Le Her and Other Problems in Probability Discussed by Bernoulli, Montmort and Waldegrave
aa r X i v : . [ s t a t . O T ] A p r Statistical Science (cid:13)
Institute of Mathematical Statistics, 2015
Le Her and Other Problems inProbability Discussed by Bernoulli,Montmort and Waldegrave
David R. Bellhouse and Nicolas Fillion
Abstract.
Part V of the second edition of Pierre R´emond de Mont-mort’s
Essay d’analyse sur les jeux de hazard published in 1713 con-tains correspondence on probability problems between Montmort andNicolaus Bernoulli. This correspondence begins in 1710. The last pub-lished letter, dated November 15, 1713, is from Montmort to NicolausBernoulli. There is some discussion of the strategy of play in the cardgame Le Her and a bit of news that Montmort’s friend Waldegrave inParis was going to take care of the printing of the book. From ear-lier correspondence between Bernoulli and Montmort, it is apparentthat Waldegrave had also analyzed Le Her and had come up with amixed strategy as a solution. He had also suggested working on the“problem of the pool,” or what is often called Waldegrave’s problem.The Universit¨atsbibliothek Basel contains an additional forty-two let-ters between Bernoulli and Montmort written after 1713, as well as twoletters between Bernoulli and Waldegrave. The letters are all in French,and here we provide translations of key passages. The trio continued todiscuss probability problems, particularly Le Her which was still underdiscussion when the
Essay d’analyse went to print. We describe theprobability content of this body of correspondence and put it in its his-torical context. We also provide a proper identification of Waldegravebased on manuscripts in the Archives nationales de France in Paris.
Key words and phrases:
History of probability, history of game theory,strategy of play.
David R. Bellhouse is Professor, Department ofStatistical and Actuarial Sciences, University ofWestern Ontario, London, Ontario N6A 5B7, Canadae-mail: [email protected]. Nicolas Fillion isAssistant Professor, Department of Philosophy, SimonFraser University, Burnaby, British Columbia V5A 1S6,Canada e-mail: nfi[email protected].
This is an electronic reprint of the original articlepublished by the Institute of Mathematical Statistics in
Statistical Science , 2015, Vol. 30, No. 1, 26–39. Thisreprint differs from the original in pagination andtypographic detail.
1. INTRODUCTION
The earliest extant correspondence between PierreR´emond de Montmort and a member of the Bernoullifamily is a letter from Montmort to Johann Bernoullidated February 27, 1703, concerning a paper on cal-culus that the latter had written for the Acad´emieroyale des sciences in Paris (Bernoulli, 1702). Theycorresponded sporadically over the next few years.On April 29, 1709, Montmort sent Bernoulli a copyof his book on probability,
Essay d’analyse surles jeux de hazard , that he recently had published(Montmort, 1708). The book is the first in a seriesof books in probability published by several othersover the years 1708 to 1718 in what Hald [(1990),page 191] calls the “Great Leap Forward” in prob- D. R. BELLHOUSE AND N. FILLION ability. Bernoulli replied with a gift of a copy ofhis nephew’s doctoral dissertation (Bernoulli, 1709),the second book in Hald’s “Great Leap Forward”;Nicolaus Bernoulli’s book dealt with applicationsof probability. Once Johann Bernoulli received hiscopy of
Essay d’analyse , he sent, on March 17, 1710,a detailed set of comments on the book. In the let-ter Bernoulli included another set of comments on
Essay d’analyse , this one by his nephew Nicolaus(Montmort (1713), pages 283–303). Thus began a se-ries of correspondence between Montmort and Nico-laus Bernoulli on problems in probability. Montmortincluded much of this correspondence in Part V ofthe second edition of
Essay d’analyse (Montmort,1713). The correspondence between Montmort andNicolaus Bernoulli after 1713, left unpublished andlargely ignored by historians, contains scientificnews and further discussion of problems in prob-ability. The major topic is a continuing discussionof issues related to the card game Le Her. Next, interms of ink spilt on probability, are discussions ofthe “problem of the pool,” or Waldegrave’s prob-lem, generalized to more than three players, and ofthe game Les ´Etrennes (which may be translated as“the gifts”). The correspondence also contains dis-cussions of various problems in algebra, geometry,differential equations and infinite series.As an aristocrat, Montmort’s network includedboth political and scientific connections. His lettersto Bernoulli contain some references to his politi-cal activities that sometimes kept him from replyingpromptly. His brother, Nicolas R´emond, was Chefde conseil for Phillipe duc d’Orl´eans, who becameregent of France after his uncle Louis XIV died in1715 (Leibniz (1887), page 599). Among the math-ematicians of the era, Montmort corresponded withIsaac Newton, Gottfried Leibniz, Brook Taylor andAbraham De Moivre, in addition to the Bernoullis aswell as several others. As a talented amateur math-ematician, his work was well regarded by the math-ematicians of his day. He was generous to his scien-tific friends. He received as guests to the Chˆateau deMontmort Nicolaus Bernoulli, Brook Taylor and oneof the sons of Johann Bernoulli. He also sent gifts ofcases of wine and champagne to both Newton andTaylor.Le Her is a game of strategy and chance playedwith a standard deck of fifty-two playing cards.The simplest situation is when two players play thegame, and the solution is not simply determined even in that situation. Montmort calls the two play-ers Pierre and Paul. Pierre deals a card from thedeck to Paul and then one to himself. Paul has theoption of switching his card for Pierre’s card. Pierrecan only refuse the switch if he holds a king (thehighest valued card). After Paul makes his decisionto hold or switch, Pierre now has the option to holdwhatever card he now has or to switch it with a carddrawn from the deck. However, if he draws a king,he must retain his original card. The player withthe highest card wins the pot, with ties going to thedealer Pierre. The game can be expanded to morethan two players. Montmort [(1708), pages 186–187]originally described the problem for four players andposed the question: What are the chances of eachplayer relative to the order in which they make theirplay?Because of the winning conditions, it is obviousthat one would want to switch low cards and keephigh ones. The key is to find what to do with themiddle cards, such as seven and eight, when twoplayers are playing the game. In other cases, cardsare clearly too low to keep or too high to switch,being much below or above the average in a randomdraw. Naturally, the threshold would be lower withmore than two players.In Part V of
Essay d’analyse , only the game withtwo players is considered. Initially, Montmort andNicolaus Bernoulli wrote back and forth about theproblem and came to the same solution. However,two of Montmort’s friends contended that this so-lution was incorrect. These were an English gentle-man named Waldegrave and an abbot whose abbeywas only a league and a half (about 5.8 kilometers)from Chˆateau de Montmort (Montmort (1713), page338). Montmort identified Waldegrave only as thebrother of the Lord Waldegrave who married thenatural daughter of King James II of England. LordWaldegrave is Henry Waldegrave, 1st Baron Walde-grave, and his wife is Henrietta FitzJames, daughterof James II, and his mistress Arabella Churchill. Theabbot is the Abb´e d’Orbais; Montmort also refers tohim as the Abb´e de Monsoury. The reason for thetwo appellations for the abbot is that his full nameis Pierre Cuvier de Montsoury, Abb´e d’Orbais. Hehas been described as “un prodige de bon coeur,d’urbanit´e et de science” (Bout (1887)). For thespelling choice between Montsoury and Monsoury,it should be noted that Montmort often spelled hisname “Monmort.”
