Conference Agenda

The famous problem of points deals with a game of chance that is played out in several rounds. In it, the players pay a stake before the game begins and agree that the first player to win $p$ rounds or points will receive the entire stake. The problem now is to divide the stakes ‘fairly’ between the players in case the game, due to force majeure, has to be stopped before one of the players has reached the number of points required for overall victory, with the player in the lead having won $g$ rounds and the other player having won $f$ rounds. For this reason, it is also known as the problem of division of the stakes.

The discussion of the problem by Pierre de Fermat, Blaise Pascal, and Christaan Huygens in the middle of the 17th century is often apostrophized as the birth of a new mathematical discipline, namely probability theory.

A few years after the aforementioned, Gottfried Wilhelm Leibniz (1646–1716) also dealt with the problem and developed his very own thoughts on it. He dedicated two manuscripts to this question, one in French in 1676 and the other in Latin in 1678. The results of an analysis of these two texts can be summarized in the following six theses:

(1) When Leibniz was dealing with the problem of points, there were already insightful contributions to its discussion he could have known about. But he obviously did not take note of these — especially not of Pascal’s probabilistically correct solution — although it would have been possible for him to do so. Instead, he started his very own reflections.

(2) In 1676 Leibniz formulated — in words, not as a formula — a first partition rule that can be considered ‘fair’ in a broader sense. It proposes a division of the stakes in the ratio of $(p+g-2f) : (p-g)$. As Struve and Struve prove, this rule has a unique feature: Every point a player is ahead is worth the same.

(3) A variant of the rule, formulated differently and less clearly, but with the same meaning, is found in his 1678 manuscript. Most historians of mathematics are only familiar with this later version, though, not with the more lucid earlier one.

(4) Again in 1676, Leibniz formulated a second ‘fair’ rule, based on a different idea. Translated into a formula, it stipulates a division of the stakes in the ratio of $(p-f)^2 : (p-g)^2$. This rule has received almost no attention in the historiography of probability theory — possibly since Leibniz hid it well in subordinate clauses, describing it in opaque words instead of giving a formula for it. More importantly, he dropped the second rule shortly after he had established it, as not even he himself attached further importance to it.

(5) This rule deserves attention nevertheless, as it regularly provides a better approximation of the ratio of winning probabilities than the first rule or any other common, simple rule, and its results are much easier to calculate than the exact solutions, which can be quite helpful.

(6) The assumption that Leibniz worked with a different notion of ‘fair partition’ than Pascal or Fermat cannot be substantiated. It is far more plausible to assume that he actually defined ‘fair’ as ‘according to probabilities’ in the modern sense. This, however, turns the problem of points from a normative to a probabilistic question, which means that Leibniz’s results are not to be characterised as creative normative solutions, but rather as probabilistic solutions that are not entirely correct.

In 2022, Glenn Shafer and I published a collective work on the history of the martingale concept. In the present contribution, I focus on the key moment when the concept gradually stabilized as a mathematical theory, through the work of three mathematicians: Paul Lévy, Joseph Doob and Jean-André Ville. Although the latter is obviously the least well-known, it was he who, in his thesis defended in 1939, gave the name martingale to a construction inspired by game theory, which Doob would later take up and develop, while Paul Lévy's work on the subject remained relatively unnoticed until the 1950s. I give some insight into the astonishing emergence of a concept which, in the second half of the twentieth century, became one of the central concepts of modern probability theory.