Casanova’s Lottery [book review]

This “history of a revolutionary game of chance” is the latest book by Stephen Stigler and is indeed of an historical nature, following the French Lottery from its inception as Loterie royale in 1758 to the Loterie Nationale in 1836 (with the intermediate names of Loterie de France, Loterie Nationale, Loterie impériale, Loterie royale reflecting the agitated history of the turn of that Century!).

The incentive for following this State lottery is that it is exceptional by its mathematical foundations. Contrary to other lotteries of the time, it was indeed grounded on the averaging of losses and gains on the long run (for the State). The French (Royal) State thus accepted the possibility of huge losses at some draws since they would be compensated by even larger gains. The reasoning proved most correct since the Loterie went providing as far as 4% of the overall State budget, despite the running costs of maintaining a network of betting places and employees, who had to be mathematically savy in order to compute the exact gains of the winners.This is rather amazing as the understanding of the Law of Large Numbers was quite fresh (on an historical scale) thanks to the considerable advances made by Pascal, Fermat, (Jakob) Bernoulli and a few others. (The book mentions the Encyclopedist and mathematician Jean d’Alembert as being present at the meeting that decided of the creation of the Loterie in 1757.)

One may wonder why Casanova gets the credit for this lottery. In true agreement with Stigler’s Law, it is directly connected with the Genoan lottery and subsequent avatars in some Italian cities, including Casanova’s Venezia. But jack-of-all-trades Casanova was instrumental in selling the notion to the French State, having landed in Paris after a daring flight from the Serenissima’s jails. After succeeding in convincing the King’s officers to launch the scheme crafted by a certain Ranieri (de’) Calzabig—not to be confused with the much maligned Salieri!—who would later collaborate with Gluck on Orfeo ed Eurydice and Alceste, Casanova received a salary from the Loterie administration and further run several betting offices. Until he left Paris for further adventures! Including an attempt to reproduce the lottery in Berlin, where Frederick II proved less receptive than Louis XIV. (Possibly due to Euler’s cautionary advice.) The final sentence of the book stands by its title: “It was indeed Casanova’s lottery” (p.210).

Unsurprisingly, given Stephen’s fascination for Pierre-Simon Laplace, the great man plays a role in the history, first by writing in 1774 one of his earliest papers on a lottery problem, namely the distribution of the number of draws needed for all 90 numbers to appear. His (correct) solution is an alternating sum whose derivation proved a numerical challenge. Thirty years later, Laplace came up with a good and manageable approximation (see Appendix Two). Laplace also contributed to the end of the Loterie by arguing on moral grounds against this “voluntary” tax, along Talleyrand, a fellow in perpetually adapting to the changing political regimes. It is a bit of a surprise to read that this rather profitable venture ended up in 1836, more under bankers’ than moralists´ pressure. (A new national lottery—based on printed tickets rather than bets on results—was created a century later, in 1933 and survived the second World War, with the French Loto appearing in 1974 as a direct successor to Casanova’s lottery.)

The book covers many fascinating aspects, from the daily run of the Loterie, to the various measures (successfully) taken against fraud, to the survival during the Révolution and its extension through (the Napoleonic) Empire, to tests for fairness thanks to numerous data from almanacs, to the behaviour of bettors and the sale of “helping” books. to (Daniel) Bernoulli, Buffon, Condorcet, and Laplace modelling rewards and supporting decreasing marginal utility. Note that there are hardly any mathematical formula, except for an appendix on the probabilities of wins and the returns, as well as Laplace’s (and Legendre’s) derivations. Which makes the book eminently suited for a large audience, the more thanks to Stephen Stigler’s perfect style.

This (paperback) book is also very pleasantly designed by the University of Chicago Press, with a plesant font (Adobe Calson Pro) and a very nice cover involving Laplace undercover, taken from a painting owned by the author. The many reproductions of epoch documents are well-done and easily readable. And, needless to say given the scholarship of Stephen, the reference list is impressive.