ROBLEMS IN PROBABILITY BY BERNOULLI, MONTMORT & WALDEGRAVE Two other problems were discussed extensively inthe correspondence. The first is the problem of thepool, a problem that Waldegrave suggested to Mont-mort and solved himself (Montmort (1713), page318). In
Essay d’analyse the problem is solved forthree players. It is often called Waldegrave’s prob-lem (Bellhouse, 2007). The “pool” is a way of gettingthree or more players to gamble against one another,when the game put into play is for two players only.In the situation for three players (Montmort uses thenames Pierre, Paul and Jacques), all three begin byputting an ante into the pot. Then Pierre and Paulplay a game against each other. The winner playsagainst Jacques and the loser puts money into thepot. The game continues until one player has beatenthe other two in a row. That player takes the pot.The game can be expanded to more than three play-ers, but that situation was not fully treated in
Essayd’analyse . The second is the problem of solving thegame Les ´Etrennes (or “estreine,” an alternative oldFrench spelling). As described by Montmort [(1713),pages 406–407], this is a strategic game between afather and his son. The father holds an odd or evennumber of tokens in his hand, which his son can-not see. When the son guesses even, he receives agift of four ´ecus (silver coins) if he is correct andnothing if wrong. When the son guesses odd, he re-ceives one ´ecu if he is correct and nothing if wrong.The discussion of this game in the correspondenceis only brought in to enlighten Le Her whose strate-gic nature is in some important respects essentiallysimilar.Montmort concludes the last letter (to Bernoulli)that appears in
Essay d’analyse with a remark thatWaldegrave had volunteered to take care of gettingthe book printed in Paris. Montmort’s letter wasdated November 15, 1713, and was written fromParis. What is also of concern to us are the lettersafter this date and how these letters relate to earlierdiscussions. The unpublished correspondence beginswith a letter from Montmort to Bernoulli dated Jan-uary 25, 1714, in which he says that he has sentBernoulli two copies of the second edition of
Es-say d’analyse . Montmort was still in Paris, wherehe claimed to have been for three months. He wasstaying at a hotel in Rue des Bernardins, which inmodern Paris is only a walk of 350 meters to theprinter, Jacques Quillau in Rue Galande. Presum-ably, Waldegrave’s help consisted mainly in dealingwith the printer and the proof sheets as they cameoff the press, thus relieving Montmort of some te-dious work.
2. THE TREATMENT OF THE GAME OF LEHER IN ESSAY D’ANALYSE
To understand the discussion of Le Her after 1713,it is necessary to describe the treatment of thegame as it appears in the second edition of
Es-say d’analyse . Hald [(1990), pages 314–322] providesa detailed description of the mathematical calcula-tions involved in assessing the game. Yet he devoteslittle space to elucidating the discussions amongBernoulli, Montmort and Waldegrave, as well as theAbb´e d’Orbais, concerning the issues surroundingthese mathematical calculations. It is the substanceof these discussions that are of interest to us.Hald’s only comment on the discussion over LeHer concerns a comment made by Waldegrave andAbb´e d’Orbais to the effect that Bernoulli’s reason-ing in obtaining his mathematical solution is faulty.After pointing out their observation that Bernoulli’ssolution fails to account for a player’s probability ofplaying in a certain way, Hald [(1990), page 315]claims:It is no wonder that Bernoulli does notunderstand the implications of this re-mark, since the writers themselves havenot grasped the full implication of theirpoint of view.It is indeed true that there was some confusion onBernoulli’s side which he deftly tried to hide.Henny [(1975), page 502] comments that he isamazed to find expressed in the letters many con-cepts and ideas that appear in modern game theory.At the same time, he is surprised to find Waldegravedefending his position so strongly against Bernoulliwho was the superior mathematician. Henny statesfurther that Waldegrave did not have the necessarymathematical skills to provide a mathematical proofof his results.As we will show in a review of the treatment ofLe Her in
Essay d’analyse and the subsequent un-published correspondence, both Hald’s and Henny’sinsights fall short of the mark. One reason they fall The same could be said of others who have passed harshjudgments on Montmort and Bernoulli. For instance, Fisher(1934) argues that “Montmort’s conclusion [that no absoluterule could be given], though obviously correct for the limitedaspect in which he viewed the problem, is unsatisfactory tocommon sense, which suggests that in all circumstances theremust be, according to the degree of our knowledge, at least onerule of conduct which shall be not less satisfactory than anyother; and this his discussion fails to provide.” Our discussionbelow will show that this assessment is misinformed.
D. R. BELLHOUSE AND N. FILLION short is that they do not consider the full range ofthe various events that were under discussion andtheir associated probabilities. Two events are natu-ral for a probabilist to consider. The first is the dis-tribution of the cards to Pierre and Paul. The secondis the randomizing device used to come up with themixed strategy prescribing when the players shouldhold and when they should switch. The randomizingdevice considered in
Essay d’analyse is a bag con-taining black and white counters or tokens (the oldFrench word used is “jetton”). The third event thatMontmort, Waldegrave and Bernoulli consider (butnot Hald or Henny) is difficult, or perhaps impos-sible, to quantify. This is the possibility that Paul,say, is a poor player and does not follow a strategythat is mathematically optimal, or the possibilitythat Paul, say, is a very good player who tries totrick Pierre into making a poor choice. This kind ofevent unfolds regularly in modern poker games.Another reason for which Hald and Henny seesome confusion in the discussions among Bernoulli,Montmort and Waldegrave is that what we are see-ing in the correspondence is the complete unfoldingof a problem from its initial statement, and discus-sions around it, to a complete solution. This is dif-ferent from a “textbook” statement of a problem fol-lowed by a solution. In the latter case, the problemand solution are both well laid out. In the formercase, there is some grappling with the problem untilit becomes clear how to proceed.We begin with the correspondence in
Essayd’analyse where Le Her is first mentioned. In Jo-hann Bernoulli’s 1710 letter to Montmort in
Es-say d’analyse , he suggests more efficient methodsto reach Montmort’s conclusions for a variety ofproblems and in some cases generalizes Montmort’sresults. There is only one reference to the problemof Le Her, which is the second of four problems pro-posed in Montmort [(1708), pages 185–187]:The second and the third [problem] seemto me amenable, but not without muchdifficulty and work, that I prefer to de-fer to you and learn the solution, thanto work long at the expense of my ordi-nary occupations that leave me scarcelyany time to apply myself to other things.In his reply to this letter, which is dated November15, 1710 (Montmort (1713), pages 303–307), Mont-mort makes no reference to this passage. Nicolaus Bernoulli’s first letter to Montmort,dated February 26, 1711, makes no reference to thegame Le Her. It is a note in Montmort’s reply toNicolaus Bernoulli, dated April 10, 1711 (Montmort(1713), pages 315–323) that initiates the discussionof Montmort’s second problem:I started some time ago to work on thesolution of problems that I propose atthe end of my book; I find that in LeHer, when there are only two playersleft, Pierre and Paul, Paul’s advantage isgreater than 1 in 85, and less than 1 in 84.This problem has difficulties of a singularnature.In a postscript to this letter, Montmort makes anadditional remark:As there are few copies of my book left,there will soon be a new edition. When Ihave decided, I will