Pierre Rémond de Montmort

b. 27 October 1678, Paris, France
d. 7 October 1719, Paris

Essay d'analyse sur les jeux de hazard

It has been suggested that the inspiration to write the Essay came from hints of the content of Jakob Bernoulli's Ars Conjectandi, accounts of which had been made known in two obituaries. Montmort himself gives his reasons at the beginning of his Preface:

"For a long time Geometers themselves brag to be able by their methods to discover in the natural Sciences, all the truths which are in the reach of the human mind; & it is certain that by the marvelous mixture that they have made since fifty years of Geometry with Physics, they have forced men to understand that this which they say to the advantage of Geometry is not without basis. What glory would there be for Science if it could yet serve to regulate judgments & to lead men into the practice of the things of life!

The eldest of the Messers Bernoulli both so known in the scholarly world, has not believed that it was impossible to carry Geometry to this point. He had undertaken to give the Rules in order to judge the probability of future events, & of which the knowledge is hidden to us, either in Games, or in the other things of life where chance alone has a part. The title of this Work must be De arte coniectandi, l'art de diviner. A premature death has not permitted him to set the last hand.

"Mister Fontenelle & Mister Saurin have each given a short Analysis of this Book; the first in the Histoire de l'Academie (1705, p. 148); the other in the Journal des Sçavans of France (1706, p. 81). Here is, according to these two Authors, what was the plan of this Work. Mr. Bernoulli divided it into four Parts; in the first three he gave the solution of diverse Problems on the Games of chance: one must find there many new things on infinite series, on combinations & changes of order, with the solution of the five Problems proposed a long time ago to Geometers by Mr. Hugens. In the fourth Part, he employed the methods that he had given in the first three, to resolve diverse moral, political & civil questions.

"No one has apprised us what are the Games of which this Author determined the parts, nor what subjects of politics & morals he had undertaken to clarify; but as surprising that this project be, there is place to believe that this scholarly Author had perfectly executed it. Mr. Bernoulli was too superior to the others in order to wish to impose on them, he was of that small number of rare men who are fit to invent, & I am persuaded that he had held all that which the title of his Book promised.

"Nothing delays more the advancement of the Sciences, & puts a greater obstacle to the discovery of hidden truths, than the mistrust in which we have of our forces. The greater part of the things which may appear impossible are only for lack of giving to the human intellect all the extent that it can have.

"Many of my Friends had prompted me, already a long time ago, to test if Algebra could not reach to determine what is the advantage of the Banker in the Game of Pharaon. I had never dared to undertake this research, because I knew that the number of all the diverse possible arrangements of fifty-two cards, surpasses more than one hundred thousand millions of times the one of the grains of sand that the globe of the earth could contain; and it did not appear possible to me to untangle, in a number so vast, the arrangements which are advantageous to the Banker, from those which are contrary or indifferent to him. I would be still in this prejudice if the success of the late Mr. Bernoulli had not invited me there some years ago to seek the different chances of this Game. I was happier than I had dared hoped, because beyond the general solution of this Problem, I perceived the routes that it was necessary to take in order to discover an infinity of similar, or even much more difficult of them. I knew that one could go quite far in this country where no person had yet been; I flattered myself that one could make an ample harvest of truths equally curious & novel; this gave to me the thought to work at the foundation on this matter, & to desire to compensate in some sort the Public from the loss that it would get if it were deprived of the excellent Work of Mr. Bernoulli. Diverse reflections have confirmed me in this decision." (pages iii-vi of the Preface.) 

Description of the Contents of the 2nd Edition

The work (2nd edition) consists of five sections:

1. A Treatise on Combinations: Sections 1 - 60 (pages 1-72).
2. Problems on Games of Chance: Sections 61 - 136 (pages 73 - 172). These include analyses of the card games Pharaon, Lansquenet, Treize, Bassette and more limited analyses of games not of pure chance Piquet, Triomphe, L'Ombre, Brelan, Imperial and Quinze. 
3. This is called "Problem on Quinquenove:"  Sections 137 - 168 (pages 173 - 215). This part examines games of dice: Quinquenove, Hazard, Esperance, TricTrac, Trois Dez, Rafle, Trois Rafles, and Noyaux.  
4. Divers Problems: Sections 169 - 217 (pages 216 - 277). Here Montmort solves, among others, the problems of Huygens, the problem of points,  and duration of play. The part terminates with four problems to be solved.
5. Correspondence (pages 282 - 414). This section is the most interesting because it makes public discussions between Montmort and two Bernoullis. In particular, the Petersburg and Waldgrave problems are introduced.

