 23, 29– 34, 78– 80, 87, 93, 108, 125– 26, 146– 47, 208, 213 fraud, 4, 31, 39, 47, 49, 61, 62, 66, 70, 71, 93, 141, 168, 199– 200 Frederick the Great, 8, 40– 41, 198 Friedman, Milton, 206– 7 Frömmichen, Karl Heinrich, 164 gambling, immorality of, 173, 201, 203 Garvia, Roberto, 224n1 (chap. 7) Gataker, Thomas, 173– 75, 202 Genoese lottery, 3, 13, 17, 19– 22 Gentile, Benedetto, 19 Ghilini, Ambrose, 105– 6 Ging, Teresa, 6 Gluck, Christoph, 16 God, 14, 173– 74 Goethe, Johann Wolfgang von, 9 Gordon, George, 227n10 (conclusion) Goudar, Ange, 60 Gourgaud, Gaspard, 227n20 Gräff, 118 Greppi, Fortunato, 103– 4 Guilbert, Claude, 196, 229 Hacking, Ian, 7 Halhed, Nathaniel, 227n17 Hamburg (lottery), 56– 57, 71, 85– 87, 152 Hamilton, Earl J., 223n2 (introduction) Harnage, Thomas, 19– 20, 223n2 (chap. 4), 224n3 (chap. 7)

Index  241 Hautefeuille, Jean de, 144– 46 Henry, Charles, 229 Henze, Norbert, 224n1 (chap. 11) Hindu law, 208– 9 Hogarth, William, 31, 58 Hoyle, Edmond, 7 Hua-Hoey lottery, 20 Huygens, Christian, 7 Huyn, Pierre Nicholas, 164, 179– 81 Hypolite, 116, 118 Ilchman, Frederick, 229 Jabbe-Minoblant, 184 Jahn, Gustav Adolph, 221 Jeanne (No. 45), 63, 111– 16, 118, 121– 22, 169 Jefferson, Thomas, 75– 76 Joe, Harry, 83 Joncourt, Pierre de, 175 jumeaux, 94– 97, 170– 71 Justus, Pascasius (Pâquier Joostens), 175– 77 Kadell, Dan, 224n1 (chap. 11) Kaigh, William D., 227n16 Kelly, Ian, 229 keno (casino game), 207 Keynes, John Maynard, 209 Kirwan, Françoise Davide de, 74 Kirwan Château (wine), 74– 76 Knight, Frank H., 209 Kolchin, Valentin F. 222 Kotzebue, August von, 108– 9 Kruskal, William Henry, 205 Kynnersley, C., 223n4 (chap. 2) La Condamine, Charles-Marie de, 3, 153– 55 Lacroix, Silvestre François, 97, 157– 58, 162, 179 Lagrange, Joseph-Louis, 222 Laplace, Pierre Simon, 119, 121, 139, 159– 65, 185– 89, 192, 202– 3, 216– 22 Laplace’s theorem, 121, 162, 165, 216– 22 La Placette, Jean, 175 Laulan, Robert, 224n1 (chap. 8) Law, John, 2, 128 Lebrun-Tossa, Jean-Antoine, 25, 107– 8 Legay, Marie-Laure, 143, 223n9 (introduc-

tion), 224, 224n7 (chap. 4), 225n2 (chap. 14), 229 Legendre, Adrien-Marie, 218– 19 Leonnet, Jean, 224n1 (chap. 9), 227n6 (conclusion), 229 Le Peletier des Forts, Michel Robert, 153– 55 Leti, Gregorio (Gregorio Capocoda), 202 Lille (loterie), 56, 67, 70, 88, 133, 137, 138– 41 Lincoln’s Inn, 173 Littrow, Joseph Johann, 164– 65, 221 livre tournois (vs. franc), 8, 38, 63 Loterie: expected and realized return, 52– 54, 71– 73, 91– 92, 128, 166– 67, 189– 93, 208; vs. lottery, 4 Loterie names: Loterie de France, 82; Loterie nationale, 68, 70; Loterie royale, 43, 70 lotteries: blanks, 29– 31, 59– 62, 126– 28, 146, 177, 198, 205, 207; classis, 146– 52; foreign competition, 4, 48, 207; Genoese, 3, 13, 17, 19– 22, 31, 55, 58– 60, 105, 174; Great Britain, 3; Klasse, 56, 198, 207; Netherlands, 3, 128, 132, 146, 152, 189; raffles, 3, 19, 131; Spain, 55; state, 1, 30, 143– 44. See also under specific cities lottery: immorality of, 43, 68, 129, 160– 61, 175, 180– 95, 201, 203; number, “age” of, 112, 115– 18 lotto, 1, 64, 92– 94, 190, 207 Louis XIV, 1– 2 Louis XV, 12, 48 Louis XVI, 67 Louis XVIII, 70, 180, 185 Lyon (loterie), 67, 70, 84, 88, 133, 137– 42, 152, 196 Lyon, questionable winning draws, 84– 85, 93, 96, 140 manuals for Loterie, 134– 35, 167– 71 Markov, A. A., 85, 222 Markov dependence, 85 Marseille, John-Baptiste, 105– 6 Marshall, Alfred, 204– 5 martingale, 114, 122– 23, 201; compensative, 123; doubling, 114, 118, 122; productive, 123 maturity of chances, law of, 112, 118– 31, 163, 181, 189, 221

