## Casanova’s Lottery [book review]

Posted in Books, Statistics, University life with tags , , , , , , , , , , , , , , , , , , , , , , , , , , , , , on January 12, 2023 by xi'an

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!]

## genuine Latinsquare rectangle

Posted in Books, pictures, Statistics, Travel, University life with tags , , , , , , , , , , , , on June 9, 2021 by xi'an

## Simulating a coin with irrational bias using rational arithmetic

Posted in Books, Statistics with tags , , , , , , on December 17, 2020 by xi'an

An arXived paper by Luis Mendo adresses the issue of simulating coins with irrational probabilities from a fair coin, somehow connected with one of the latest riddles. Where I realised only irrational coins could be simulated in a fixed and finite number of throws! The setting of the paper is however similar to the one of a Bernoulli factory in that an unlimited number of coins can be generated (but it relies on a  fair coin). And the starting point is a series representation of the irrational ζ as a sum of positive and rational terms. As well as an earlier paper by the Warwick team of Łatuszyński et al. (2011). The solution is somewhat anticlimactic in that the successive draws of the fair coin lead to a sequence of intervals with length divided by 2 at each step. And stopping when a certain condition is met, requiring some knowledge on the tail error of the series. The paper shows further that the number of inputs used by its algorithm has an exponential tail. The examples provided therein are Euler’s constant

$\gamma =\frac{1}{2} + \sum_{i=1}^\infty \frac{B(i)}{2i(2i+1)(2i+2)}$

where B(j) is the number of binary digits of j, and π/4 which can be written as an alternating series. An idle question that came to me while reading this paper is the influence of the series chosen to represent the irrational ζ as it seems that a faster decrease in the series should lead to fewer terms being used. However, the number of iterations is a geometric random variable with parameter 1/2, therefore the choice of the series curiously does not matter.

## Riddler collector

Posted in Statistics with tags , , , , , , , on September 22, 2018 by xi'an

Once in a while a fairly standard problem makes it to the Riddler puzzle of the week. Today, it is the coupon collector problem, explained by W. Huber on X validated. (W. Huber happens to be the top contributor to this forum, with over 2000 answers, and the highest reputation closing on 200,000!) With nothing (apparently) unusual: coupons [e.g., collecting cards] come in packs of k=10 with no duplicate, and there are n=100 different coupons. What is the expected number one has to collect before getting all of the n coupons?  W. Huber provides an R code to solve the recurrence on the expectation, obtained by conditioning on the number m of different coupons already collected, e(m,n,k) and hence on the remaining number of collect, with an Hypergeometric distribution for the number of new coupons in the next pack. Returning 25.23 packs on average. As is well-known, the average number of packs to complete one’s collection with the final missing card is expensively large, with more than 5 packs necessary on average. The probability distribution of the required number of packs has actually been computed by Laplace in 1774 (and then again by Euler in 1785).

## a funny mistake

Posted in Statistics with tags , , , , , , , , , , , on August 20, 2018 by xi'an

While watching the early morning activity in Tofino inlet from my rental desk, I was looking at a recent fivethirthyeight Riddle, which consisted in finding the probability of stopping a coin game which rule was to wait for the n consecutive heads if (n-1) consecutive heads had failed to happen when requested, which is

p+(1-p)p²+(1-p)(1-p²)p³+…

or

$q=\sum_{k=1}^\infty p^k \prod_{j=1}^{k-1}(1-p^j)$

While the above can write as

$q=\sum_{k=1}^\infty \{1-(1-p^k)\} \prod_{j=1}^{k-1}(1-p^j)$

or

$\sum_{k=1}^\infty \prod_{j=1}^{k-1}(1-p^j)-\prod_{j=1}^{k}(1-p^j)$

hence suggesting

$q=\sum_{k=1}^\infty \prod_{j=1}^{k-1}(1-p^j) - \sum_{k=2}^\infty \prod_{j=1}^{k-1}(1-p^j) =1$

the answer is (obviously) false and the mistake in separating the series into a difference of series is that both terms are infinite. The correct answer is actually

$q=1-\prod_{j=1}^{\infty}(1-p^j)$

which is Euler’s function. Maybe nonstandard analysis can apply to go directly from the difference of the infinite series to the answer!