ask you permission,and your uncle, to insert your beautifulletters which will make the principal em-bellishment.It is this announcement that may have motivatedNicolaus Bernoulli to continue his correspondencewith Montmort and to send him much interestingmaterial. Publishing mathematical material outsidea scientific society or without a patron to cover thecosts was an expensive proposition, one that Mont-mort could afford. Because of the specialized typethat was used and the accompanying necessary skillof the typesetter, the cost of a mathematical pub-lication was well above the norm for less technicalbooks. Bernoulli could get his results in print at nocost to himself.Bernoulli responded with a long letter, datedNovember 10, 1711 (Montmort (1713), pages 323–337). In this letter, he announces, among many otherthings, that he has also solved the two-person casefor Le Her (Montmort (1713), page 334):I also solved the problem on Le Her inthe simplest case; here is what I found.If we suppose that each player observesthe conduct that is most advantageous tohim, Paul must only hold to a card that ishigher than a seven and Pierre to one thatis higher than an eight, and we find underthis supposition that the lot of Pierre willbe to that of Paul as 2697 is to 2828. Sup-posing that Paul also holds to a seven,
ROBLEMS IN PROBABILITY BY BERNOULLI, MONTMORT & WALDEGRAVE then Pierre must hold to an eight, andtheir lots will still be as 2697 to 2828. Nev-ertheless it is more advantageous for himnot to hold to a seven than to hold to it,which is a puzzle that I leave you to de-velop.This passage is carefully worded, yet it will be misin-terpreted by Montmort and Waldegrave. As we willsee, a key aspect that is neglected by Montmort andWaldegrave is the antecedent of Bernoulli’s condi-tional statement starting with “If we suppose thateach player observes the conduct. . . ”Montmort’s reply, dated March 1, 1712 (Mont-mort (1713), pages 337–347), highly praises Ber-noulli’s prior letter. He complains that, being inParis, he has had no time and peace to think on hisown and, as a consequence, the main object of hisletter is to report progress made by his two friends,the Abb´e d’Orbais and Waldegrave, on a problemproposed by Bernoulli, and on the problem of LeHer. On the latter, Montmort reports that “theydare however not submit to your decisions” (Mont-mort (1713), page 338). However, as he says in apassage that is key to understanding the forthcom-ing controversy, the Abb´e d’Orbais also previouslydisagreed with Montmort:When I worked on Le Her a few years ago,I told M. l’Abb´e de Monsoury what I hadfound, but neither my calculations nor myarguments could convince him. He alwaysmaintained that it was impossible to de-termine the lot of Pierre and Paul, be-cause we could not determine which cardPierre must hold to, and vice versa, whichresults in a circle, and makes in his opin-ion the solution impossible. He added aquantity of subtle reasonings which mademe doubt a little that I had caught thetruth. That is where I was when I pro-posed that you examine this problem; mygoal was to make sure from you of thegoodness of my solution, without havingthe trouble of recalling my ideas on thiswhich were completely erased.Montmort then claims that Bernoulli’s solution con-firms what he had found, a decision that prompts areply from Waldegrave objecting to Bernoulli’s so-lution, quoted at length in Montmort (1713), pages339–340. According to Waldegrave and the Abb´e d’Orbais,it is not true that Paul must hold only to an eightand Pierre to a nine. Rather, that Paul should beindifferent to hold to a seven or to switch, and thatPierre should be indifferent to hold to an eight or toswitch. Waldegrave wrote the following to Montmort(Montmort (1713), page 339):We argue that it is indifferent to Paulto switch or hold with a seven, and toPierre to switch or hold with an eight. Toprove this, I must first explain their lotin all cases. That of Paul having a seven,is × when he switches, and when heholds on to it his lot is × if Pierreholds on to an eight, and × if Pierreswitches with an eight. The lot of Pierrehaving an eight is × if he holds on toit, and × if he switches in the case thatPaul only holds on to a seven; and × by holding on to it, and × by switch-ing in the case that Paul holds on to aseven, so here they are. The lots of Paul
780 or 720 or 81650 × , those of Pierre
150 or 21023 × or
350 or 31427 × .Based on the numbers he obtains, Waldegrave ob-serves that “720 being more below 780 than 816 isabove, it appears that Paul must have a reason toswitch with 7” (Montmort (1713), page 339). Thedifferences, 780 −
720 and 816 − A , and the weight thatleads Pierre to switch be B . And he argues that thesame weights lead Paul and Pierre to both strate-gies. A leads Paul to switch with 7 and, as a conse-quence, it also leads Pierre to switch his 8; but whatleads Pierre to switch his 8 must lead Paul to holdwith 7. So, A leads Paul to both switch with a 7 andhold on to it. The same goes for Pierre. Therefore,“it is false that Paul must only hold on to an 8, andPierre to a 9,” which was Bernoulli’s claimed solu-tion. The word “probability” comes up only oncein this discussion, in the conclusion of the excerptfrom Waldegrave’s letter to Montmort. Waldegravewrites (Montmort (1713), page 340):Apparently Mr. Bernoulli was simplylooking at the fractions that express the D. R. BELLHOUSE AND N. FILLION
Table 1
Probabilities that Paul wins depending on the strategies ofplay ❍❍❍❍❍
Paul Pierre Switch the 8 Hold the 8(and under) (and over)
Switch the 7 (and under)Hold the 7 (and over) different lots of Pierre and Paul, with-out paying attention to the probabilityof what the other will do.Montmort leaves the discussion there without fur-ther comment.Upon receiving Montmort’s letter, Bernoulli agreeswith these figures, saying that “the lots they foundfor Pierre and Paul are very right” (Montmort(1713), page 348). And yet, when Bernoulli pro-poses his solution, and when Montmort eventu-ally publishes a table of probabilities as an ap-pendix to
Essay d’analyse (Montmort (1713), page413), the numbers are different. The Bernoulli–Montmort probabilities are shown in Table 1, whichappears in Hald (1990), page 318. None of theparties in this debate actually explain their calcu-lations. Waldegrave’s probabilities are justified inTodhunter (1865), pages 107–110; the Bernoulli–Montmort probabilities are in Hald (1990), pages315–318. The difference in the probabilities is thatWaldegrave’s probabilities are conditional on Paulhaving a seven in his hand and the Bernoulli–Montmort probabilities are the marginal probabili-ties for all cards that Paul may hold.In a letter dated June 2, 1712, Bernoulli repliesto Waldegrave’s argument by accusing him of com-mitting a fallacy. He argues that if we suppose that A leads Paul to switch with a seven, and so leadsPierre to switch with an eight (if Pierre knows Paulswitches with seven), then it also leads Paul to holdon to a seven. Therefore, A both leads Paul to switchwith a seven and to hold on to a seven. His conclu-sion is that (Montmort (1713), page 348):we are supposing two contradictory thingsat the same time; that is, that Paul knowsand ignores at the same time what Pierrewill do, and Pierre what Paul will do.Bernoulli explains that if we do not commit this fal-lacy regarding what Paul and Pierre know about the other’s intent, we are led to reasoning in a cir-cle, which shows that Waldegrave’s argument can-not show anything. This argument is peculiar, andseems to suggest that Bernoulli does not understandWaldegrave’s point. It might, however, be simplya misinterpretation of Waldegrave’s argument, forit is expressed in terms of weight rather than interms of probability. The word “weight” or “poids”in French offers more opportunity for misinterpre-tation. Moreover, Bernoulli admits having writtenhis letter hastily, as he was preparing for a long tripthrough the Netherlands and England. As a resultof this travel, some subsequent letters are delayed,and the arguments they contain do not follow thechronological