Relationship to the Doctrine of Chances

In 1711, Abraham Moivre presented his paper, De Mensura Sortis, to the Royal Academy in London. When Montmort received a copy from him and had an opportunity to examine it, he felt that his own work had been slighted by Moivre. We have here correspondence with Nicholas Bernoulli critiquing the De Mensura Sortis taken from the 1913 edition.

The Avertissement of the 1913 edition contains a discussion of the controversy with Moivre and a history of the literature until 1713 which is well worth reading.

Analysis of Games and Exercises

The Essay contains the analysis of several card games. Because other mathematicians also analyzed these same, translations of the treatment by Montmort are included here. In particular there are analyses of the games known as Bassette, Pharaon and Treize. The first analysis of Bassette was by Sauveur in 1679. See in the Journal of Sçavans for that year Supputation des avantages du Banquier dans le jeu de la Bassette. Other analyses were by Moivre and Jakob Bernoulli.(See Book III Chapter x of the Ars Conjectandi.) Montmort's table for Pharaon is incorrect and has been amended here. Pharaon and Treize were analyzed by Moivre in his Doctrine of Chances as well and it is interesting to compare treatments. Furthermore, Daniel Bernoulli in his youth analyzed the game. Euler himself wrote papers concerning questions derived from these latter two games.

Similar to Huygens, Montmort posed four exercises. Note the comments by Nicolas Bernoulli on these. The problems are:

1. On the game of Treize. To determine generally what is the advantage in this game of the one who holds the cards.
2. On the game called Her. Three Players, Pierre, Paul & Jacques are the only Players who remain, & they each have no more than one token. Pierre holds the cards, Paul is to his right, & Jacques next. One demands what is their lot with respect to the different place that they occupy, & with what proportion should the money of the pool be divided, this will be, for example, ten pistoles, if they would wish to divide among them without finishing the game. This game is particularly interesting because it calls for a mixed strategy for optimal play. Related to this are comments made by Montmort on a game of Even or Odd. See also Trembley 's memoir "Observations sur le calcul d'un Jeu de hasard."
3. On the game of the Ferme (Bank). Being supposed a certain determined number of Players, for example, two Players Pierre & Paul, & that the price of the tokens is twenty sols: One demands how much must be the price of the Ferme, so that one is able to hold it with neither profit nor disadvantage.
4. On the game of Tas (Heap).  To determine what is the advantage or the disadvantage of the one who undertakes to make the heap, if it is with a game of Piquet, or with a game of Ombre, or with an entire game, & if it the way to conduct his game is the most advantageous that it is possible. 

Special Problems

In addition, there are interesting treatments of the essai_prop_XLI_187_194.pdf

• Problem of Points, 
• Duration of Play, 
• In the Essai, Montmort discusses the Lottery of Lorraine, reprinting what was published in the Journal des Sçavans in 1711, but including correspondence,  
• Game of Bowls and associated tables (formatted for legal size paper),
• Correspondence with Nicolas Bernoulli on the St. Petersburg Problem,
• Correspondence with Nicolas Bernoulli on Waldegrave's Problem, (Pool)
• Correspondence with Nicolas Bernoulli on the paper by Arbuthnot on Divine Providence

All extracts are taken from the 1713 edition.

Correspondence Appended to the Second Edition

The following table serves to indicate the questions discussed by Montmort and Nicolas Bernoulli over the course of two years.  See the correspondence with Jean Bernoulli and the correspondence with Nicolas Bernoulli.   Note, in particular, the persistence of the exchange on Her.

Correspondents	Date	Pages	Treize	Her	Pool	Lotttery	Tennis	Moivre	Births	Petersburg
Jean I Bernoulli to Montmort with remarks by Nicolaus Bernoulli	17 March 1710	283-303	X
Montmort to Jean I Bernoulli	15 November 1710	303-307
Nicolas Bernoulli to Montmort	26 February 1711	308-314	X			X
Montmort to Nicolas Bernoulli	10 April 1711	315-323	X	X	X	X
Nicolas Bernoulli to Montmort	10 November 1711	323-337	X	X	X	X	X
Montmort to Nicolas Bernoulli	1 March 1712	337-347		X	X	X	X
Nicolas Bernoulli to Montmort	2 June 1712	348-352		X	X		X
Montmort to Nicolas Bernoulli	8 June 1712	352-360		X			X	X
Montmort to Nicolas Bernoulli	5 September 1712	361-370		X	X			X
Nicolas Bernoulli to Montmort	11 October 1712	371-375					X	X	X
Nicolas Bernoulli to Montmort	30 December 1712	375-387		X	X			X
Nicolas Bernoulli	23 January 1713	388-393							X
Montmort to Nicolas Bernoulli	20 August 1713	395-400		X				X
Nicolas Bernoulli to Montmort	9 September 1713	401-402		X						X
Montmort to Nicolas Bernoulli	15 November 1713	403-414		X						X