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!]

an hypothetical chain of transmissions

El asiedo [book review]

Just finished this long book by Arturo Pérez-Reverte that I bought [in its French translation] after reading the fascinating Dos de Mayo about the rebellion of the people of Madrid against the Napoleonian occupants. This book, The Siege, is just fantastic, more literary than Dos de Mayo and a mix of different genres, from the military to the historical, to the criminal, to the chess, to the speculative, to the romantic novel..! There are a few major characters, a police investigator, a trading company head, a corsair, a French canon engineer, a guerilla, with a well-defined unique location, the city of Cádiz under [land] siege by the French troops, but with access to the sea thanks to the British Navy. The serial killer part is certainly not the best item in the plot [as often with serial killer stories!], as it slowly drifts to the supernatural, borrowing from Laplace and Condorcet to lead to perfect predictions of where and when French bombs will fall. The historical part also appears to be rather biased against the British forces, if this opinion page is to be believed, towards a nationalist narrative making the Spanish guerilla resistance bigger and stronger than it actually was. But I still read the story with fascination and it kept me awake past my usual bedtime for several nights as I could not let the story go!

Who’s #1?

First, apologies for this teaser of a title! This post is not about who is #1 in whatever category you can think of, from statisticians to climbs [the Eiger Nordwand, to be sure!], to runners (Gebrselassie?), to books… (My daughter simply said “c’est moi!” when she saw the cover of this book on my desk.) So this is in fact a book review of…a book with this catching title I received a month or so ago!

“We decided to forgo purely statistical methodology, which is probably a disappointment to the hardcore statisticians.” A.N. Langville & C.D. Meyer, Who’s #1? The Science of Rating and Ranking (page 225)

This book may be one of the most boring ones I have had to review so far! The reason for this disgruntled introduction to “Who’s #1? The Science of Rating and Ranking” by Langville and Meyer is that it has very little if any to do with statistics and modelling. (And also that it is mostly about American football, a sport I am not even remotely interested in.) The purpose of the book is to present ways of building rating and ranking within a population, based on pairwise numerical connections between some members of this population. The methods abound, at least eight are covered by the book, but they all suffer from the same drawback that they are connected to no grand truth, to no parameter from an underlying probabilistic model, to no loss function that would measure the impact of a “wrong” rating. (The closer it comes to this is when discussing spread betting in Chapter 9.) It is thus a collection of transformation rules, from matrices to ratings. I find this the more disappointing in that there exists a branch of statistics called ranking and selection that specializes in this kind of problems and that statistics in sports is a quite active branch of our profession, witness the numerous books by Jim Albert. (Not to mention Efron’s analysis of baseball data in the 70’s.)

“First suppose that in some absolutely perfect universe there is a perfect rating vector.” A.N. Langville & C.D. Meyer, Who’s #1? The Science of Rating and Ranking (page 117)

The style of the book is disconcerting at first, and then some, as it sounds written partly from Internet excerpts (at least for most of the pictures) and partly from local student dissertations… The mathematical level is highly varying, in that the authors take the pain to define what a matrix is (page 33), only to jump to Perron-Frobenius theorem a few pages later (page 36). It also mentions Laplace’s succession rule (only justified as a shrinkage towards the center, i.e. away from 0 and 1), the Sinkhorn-Knopp theorem, the traveling salesman problem, Arrow and Condorcet, relaxation and evolutionary optimization, and even Kendall’s and Spearman’s rank tests (Chapter 16), even though no statistical model is involved. (Nothing as terrible as the completely inappropriate use of Spearman’s rho coefficient in one of Belfiglio’s studies…)

“Since it is hard to say which ranking is better, our point here is simply that different methods can produce vastly different rankings.” A.N. Langville & C.D. Meyer, Who’s #1? The Science of Rating and Ranking (page 78)

I also find irritating the association of “science” with “rating”, because the techniques presented in this book are simply tricks to turn pairwise comparison into a general ordering of a population, nothing to do with uncovering ruling principles explaining the difference between the individuals. Since there is no validation for one ordering against another, we can see no rationality in proposing any of those, except to set a convention. The fascination of the authors for the Markov chain approach to the ranking problem is difficult to fathom as the underlying structure is not dynamical (there is not evolving ranking along games in this book) and the Markov transition matrix is just constructed to derive a stationary distribution, inducing a particular “Markov” ranking.

“The Elo rating system is the epitome of simple elegance.” A.N. Langville & C.D. Meyer, Who’s #1? The Science of Rating and Ranking (page 64)

An interesting input of the book is its description of the Elo ranking system used in chess, of which I did not know anything apart from its existence. Once again, there is a high degree of arbitrariness in the construction of the ranking, whose sole goal is to provide a convention upon which most people agree. A convention, mind, not a representation of truth! (This chapter contains a section on the Social Network movie, where a character writes a logistic transform on a window, missing the exponent. This should remind Andrew of someone he often refer to in his blog!)

“Perhaps the largest lesson is not to put an undue amount of faith in anyone’s rating.” A.N. Langville & C.D. Meyer, Who’s #1? The Science of Rating and Ranking (page 125)

In conclusion, I see little point in suggesting reading this book, unless one is interested in matrix optimization problems and/or illustrations in American football… Or unless one wishes to write a statistics book on the topic!