order of when the letters were writ-ten.A letter to Bernoulli, dated September 5, 1712(Montmort (1713), pages 361–370), announces thatWaldegrave and the Abb´e d’Orbais have seen Ber-noulli’s reply in which he accuses them of com-mitting a fallacy. Montmort includes a note fromthe Abb´e d’Orbais in which he claims that Walde-grave has written a beautiful and precise reply toBernoulli’s objection; the rebuttal, however, is notincluded. In this note, the Abb´e d’Orbais also en-joins Montmort to take a side in this dispute be-tween them. This suggests that, even if Montmortthanked Bernoulli for his solution, which he claimedagreed with his own, Montmort has not yet made uphis mind as to whether Bernoulli really solved theproblem.The next letter concerning Le Her is from Bernoullito Montmort, dated December 30, 1712 (Montmort(1713), pages 375–394). Adding important pieces tothe puzzle, it contains a three-page discussion ofLe Her (Bernoulli mentions having just received theJune 2 letter, since it was sent from Switzerlandto Holland, then to England, and finally back toSwitzerland). Bernoulli insists that, despite Walde-grave’s arguments, Paul does not do as well by abid-ing to the maxim of holding to a seven, than that ofswitching with a seven. Bernoulli then says (Mont-mort (1713), page 376):If it were impossible to decide this prob-lem, Paul having a seven would not knowwhat to do; and to rid himself [from decid-ing], he would subject himself to chance,for example, he would put in a bag anequal number of white tokens and blacktokens, with the intent of holding to a ROBLEMS IN PROBABILITY BY BERNOULLI, MONTMORT & WALDEGRAVE seven if he draws a white one, & to switchwith a seven if he draws a black one; be-cause if he put an unequal number hewould be lead more to one party than tothe other, which is against the assump-tion. Pierre with an eight would do thesame thing to see whether he must switchor not.This comment introduces with clarity the idea ofchance by “the way of tokens” (as they will saylater). What Bernoulli says here seems to confirmthat, at first, when he accused Waldegrave of com-mitting a fallacy, he did not interpret Waldegrave’sweights as probabilities. Nonetheless, he suggeststhat the only probability allocation compatible withthe supposed state of ignorance of the players is thateach player chooses a strategy with probability .Under these choices, he computes the lot of Paul(which is then × ) and concludes that it would bea bad thing for Paul to randomize in this way, sincehe could guarantee himself a lot of × . Therefore,Bernoulli concludes Paul must always switch witha seven. As Bernoulli says (Montmort (1713), page376), “it is better to make the choice where we riskless.” He then explains the reasoning that he had leftout of his hastily written letter from June 2. In con-temporary terms, he calculated the unconditionalprobability of winning under each pure strategy pro-file (without assuming that any card has been dealtyet). He displays a refined version of the reasoningthat led to accusing Waldegrave of a fallacy, yet itdoes not do full justice to Waldegrave’s idea.Eight months later, on August 20, 1713, Mont-mort [(1713), pages 395–400] finally replies toBernoulli, complaining that he has, despite his philo-sophical inclinations, been involved in political ac-tivities, and so he did not have the leisure for intel-lectual work. Thus, his letter only contains scientificnews. There is only one brief mention of Le Her; hetells Bernoulli that, despite his last effort to pro-vide a thorough and precise argument, Waldegraveand the Abb´e d’Orbais are still unconvinced by hisclaimed solution. Shortly after, in a letter datedSeptember 9, 1713, Bernoulli also asks Montmortto explain his own views on the dispute. Montmortobliges him in his letter dated November 15, 1713.This is the last letter published in the second edi-tion of Essay d’analyse (Montmort (1713), pages403–413). The letter also contains an excerpt of aletter from Waldegrave and a table of the lots of Paul and Pierre for the four crucial combinations ofstrategies, which are summarized in Table 1.Here, then, is Montmort’s understanding of thesituation. To begin with, he agrees with Bernoullithat it is not indifferent to Paul to switch or holdwith a seven, and to Pierre to switch or hold with aneight, because of Bernoulli’s calculations of the un-equal chances for each strategy. (This shows thatBernoulli and Montmort use “indifferent” in thesense of having the same probability of winning.For Waldegrave and d’Orbais, however, “indifferent”seems to mean, perhaps more awkwardly, that nostrategy dominates the other in probability.) Thisbeing said, Montmort nonetheless disagrees withBernoulli that this establishes the strategy as amaxim, that is, as a rule of conduct that must beobeyed invariably to obtain the best results. Rather,he thinks that it is impossible to establish such amaxim (Montmort (1713), page 403):[T]he solution of the problem is impossi-ble, that is, we cannot prescribe to Paulthe conduct that he must adopt when hehas a seven, and to Pierre the conduct hemust adopt when he has an eight.He grants that, if one is to choose a fixed and de-termined maxim, then switching on seven, for Paul,will be better than any other, yet Paul can hope tomake his lot better.Why, then, would a solution be impossible? Wouldthe solution not be the optimum that one can reachin Paul’s hope of making his lot better? Montmortclaims that, whereas he used to think that the useof black and white tokens to randomize strategiescould avoid the “circle,” he does not think that any-more. He gives a general formula to find the prob-ability of winning with a certain probability alloca-tion for what we call a mixed strategy:2828 ac + 2834 bc + 2838 ad + 2828 bd · · a + b + c + d ) , where a is Paul’s probability of switching with seven, b is Paul’s probability of holding the seven, c isPierre’s probability of switching with an eight, and d is Pierre’s probability of holding on to an eight.But how should the probabilities be chosen? Mont-mort claims that any argument will only inform usof what Paul must do conditionally to what Pierredoes and vice versa, which leads us into a circle onceagain. He concludes that Bernoulli’s arguments to D. R. BELLHOUSE AND N. FILLION show that a circle does not occur are wrong, and in-stead formulates this thesis (Montmort (1713), page404):[W]e must suppose that both players areequally subtle, and that they will choosetheir conduct only based on their knowl-edge of the conduct of the other player.However, since there is here no fixed point,the maxim of a player depends on the yetunknown maxim of the other, so that ifwe establish one, we draw from this sup-position a contradiction that shows thatwe must not have established it.Montmort also disagrees with Bernoulli that, underpain of contradiction, if we are to use white andblack tokens to randomize, we must use an equalnumber of tokens. Instead, he thinks that the prob-ability of winning calculated for the fixed and deter-mined maxims shows that Paul must switch moreoften with a seven than hold on to it. Yet, he main-tains (Montmort (1713), page 405):But how much more often must he switchrather than hold, and in particular whathe must do (here and now) is the principalquestion: the calculation does not teachus anything about that, and I take thisdecision to be impossible.Thus, Montmort believes, it seems, that there is nooptimal probability allocation.But he has another reason for believing that thesolution of the game is impossible. He has in mindthe game Les ´Etrennes (Montmort (1713), pages406–407). Montmort also believes that it is impossi-ble to prescribe any strategy of play in Les ´Etrennesbecause the players might always try, and indeedgood players will try, to deceive other players intothinking that they will play something they are notplaying, thus trying to outsmart each other (“jouerau plus fin” is the phrase used in French).As he was finishing his letter, Montmort receivedone from Waldegrave and quoted extensively from itto Bernoulli.
