Le Her and Other Problems in Probability Discussed by Bernoulli, Montmort and Waldegrave

TL;DR

This paper explores early 18th-century correspondence among Montmort, Bernoulli, and Waldegrave discussing probability problems, including the game Le Her, providing historical context and translations of key letters.

Contribution

It uncovers the historical discussion of probability strategies in the correspondence, clarifies Waldegrave's role, and contextualizes early game theory development.

Findings

Analysis of historical letters reveals early probability strategies.

Identification of Waldegrave's contributions to game theory.

Contextualization of Le Her in the history of probability.

Abstract

Part V of the second edition of Pierre R\'{e}mond de Montmort's Essay d'analyse sur les jeux de hazard published in 1713 contains correspondence on probability problems between Montmort and Nicolaus Bernoulli. This correspondence begins in 1710. The last published letter, dated November 15, 1713, is from Montmort to Nicolaus Bernoulli. There is some discussion of the strategy of play in the card game Le Her and a bit of news that Montmort's friend Waldegrave in Paris was going to take care of the printing of the book. From earlier correspondence between Bernoulli and Montmort, it is apparent that Waldegrave had also analyzed Le Her and had come up with a mixed 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\"{a}tsbibliothek Basel contains an additional forty-two letters between Bernoulli and Montmort written after 1713, as well as two letters between Bernoulli and Waldegrave. The letters are all in French, and here we provide translations of key passages. The trio continued to discuss probability problems, particularly Le Her which was still under discussion when the Essay d'analyse went to print. We describe the probability content of this body of correspondence and put it in its historical context. We also provide a proper identification of Waldegrave based on manuscripts in the Archives nationales de France in Paris.