Stephen Stigler awarded Neumann Prize

The British Society for the History of Mathematics announced that Stephen Stigler, Ernest DeWitt Burton Distinguished Service Professor Emeritus of Statistics, has won the 2023 Neumann Prize for his book Casanova’s Lottery: The History of a Revolutionary Game of Chance (University of Chicago Press, 2022). The prize is awarded for a book in English (including books in translation) dealing with the history of mathematics, aimed at a non-specialist readership. Read more about the prize at https://physicalsciences.uchicago.edu/news/article/stephen-stigler-awarded-neumann-prize/.

We asked Steve Stigler to write a bit about the book, which was the culmination of a 25-year-long project. He said:

My book Casanova’s Lottery tells the story of a lottery that flourished in France from 1758 to 1836. It was an early version of modern lotto, operated as a state lottery. In its first year it offered one drawing a month; by 1801 there were 15 drawings a month, three in each of five cities, providing up to 4% of France’s state income, an amazingly large experiment in applied probability.

Two popular theorems in probability played significant roles in the story. One was the Law of Large Numbers (LLN): Unlike modern lotto, the state faced the possibility of ruin on each drawing, protected (like some insurance companies) only by the LLN. The protection was substantial and the risk was calculable, but the finance ministers were risk adverse. One of the book’s conclusions is that states are more willing to undertake poorly understood risk than a risk subject to exact calculation.

The other theorem was the Law of the Maturity of Chances (LMC). It states that if a particular outcome of a fair game has not occurred for a number N of trials, its chance of occurring increases as N increases: Its chance grows with “maturity”. No probabilist of my acquaintance believes this is true; most gamblers take it on faith. With an eager market of gamblers looking for reassurance, probabilists have presented theorems that speak to the LMC obliquely. One example presented in an appendix is one of Pierre Simon Laplace’s earliest theorems. His formal theorem was correct, but it was nearly 40 years before he could give numerical answers for the French lottery. Others, including Leonard Euler, considered the question, and some helpfully designed martingale betting systems that permitted gamblers to remain solvent longer.

A major part of the book reports on the results of a rigorously randomized survey of bettors that was carried out inadvertently more than two centuries ago. If the drawing of numbers was done fairly (and this is demonstrated), the winners of large prizes are a random sample of those betting on long shots; we then have information on those winners and can learn about the general population of bettors from the sample. Other studies look at the choices of numbers the bettors made and the advice they received from many authors of books advising what choices should be made to best suit the individual gambler (“personalized betting advice”).

The lottery was a boon to mathematical education, particularly in combinatorial probability. Every textbook considered the calculation of odds for the lottery, usually while pointing out what a poor gamble it was, often with an attempt to emphasize the remoteness of a chance for a big win (“less than the risk of a man of age 50 of dying of apoplexy in an hour”). The lottery was also a boon to moral philosophers, who railed at the evils of gambling of any sort. Of course, then as now, hypocrisy reigned supreme: on the few occasions that the lottery was briefly banned (ostensibly on moral grounds), there was always an exception for lotteries run by the Church.

Why would any rational person gamble in an unfair game? That question was raised even earlier and most mathematicians answered that it was not rational, invoking Daniel Bernoulli’s 1738 argument for decreasing marginal utility. Still, widespread gambling persisted, and the question remained. The best answer, to my eye, was provided only in 1948 by two members of the IMS, Jimmie Savage and Milton Friedman.

All of these issues are addressed in the book, as well as some odd excursions, such as a little-known adventure by the young philosopher Voltaire and his friend La Condamine in the late 1720s, where they won millions in an ill-conceived state lottery with no risk of loss, and the “big data” problems the French lottery encountered in data security and fraud detection as it grew. The book is based upon two decades of research in archives, scrapbooks, and many libraries, and in addition to those already mentioned the cast of characters includes Casanova (yes, that Casanova), Lagrange, Napoleon, Talleyrand, John Law, and Madame de Pompadour.