Essay d’analyse essentially concludeswith Waldegrave’s letter. Waldegrave refers to a for-mula, which is not included by Montmort; it pre-sumably is the formula displayed above. He explainsthat, if a = 3 and b = 5 (so that the probability ofPaul switching with a seven is 0 . + · no matter what c and d are. This shows that + · is Paul’s minimum lot. He can only adopt another conductin the hope of making his lot better. This shows,he claims, that both Bernoulli and (formerly) him-self were wrong to claim that the lots of Paul wasto that of Pierre as 2828 is to 2697; if both play-ers play in the most advantageous way, Paul’s lotis + · . Waldegrave is convinced that this issomething that both Bernoulli and Montmort willagree to, now that it is agreed that one can use arandomized strategy. He also explains that, if Pierreuses c = 5 and d = 3, then + · will also bePaul’s maximum lot.Waldegrave also asserts that it is impossible toestablish a maxim; he grants, however, that it is im-possible for him to show this with the same levelof evidence. This is often taken incorrectly as evi-dence of a lack of Waldegrave’s mathematical abili-ties. Waldegrave is instead referring to the situationin which players may try to outsmart each other.Waldegrave agrees that if Paul does not use a = 3and b = 5, then it is possible for Paul to do bet-ter than + · provided that Pierre does notplay in the best way. On the other hand, it would beworse if Pierre plays correctly. Furthermore, Walde-grave remarks (Montmort (1713), page 411):What means are there to discover the ra-tio of the probability that Pierre will playcorrectly to the probability that he willnot? This appears to me to be absolutelyimpossible, and thus leads us into a circle.As with Montmort, his main concern is that it isalways possible for the players to try to outsmarteach other (“jouer au plus fin”).
3. ISSUES ARISING FROM THE PUBLISHEDCORRESPONDENCE
Examining the detailed arguments provided byMontmort, Bernoulli and Waldegrave reveals a pic-ture that contrasts with the judgment that theywere essentially confused on the fundamental con-cepts and methods required to solve a strategic gamesuch as Le Her. In fact, we maintain that they un-derstood most of the aspects of the problem withclarity. There are, however, a number of importantoutstanding issues left unresolved in the correspon-dence on Le Her as it appears in
Essay d’analyse .Let us review them briefly.It is true that the letters reveal a certain typeof misunderstanding; however, it is not conceptual
ROBLEMS IN PROBABILITY BY BERNOULLI, MONTMORT & WALDEGRAVE confusion, but rather mutual misinterpretation dueto using terms differently. An instance of this iswhether it is indifferent to Paul to switch or holdto a seven. On the one hand, both Montmort andBernoulli claim that it is not indifferent to Paul be-cause the chances of winning are not identical. Onthe other hand, Waldegrave claims that it is indiffer-ent, and the reason for that seems to be that neitherpure strategy dominates the other in probability.Another instance of this is the disagreement theyappear to have on the existence of a circularity inthe analysis of the game. Montmort and Waldegraveassert that there is a vicious circle that preventsone from establishing a maxim; the circle they dis-cuss, however, is really a regression ad infinitum ,that is, to establish a maxim, we always need togo one step further in the “ A must know what B does” loop (Bernoulli agrees with this point). How-ever, Bernoulli claims that there is a circle in Walde-grave’s argument, in the sense that either his ar-gument is contradictory or a petitio principii (butBernoulli is not considering randomizing strategiesat this point). Again, they are only contradictingeach other in the wording, not in the idea.Finally, a third instance is that Montmort andWaldegrave claim that the solution of the game isimpossible, whereas Bernoulli does not. Here again,they disagree on what it means to “solve” the gameLe Her. Bernoulli claims that the solution is thestrategy that guarantees the best minimal gain—what we would call a minimax solution—and thatas such there is a solution. However, despite un-derstanding this “solution concept,” Montmort andWaldegrave refuse to affirm that it “solves” thegame, since there are situations in which it mightnot be the best rule to follow, namely, if a playeris weak and can be taken advantage of. Clearly, theconcept of solution they have in mind differs fromthe minimax concept of solution. This latter con-cept, in addition to the probability of gain with apure strategy and the probability allocation requiredto form mixed strategies, requires that we know theprobability that a player will play an inferior strat-egy. But, they assert, this cannot be analyzed bycalculations, so the game cannot be solved.This being said, there are a number of things thatare said that suggest a certain level of confusion at aconceptual level. The two most important are these.First, Bernoulli appears to have some difficulty withthe relation between the knowledge of the playersand the probabilities involved in mixing strategies. His circularity objection to Waldegrave is awkwardand somewhat mystifying. Moreover, his argumentthat, if we allow randomized strategies with blackand white tokens, it must be because neither playerknows what the other player will do, and that as aresult the only acceptable probability allocation of is problematic. This kind of mistaken argumenthas been repeated over the centuries by some of thegreatest minds in probability, statistics and gametheory. Second, Montmort understands very well theidea of randomizing strategies, but he nonethelessclaims that there is no optimal probability alloca-tion that can be calculated. This claim, however,was made before consulting Waldegrave’s letter inwhich he reveals the optimal probability.