242  I n d e x Mavidal, M. J., 226n16 (chap. 16) May, Johann Carl, 57, 224n2 (chap. 7) McCullagh, Peter, 224n3 (chap. 10) Ménestrier, Claude-François, 143 Menut de St. Mesmin, M., Almanach Romain sur la loterie de France, 5– 6, 78– 82, 93– 96, 103, 132– 36, 190 Mercier, Louis-Sébastien, 200 Mercure de France (gazette), 179– 80, 183 Merton, Robert K., 227n18 (conclusion) Mettra, Louis-François, 226n12 (chap. 16) Mississippi Bubble, 2 Molière ( Jean-Baptiste Poquelin), 155 Moniteur Universel (newspaper), 62, 72, 104, 183, 192– 94 Montmort, Pierre Reymond de, 7 moral certainty, 178 Morgenstern, Oscar, 206 Mozart, Wolfgang Amadeus, 9 Murphy, Antoin, 223nn1– 2 (introduction) muskets, 146 names, bet on, 103– 5 Napoleon, 56, 70, 100, 105– 6, 132, 185– 86, 189, 207 Nelder, John, 224n3 (chap. 10) Newberry Library, 97, 212 North, Lord, 59– 60 Nougaret, Pierre-Jean-Baptiste, 107 number of tickets sold, 110 orphans, 30– 33, 197 parimutuel pool, 208 Pâris-Duverney, Joseph de, 11– 16, 26 Parisot, Sébastien Antoine, 81, 119– 25, 163 Pascal, Blaise, 7 payoffs, 24, 26, 47, 52, 136 Pereire, Émile, 194– 95 Petite Loterie Parisienne, 68– 69 Piombi (Leads), 10– 11, 16 playing cards, 12– 13, 174 poem, 48 Poisson, Siméon-Denis, 197 Polo, Marco, 21– 22 Pompadour, Madame de, 12, 16, 40

Porter, Theodore M., 227n4 (conclusion) Prévost, Marie-Laure, 229 primes gratuites, 50– 54, 144 probabilities, small, 158– 59 Pythagoras, 103 quaterne (defined), 46 Quetelet, Adolphe, 164, 166– 67 quick picks, 94, 96– 97, 199 quine: defined, 46; discontinued, 47, 199– 200 randomized survey, 6, 92– 96, 110, 132 randomness of draws, 77– 85, 88– 89, 93 randomness tested, 81– 85 Raven, James, 227n8 (conclusion) Riedwyl, Hans, 224n1 (chap. 11) Robinet, J.-B.-R., 202 Rouse, William, 164 rule of three, 156 Sabbagh, Gabriel, 60, 225n4 (chap. 14) salaries, 28, 67, 182, 186, 193 Sales, François de, 174– 75 Savage, L. Jimmie, 206– 7 Schädler, Ulrich, 229 Schuchard, Marsha Keith, 224n5 (chap. 4) scrapbooks, 6– 7, 58, 92, 110– 11, 115, 152 Sevast’yanov, Boris, 222 Seymour, Richard, 7 Shafer, Glenn, 179, 225n1 (chap. 12), 225n9 (chap. 12) Sills, David L., 227n18 (conclusion) Simpson, Thomas, 7 Smith, Adam, 89, 158, 177– 79, 203 sou or sol, 8, 52 Spang, Rebecca L., 223n2 (introduction) Stampa, Giuseppe, 107 Stark, Werner, 227n14 (conclusion) Stefanovska, Malina, 229 Stigler, Stephen M., 223n6 (introduction), 225n3 (chap. 12), 225n18 (chap. 12), 226n1 (chap. 15), 226n10 (chap. 15) Strasbourg (loterie), 67, 70, 84, 88, 96, 137– 42 Struyck, Nicolas, 128 suicides, claims of, 182, 188, 201– 2 suppression, 172

Index  243 Swijtink, Zeno G., 223n4 (chap. 1), 225n4 (chap. 12) Talleyrand-Périgord, Charles-Maurice de, 180– 85, 187– 90 tax on stupidity, 12, 90– 92, 167, 188, 202 terne (defined), 24 Terror, 67, 172 Thiers, Jean Baptiste, 226n5 (chap. 16) Thomas, Chantel, 229 Thomas, Peter D. G., 224n7 (chap. 7) Thomas, W. I., 209 Thuillier, Guy, 224n5 (chap. 10) tickets, insurance for, 126– 28 Tissier, François, 201 Tolot, Madama, 107– 8 Tomkins Picture Lottery, 149– 52 Trembley, Jean, 220 Tuer, Andrew White, 58 Turgot, Anne Robert Jacques, 42– 43, 88

utility, 159– 61, 178– 79, 203– 9 Van Houdt, Toon, 226n7 (chap. 16) Venice, 2, 9– 13, 21– 22 Verne, Jules, 130– 31 Voltaire (François-Marie Arouet), 3, 9, 153– 55 von Neumann, John, 206 Wandsbek (lottery), 56, 85– 87 Wennerlind, Carl, 66, 223n1 (introduction) wheel of fortune. See fortune, wheel of Wilson, Edmund, 10, 229 wolf and lamb (fable), 76 women’s names on tickets, 63– 66 Ylvisaker, Don, 224n1 (chap. 11) Zollinger, Manfred, 224n6 (chap. 4), 227n8 (conclusion), 229 Zweig, Stefan, 10, 229