The book is testament to the remarkable skills of Stephen who searched for material over thirty years, from Parisian specialised booksellers to French, English, and American archives. He manages to bring into the story a wealth of connections and characters, as for instance Voltaire’s scheme to take advantage of an earlier French State lottery aimed at reimbursing State debtors. (Voltaire actually made a fortune of several millions francs out of this poorly designed lottery.) For my personal instructions, the book also put life to several Métro stations like Pereire and Duverney. But the book‘s contents will prove fascinating way beyond Parisian locals and francophiles. Enjoy!

[Disclaimer about potential self-plagiarism: this post or an edited version will eventually appear in my Books Review section in CHANCE. As appropriate for a book about capitalising on chance beliefs!]

Casanova’s lottery is out!

down with Galton (and Pearson and Fisher…)

In the last issue of Significance, which I read in Warwick prior to the conference, there is a most interesting article on Galton’s eugenics, his heritage at University College London (UCL), and the overall trouble with honouring prominent figures of the past with memorials like named building or lectures… The starting point of this debate is a protest from some UCL students and faculty about UCL having a lecture room named after the late Francis Galton who was a professor there. Who further donated at his death most of his fortune to the university towards creating a professorship in eugenics. The protests are about Galton’s involvement in the eugenics movement of the late 18th and early 19th century. As well as professing racist opinions.

My first reaction after reading about these protests was why not?! Named places or lectures, as well as statues and other memorials, have a limited utility, especially when the named person is long dead and they certainly do not contribute in making a scientific theory [associated with the said individual] more appealing or more valid. And since “humans are [only] humans”, to quote Stephen Stigler speaking in this article, it is unrealistic to expect great scientists to be perfect, the more if one multiplies the codes for ethical or acceptable behaviours across ages and cultures. It is also more rational to use amphitheater MS.02 and lecture room AC.18 rather than associate them with one name chosen out of many alumni’s or former professors’.

Predictably, another reaction of mine was why bother?!, as removing Galton’s name from the items it is attached to is highly unlikely to change current views on eugenism or racism. On the opposite, it seems to detract from opposing the present versions of these ideologies. As some recent proposals linking genes and some form of academic success. Another of my (multiple) reactions was that as stated in the article these views of Galton’s reflected upon the views and prejudices of the time, when the notions of races and inequalities between races (as well as genders and social classes) were almost universally accepted, including in scientific publications like the proceedings of the Royal Society and Nature. When Karl Pearson launched the Annals of Eugenics in 1925 (after he started Biometrika) with the very purpose of establishing a scientific basis for eugenics. (An editorship that Ronald Fisher would later take over, along with his views on the differences between races, believing that “human groups differ profoundly in their innate capacity for intellectual and emotional development”.) Starting from these prejudiced views, Galton set up a scientific and statistical approach to support them, by accumulating data and possibly modifying some of these views. But without much empathy for the consequences, as shown in this terrible quote I found when looking for more material:

“I should feel but little compassion if I saw all the Damaras in the hand of a slave-owner, for they could hardly become more wretched than they are now…”

As it happens, my first exposure to Galton was in my first probability course at ENSAE when a terrific professor was peppering his lectures with historical anecdotes and used to mention Galton’s data-gathering trip to Namibia, literally measure local inhabitants towards his physiognomical views , also reflected in the above attempt of his to superpose photographs to achieve the “ideal” thief…

The Seven Pillars of Statistical Wisdom [book review]

I remember quite well attending the ASA Presidential address of Stephen Stigler at JSM 2014, Boston, on the seven pillars of statistical wisdom. In connection with T.E. Lawrence’s 1926 book. Itself in connection with Proverbs IX:1. Unfortunately wrongly translated as seven pillars rather than seven sages.

As pointed out in the Acknowledgements section, the book came prior to the address by several years. I found it immensely enjoyable, first for putting the field in a (historical and) coherent perspective through those seven pillars, second for exposing new facts and curios about the history of statistics, third because of a literary style one would wish to see more often in scholarly texts and of a most pleasant design (and the list of reasons could go on for quite a while, one being the several references to Jorge Luis Borges!). But the main reason is to highlight the unified nature of Statistics and the reasons why it does not constitute a subfield of either Mathematics or Computer Science. In these days where centrifugal forces threaten to split the field into seven or more disciplines, the message is welcome and urgent.