4. DISCUSSION OF THE GAME OF LE HERAFTER 1713
Referring to a letter from Bernoulli to Montmortdated February 20, 1714, Henny (1975), in his treat-ment of Le Her, mentions only that Bernoulli ac-cepted Waldegrave’s solution to the problem. How-ever, Bernoulli had other things to say about LeHer in that same letter. Henny also refers to a letterof January 9, 1715, from Waldegrave to Bernoulliin which Waldegrave seemingly admits to Bernoullithat he does not have the mathematical skills to ac-tually prove his results. What Henny leaves out isthat the letter was written in reply to a detailedcriticism of the solutions to Le Her that Bernoullihad sent earlier to Montmort.After some personal news and apologies for notwriting sooner, in his letter of February 20, 1714,to Montmort, Bernoulli initially thanks Montmortfor correcting, editing and making clearer his let-ters that Montmort had printed in
Essay d’analyse .Then follows the discussion of Le Her that Henny(1975) only briefly mentions. Initially, Bernoulli sug-gests that the controversy is essentially over:Concerning Le Her, I seem to have fore-seen that in the end we would all be right.However, I congratulate Mr. de Walde-grave who has the final decision on thisquestion, and I willfully grant him thehonor of closing this affair. . .Despite this, Bernoulli still claims that he disagreeson a few minor points, and these point directly tothe outstanding issues we mentioned above. Themain concern is the relation between “establishinga maxim” and solving the problem of Le Her posedby Montmort in his book. Bernoulli states: D. R. BELLHOUSE AND N. FILLION
One can establish a maxim and propose arule to conduct one’s game, without fol-lowing it all the time. We sometimes playbadly on purpose, to deceive the oppo-nent, and that is what cannot be decidedin such questions, when one should makea mistake on purpose.This point was raised before by Montmort andWaldegrave, but they do not consider that such aplay would be necessarily a mistake. Whether or notthis kind of play is a mistake, we saw that from thesame consideration, Montmort and Waldegrave con-clude that solving the problem is impossible. How-ever, Bernoulli now phrases things more carefully:Mr. de Waldegrave wrongs me on p. 410by claiming that I once said that the lot ofPaul is to that of Pierre as 2828 : 2697. Ifyou carefully read my letter from Oct. 10,1711, you will find that I did not say it ab-solute and without restriction. I beg youto consider those words: once we have de-termined or rather supposed what are thecards to which the players will hold, etc.And the following words. You will see thatI there supposed that the players want tohold to a fixed and determined card, andindeed I had not thought about the wayof tokens, which, as Mr. de Waldegravesaid, is not among the ordinary rules ofthe game.Bernoulli essentially says that he was misinterpretedand that he only computed the best odds of win-ning with a pure strategy, not that he establishedwhat a player should do in an actual game. More-over, if we grant his supposition, then he has foundthe most advantageous maxim. After this correction,Bernoulli thinks the discussion is over, saying, “Weare thus all agreeing, and we have made peace; cana-mus receptui [sing retreat].”In his response to Bernoulli, dated March 24,1714, Montmort concurs by writing, “I am quitepleased that we are all together by and large agree-ing.” In this letter Montmort claims that he dis-agreed with Bernoulli on some aspect of the cor-rected interpretation of his position, but he leavesit to a later letter to explain. However, in his nextletter to Bernoulli, November 21, 1714, Montmortdoes little to clarify. He says, “if it is ever permit-ted to say to two persons maintaining contradictory claims that they are both right, it is assuredly atthis occasion in our dispute.” Montmort emphasizesthat what he seeks is the correct advice that shouldbe given to the players, but the discussion does notgo much further.On August 15, 1714, Montmort sent a letter toBernoulli containing a two-page “supplement” thatreignites the debate. He makes six points. First, heclaims that telling Paul always to switch with aseven is bad advice, since his minimum lot is then2828. Second, that it would be better advice to tellhim to do whatever he pleases with a seven, so thathe can look at both options indifferently. Third, wecannot say that this would be the best advice ei-ther, for knowing that, Pierre would switch with aneight, in which case Paul should certainly have heldon to a seven. This leads to a vicious circle. Fourth,if we admit the way of tokens, the best advice thathe knows is to tell Paul to have the ratio 3 : 5 forswitching with a seven. But even then, he does notthink that we can demonstrate that it is the bestadvice. Fifth, he claims that it is impossible at thisgame to determine the lot of Paul, because one can-not determine what manner of playing is the mostadvantageous to each player, even when we admit arandomized strategy. This point makes explicit forthe first time Montmort’s (and presumably Walde-grave’s) idea that you can only claim that you havefound the lot of a player (which is what Montmort’sproblem in
Essay d’analyse demanded) if we candetermine what is the best way to play. Moreover,determining the best way to play demands knowingmore than the optimal token ratio for the random-ized strategy. He adds that, of course, some meth-ods of playing are better than others, as informedby the chances that have previously been calculated.He concludes, sixth, that he would not know whatadvice to give Paul if he had to. This letter sharpensthe debate, in that it makes explicit the connectionbetween “solving” a game and giving advice for playin actual situations.In a long letter to Montmort dated August 28,1714, along with a “supplement” dated November 1,1714, Bernoulli replies to Montmort point by point.He asks Montmort a question that is meant to dis-miss his argument:If, admitting the way of tokens, the optionof 3 to 5 for Paul to switch with a sevenis the best you know, why do you wantto give Paul other advice in article 6? It
ROBLEMS IN PROBABILITY BY BERNOULLI, MONTMORT & WALDEGRAVE suffices for Paul to follow the best maximthat he could know. It is not enough toclaim that there is still a circle despite myreasons, one must fight my reasons.And he continues: “It is not impossible at this gameto determine the lot of Paul.” To counter Mont-mort’s previous argument, he once again insists thateither Paul knows what Pierre will do, in which casehis maxim is clear, or he does not, in which casePaul should use the probability in the randomizedstrategy to determine what to do. As he admits, thisis the exact same position he had at the beginningof the discussion, supported by the exact same ar-gument. Thus, it seems that Bernoulli has missedthe point Montmort made explicit in his August 15letter.It is at this point that Waldegrave reenters the de-bate at Montmort’s request. In a letter dated Jan-uary 9, 1715, Waldegrave reiterates the six pointsthat Montmort had laid out for Bernoulli in his let-ter of August 15. For each of the six points, Walde-grave’s arguments are longer and more detailed thanwhat Montmort had previously given.It is not until March 22, 1715, that Montmortreplies to Bernoulli on this dispute. It is part of avery long letter that also contains the main topic fortheir further correspondence, infinite series. In thisletter, Montmort writes once again about his views.They are the same as what we have seen already.However, Montmort stresses that a lot of what re-mains under discussion is based on inconsistent ter-minology and misinterpretation. In essence, he be-lieves that the outstanding disagreements are onlyapparent contradictions. Nonetheless, he introducesone more element to clearly articulate his view. Hedistinguishes between the advice that he would putin print, or give to Paul publicly, and the advice hewould give so that only Paul hears it. Montmortclaims that, for the former, he would choose themixed strategy with a = 3 and b = 5, since it is theone that demonstrably brings about the lesser preju-dice. However, he explains that, in practice, if Paul isplaying against an ordinary player and not a mathe-matician, he would quietly give different advice thatcould allow Paul to take advantage of his opponent’sweakness. As he explains, the objective of this sortof analysis is not only to provide a maxim to other-wise ignorant players, but also to warn them aboutthe potential advantages of using finesse. However,this latter part is not possible to establish, and it is in this sense that there is no possible solution tothis problem.The next letter, sent by Bernoulli to Montmorton May 4, 1715, disregards Montmort’s nuance. Tobegin, Bernoulli “is forced to admit that he doesnot precisely know on what point [they] contradicteach other.” Nonetheless, Bernoulli explains that, inhis view, the distinction between public and privateadvice, the possibility of using finesse, or somethingsimilar, does not alter the fact that a = 3 and b = 5is the best solution, and that it determines the lotof Paul (so that not only is the game solvable, butit is indeed solved).Despite Bernoulli’s explanation, Montmort’s nextletter, dated June 8, 1715, once again reiterates that“you have badly solved the proposed question, oryou have not solved it at all.” He makes explicitwhat he takes the proposed question to be:The question is and has always been toknow whether we can establish the lotsand as a result the advantage of playingfirst under the supposition not that Pierreand Paul follow this or that maxim (thiswould have no utility, no difficulty), butthat both of them having the same skills,each follow the conduct that is the mostadvantageous.Montmort then says that this dispute is beginningto bore him. He considers that furthering it will notmake them learn anything new and that in the endthe dispute must be about some other thing.Our presentation of the correspondence makes itclear that they are using different concepts of so-lution; Bernoulli’s in essence is the concept of theminimax solution, whereas Montmort’s further de-pends on the probability of imperfect play (i.e., onthe skill level of the players).Around this time, Montmort’s interest shifts fromprobability and its applications to infinite series. Infact, most of the remaining correspondence withBernoulli turns to that topic. At the same time,Montmort began an extensive correspondence withBrook Taylor, also mainly on infinite series (St.John’s College Library, Cambridge, TaylorB/E4).Although the dispute with Bernoulli seems to havepetered out, Montmort was not yet done with it. Ina letter dated July 4, 1716, Montmort asked Tay-lor to examine his dispute with Bernoulli about LeHer and to express his opinion on who was right. He D. R. BELLHOUSE AND N. FILLION referred Taylor only to the correspondence that ap-pears in
Essay d’analyse . Taylor apparently wroteback but with the wrong impression about whatMontmort wanted. Montmort replied to Taylor onAugust 4, 1716, that he did not want any new re-search into the problem but only to examine, at hisleisure, which of Bernoulli or Montmort was right. Ina letter to Taylor dated November 10, 1717, Mont-mort thanked Taylor for his opinion on the disputeand concluded the letter by saying that Waldegravewould write him about Le Her as well as anothergame. Unfortunately, neither Taylor’s reply express-ing his opinion nor Waldegrave’s letter to Taylor areextant.
5. THE PROBLEM OF THE POOL ANDOTHER PROBABILITY PROBLEMS
Compared to the discussion of Le Her, the re-maining discussion in the post-1713 correspondencewith regard to probability problems is relatively mi-nor. For example, after the remarks on Le Her thatBernoulli made in his letter of February 20, 1714, toMontmort, Bernoulli comments that he thinks thereis an error in Montmort’s solution to a problem re-lated to the jeu du petit palet in
Essay d’analyse .He asks Montmort to check his solution. The prob-lem appears to be Probl`eme IV in Montmort (1713),page 254. The jeu du petit palet is a game in whichplayers toss coins or flat stones (the “palets”) to-ward a target set on the ground or a table. Theplayer with the most coins or stones on the targetwins. The English equivalent game is called chuck-farthing or chuck-penny.What takes up much of the discussion, other thanLe Her, is news about Abraham De Moivre’s work.De Moivre corresponded with both Montmort andBernoulli until about 1715 when he ceased corre-sponding with either of them. Prior to this discus-sion, Bernoulli had sent De Moivre a general solu-tion to the problem of the pool on December 30,1713 (Bellhouse (2011), pages 106–107).A report on De Moivre’s activities in probabilitytakes up part of a letter from Bernoulli to Mont-mort dated April 4, 1714. Bernoulli mentions thatDe Moivre has sent him a long letter with reports ofnew solutions that will appear in a much expandedversion of his treatise
De mensura sortis (De Moivre,1711). De Moivre’s new work, which was entitled
The Doctrine of Chances , did not appear until 1718(De Moivre, 1718). No details are given to Mont-mort other than that De Moivre has made inroads in three areas. First, De Moivre used his own methodfor the solution of the problem of the pool to gen-eralize it to more than three players. Second, he de-veloped a new kind of algebra to solve probabilityproblems. Finally, Bernoulli reports that De Moivreconsidered that nearly all problems in probabilitycan be reduced to series summations. Not only didDe Moivre report that he had generalized the prob-lem of the pool, but he also sent Bernoulli his solu-tion to the problem. At the time of his writing toMontmort, Bernoulli had not read the solution anddid not pass the solution on to Montmort. The newalgebra is probably the one that De Moivre devel-oped for finding probabilities of compound events.See, for example, Hald [(1990), pages 336–338] for amodern discussion of this topic. This part of the let-ter ends with what might be interpreted as a nastycomment about De Moivre:I will share here in confidence what hewrote to me concerning you. Here is whathe told me about your comments that Ihad sent him. ‘I cannot stop myself etc.Our Society etc. I just received etc. kind[regards].’ After the letter I find writtenthere these words: ‘in a sense,’ that mademe laugh.It is difficult to know what exactly Bernoulli is say-ing here. It appears that he sent De Moivre Mont-mort’s severe criticism of
De mensura sortis thatMontmort published in
Essay d’analyse (Montmort(1713), pages 363–369).Later that month, Montmort reported back toBernoulli that he received a very polite and fair let-ter from De Moivre in which De Moivre announcedthat he had found a new solution to the problemof the duration of play. See Bellhouse [(2011), pages111–114] for a discussion of the publication of thissolution. De Moivre sent reports about more of hisresults in probability to Montmort and Montmortsent on a pr´ecis of these results to Bernoulli in aletter dated August 15, 1714. Many of the resultsthat Montmort mentions found their way into
TheDoctrine of Chances , including what is called Wood-cock’s problem discussed in Bellhouse [(2011), pages125–126].On August 28, 1714, Bernoulli finally wrote toMontmort enclosing a copy of De Moivre’s generalsolution to the problem of the pool. In the let-ter, Bernoulli asks Montmort to tell him what he
ROBLEMS IN PROBABILITY BY BERNOULLI, MONTMORT & WALDEGRAVE Fig. 1.
Waldegrave signatures from various sources. thinks of the solution. He further states that it ap-pears that De Moivre is using Bernoulli’s approachto the solution for three and four players that ap-pears in
Essay d’analyse (Montmort (1713), pages380–387). At the same time he is using an analyti-cal approach rather than infinite series (De Moivreactually used a recursive method for his general so-lution). Montmort replied on March 22, 1715, thathe agrees with Bernoulli’s assessment. On return-ing from a trip to England, Montmort reported toBernoulli in a letter dated June 8, 1715, that one ofBernoulli’s solutions to the problem of the pool hadjust been printed in the
Philosophical Transactions (Bernoulli, 1714). Bernoulli had sent De Moivre twosolutions; De Moivre claimed he had found an errorin the first solution.