Here are Stephen’s pillars (some comments being already there in the post I wrote after the address):

1. aggregation, which leads to gain information by throwing away information, aka the sufficiency principle. One (of several) remarkable story in this section is the attempt by Francis Galton, never lacking in imagination, to visualise the average man or woman by superimposing the pictures of several people of a given group. In 1870!
2. information accumulating at the √n rate, aka precision of statistical estimates, aka CLT confidence [quoting  de Moivre at the core of this discovery]. Another nice story is Newton’s wardenship of the English Mint, with musing about [his] potential exploiting this concentration to cheat the Mint and remain undetected!
3. likelihood as the right calibration of the amount of information brought by a dataset [including Bayes’ essay as an answer to Hume and Laplace’s tests] and by Fisher in possible the most impressive single-handed advance in our field;
4. intercomparison [i.e. scaling procedures from variability within the data, sample variation], from Student’s [a.k.a., Gosset‘s] t-test, better understood and advertised by Fisher than by the author, and eventually leading to the bootstrap;
5. regression [linked with Darwin’s evolution of species, albeit paradoxically, as Darwin claimed to have faith in nothing but the irrelevant Rule of Three, a challenging consequence of this theory being an unobserved increase in trait variability across generations] exposed by Darwin’s cousin Galton [with a detailed and exhilarating entry on the quincunx!] as conditional expectation, hence as a true Bayesian tool, the Bayesian approach being more specifically addressed in (on?) this pillar;
6. design of experiments [re-enters Fisher, with his revolutionary vision of changing all factors in Latin square designs], with an fascinating insert on the 18th Century French Loterie,  which by 1811, i.e., during the Napoleonic wars, provided 4% of the national budget!;
7. residuals which again relate to Darwin, Laplace, but also Yule’s first multiple regression (in 1899), Fisher’s introduction of parametric models, and Pearson’s χ² test. Plus Nightingale’s diagrams that never cease to impress me.

The conclusion of the book revisits the seven pillars to ascertain the nature and potential need for an eight pillar.  It is somewhat pessimistic, at least my reading of it was, as it cannot (and presumably does not want to) produce any direction about this new pillar and hence about the capacity of the field of statistics to handle in-coming challenges and competition. With some amount of exaggeration (!) I do hope the analogy of the seven pillars that raises in me the image of the beautiful ruins of a Greek temple atop a Sicilian hill, in the setting sun, with little known about its original purpose, remains a mere analogy and does not extend to predict the future of the field! By its very nature, this wonderful book is about foundations of Statistics and therefore much more set in the past and on past advances than on the present, but those foundations need to move, grow, and be nurtured if the field is not to become a field of ruins, a methodology of the past!

random walk on a torus [riddle]

The Riddler of this week(-end) has a simple riddle to propose, namely given a random walk on the {1,2,…,N} torus with a ⅓ probability of death, what is the probability of death occurring at the starting point?

The question is close to William Feller’s famous Chapter III on random walks. With his equally famous reflection principle. Conditioning on the time n of death, which as we all know is definitely absorbing (!), the event of interest is a passage at zero, or any multiple of N (omitting the torus cancellation), at time n-1 (since death occurs the next time). For a passage in zero, this does not happen if n is even (since n-1 is odd) and else it is a Binomial event with probability

For a passage in kN, with k different from zero, kN+n must be odd and the probability is then

which leads to a global probability of

i.e.

Since this formula is rather unwieldy I looked for another approach in a métro ride [to downtown Paris to enjoy a drink with Stephen Stiegler]. An easier one is to allocate to each point on the torus a probability p[i] to die at position 1 and to solve the system of equations that is associated with it. For instance, when N=3, the system of equations is reduced to

which leads to a probability of ½ to die at position 0 when leaving from 0. When letting N grows to infinity, the torus structure no longer matters and the probability of dying at position 0 implies returning in position 0, which is a special case of the above combinatoric formula, namely

which happens to be equal to

as can be [unnecessarily] checked by a direct R simulation. This √5 is actually the most surprising part of the exercise!