6. WALDEGRAVE IDENTIFIED
Many in the past have tried unsuccessfully to iden-tify the Waldegrave who solved the problem of LeHer and who suggested the problem of the pool, of-ten called Waldegrave’s problem. Bellhouse (2007)reviewed these attempts at identification and nar-rowed the field down to Charles, Edward or Fran-cis Waldegrave, the three brothers of Henry Walde-grave, 1st Baron Waldegrave. Bellhouse argued forCharles Waldegrave, but in view of new informa-tion his choice was incorrect. Key to the properidentification is that several Waldegraves—siblings,cousins and at least one uncle of Henry—followedKing James II into exile in France after James wasdeposed in 1688.The proper identification of the Waldegrave of in-terest may be found in legal papers in the Archivenationales de France in conjunction with a letterfrom Waldegrave to Nicolaus Bernoulli; the letter to Bernoulli is signed only “Waldegrave” and isthe only known letter in Waldegrave’s hand thatis extant (Universit¨atsbibliothek Basel L Ia 22,Nr. 261). Other Waldegrave signatures to compareto the one on Bernoulli’s letter can be found onvarious legal documents, two in France (Archivesnationales de France MC/ET/XVII/486 and 514)and one in England (House of Lords Record OfficeHL/PO/JO/10/1/439/481). See Figure 1. From thesignatures, it is obvious that Francis is the Walde-grave of interest. From these records, it is also appar-ent that Charles Waldegrave handled the family’saffairs in England while Francis Waldegrave tookcharge of them in France.What little is known of the life of Francis Walde-grave comes mostly from Montmort’s correspon-dence with Brook Taylor and Nicolaus Bernoulli.Montmort reported to Taylor one of Waldegrave’spolitical activities. Waldegrave was planning to takepart in the Jacobite uprising in England in 1715.He was to be part of an invasion force led by theson of James II, James Stuart. The uprising in Eng-land fizzled out, James Stuart remained in Franceand Waldegrave fell ill just prior to the time whenthe planned invasion was to occur. Montmort calledWaldegrave’s illness apoplexy; it was probably astroke. From time to time, Montmort commentedto Taylor and Bernoulli about Waldegrave’s illness,recovery and setbacks. At one point, for a cure or arest, Waldegrave took the waters at a spa in France.He also spent time at Montmort’s chateau. Thoughill, he was alive in France in 1719 when Montmortdied so that the flow of information to Bernoulliand Taylor about Waldegrave stopped. Presumably,Waldegrave died in France.How Waldegrave obtained his mathematical train-ing is unknown. In whatever way he was educated, D. R. BELLHOUSE AND N. FILLION he was an adept amateur mathematician. This iscontrary to Henny’s interpretation of Waldegrave’sskills. For example, Henny [(1975), page 502] claimsthat Waldegrave did not have the mathematicalskills to work out a general method of calculationin Le Her. On the contrary, there is a hint of thefairly high level of Waldegrave’s mathematical abil-ities in a letter from Montmort to Bernoulli datedMarch 24, 1714. There Montmort says that he is get-ting Waldegrave to read L’Hˆopital’s (1696) calculusbook
Analyse des infiniment petits and that Walde-grave has a natural aptitude for mathematics.At the time that Montmort was sending
Essayd’analyse to his publisher, Francis Waldegrave wasliving in Rue Princesse near ´Eglise Saint-Sulpice inParis. In modern Paris, it is a 1 . Philosophical Transactions (Montmort, 1717).In a letter dated June 15, 1717, Montmort gave Tay-lor complete editorial control over the paper thatincluded having Taylor translate the results fromFrench into Latin (St. John’s College Library, Cam-bridge TaylorB/E4). Taylor replied August 9, 1717,saying that he had made many changes and cor-rections to the paper (St. John’s College Library,Cambridge, TaylorB/E5).
7. DISCUSSION AND CONCLUSIONS
The unpublished letters between Bernoulli andMontmort reveal a much more complex story thaneither Henny (1975) or Hald (1990) have described.The entire group—Bernoulli, Montmort and Wal-degrave—were for the most part clear about the is-sues at the conceptual level. In the end it came downto a disagreement about what it meant to solve aproblem. Further, Henny recognized many moderngame theory concepts, but we show that the group’sunderstanding of the modern notions is deeper thanwhat Henny realized.Apart from the technical and conceptual aspectsof Le Her and other probability problems, we alsoget a glimpse into the social side of a rich ama-teur mathematician at work. Montmort was a good mathematician, but mathematics was his hobby andat times he did not have time to pursue his hobby.There is a bit of quid pro quo in his relationshipswith Bernoulli, Taylor and Waldegrave. Montmortacquires some status through his connections toartists, philosophers and scientists. He can imposeon his scientific friends to do some of the more me-nial work for him in getting his research to print. Onthe other side, his scientific friends enjoy his hospi-tality, his gifts and the benefits of any political andscientific connections that he may have.Traditionally, the mixed strategy solution with a = 3 and b = 5 for Le Her has been attributed toWaldegrave. It certainly appears to be the correctattribution based on the correspondence in the sec-ond edition of Essay d’analyse . However, in the longletter from Montmort to Bernoulli dated March 22,1715, that covers discussions of Le Her, De Moivreand other topics, Montmort appears to claim pri-ority of solution. As part of the discussion of LeHer, he says, “although I first found the determi-nation of the numbers a and b , c and d . . . ” Mont-mort’s suggestion of priority could have come aboutas a result of a conversation between Waldegraveand Montmort, with Waldegrave putting pen to pa-per. This illustrates Fasolt’s (2004) claims about thelimits of history. Our data from the past is what hasbeen written, not what has been spoken. Further, wecan never know the tone behind what was written,such as Bernoulli’s apparently nasty comments toMontmort about De Moivre in his letter of April 14,1714. Instead of coming up in conservation, Mont-mort may be claiming priority because he found thegeneral formula in a , b , c and d ; the numbers wereonly a special case. Or it could be something else.Like Le Her itself, depending on how the problem isapproached, the assignment of priority is a problemwith no solution. ACKNOWLEDGMENTS
We would like to thank Dr. Fritz Nagel of Uni-versit¨at Basel for giving us access to the corre-spondence between Nicolaus Bernoulli and PierreR´emond de Montmort. We also thank Kathryn Mc-Kee and Jonathan Harrison of St. John’s CollegeLibrary, Cambridge, for providing us with copiesof the correspondence between Brook Taylor andMontmort.The originals of the letters of Bernoulli, Mont-mort and Waldgrave are in Universit¨atsbibliothek
ROBLEMS IN PROBABILITY BY BERNOULLI, MONTMORT & WALDEGRAVE Basel. The letters from Montmort to Bernoulli arecatalogued Handschriften L Ia 22:2 Nr.187–206 andfrom Bernoulli to Montmort are L Ia 21:2 Bl.209–275. The letter from Bernoulli to Waldegrave is cat-alogued L Ia 21:2 Bl.229v–232r and the letter fromWaldegrave to Bernoulli is L Ia 22, Nr. 261. Whenreferencing these letters, we have to do so by thedate, writer and recipient, rather than the catalognumbers. REFERENCES
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