How it all started: Nikolaus’s challenge and the first discussions

The long-standing debate on what was called later the “Saint Petersburg paradox” (see, for example, Samuelson 1977; Jorland 1987; Daston 1988: ch. 2; Dutka 1988; Martin 2014) started with a letter of Nikolaus to Montmort, dated 9 September 1713 and immediately published by Montmort, with some other correspondence with Johann and Nikolaus, in the fifth part of the second edition of his Essay (Remond de Montmort

1713: 401-2).

Fourth problem. A promises B to give him one ecu if, with a normal dice, he obtains six points at the first roll, two ecus if he succeeds at the second roll, three ecus if he succeeds at the third roll, four ecus at the fourth roll, and so on; one asks, what is B’s expectation. Fifth problem. The same thing is asked if A promises B to give him ecus in this progression: 1, 2, 4, 8, 16, etc. or 1, 3, 9, 27, etc.... instead of 1, 2, 3, 4, 5, etc. like before. (Ibid.: 402)

Montmort answered (ibid.: 407) that the solution, based on the calculus of the limit of infinite series developed by Nikolaus’s uncle Jakob, was easy to find. In a subsequent correspondence however (Dutka 1988: 19; Meusnier 2006: 9-11), Nikolaus pointed out to Montmort two difficulties that the latter had disregarded. Two methods could be used to solve the problem: B^,s expected gain could be found either as the sum of the terms of an infinite series, as stressed by his correspondent, or using mathematical induction. But while the solution to the fourth problem poses no problem (B’s expectation is 6) whatever the method employed, two important discrepancies arise instead in the case of the fifth problem - discrepancies which shake the belief in the meaning and relevance of (mathematical) expectation. In the first place, in the case of the first progression for example - 1, 2, 2², 2³,...

which is infinite, while, with the second method, the result is different: it is finite and moreover negative: -1/4 (Dutka 1988: 19; Meusnier 2006: 10-11). In the second place, since in games of chance the fair stake was defined as the player’s expected gain, another discrepancy arises with what “good sense” would advise: no reasonable player is sup­posed to pay an infinitely large sum of money - or even only an important sum, com­pared to his wealth - to play this game.

The first difficulty was very embarrassing and forms the real paradox in this story. After Gabriel Cramer rekindled the debate in 1728 (Montmort had died in 1719), the discussion concentrated on the second difficulty - which was not really a paradox, in spite of the name given to it, but a discrepancy between theory and reality. It involved Cramer, in correspondence with Nikolaus and Georges-Louis Leclerc de Buffon, and Nikolaus with Daniel. An excerpt of this correspondence between Nikolaus and Cramer is quoted at the end of Daniel’s essay (Bernoulli 1738 [1954]: 33-5), and the exchange of letters between Buffon and Cramer is recalled in Buffon’s “Essai d’arithmetique morale” (Leclerc de Buffon 1777: 75-7). The game considered is now the toss of a coin, with the same progression of possible gains (1, 2, 2², 2³,... 2ⁿ...). The game goes on if B obtains “tails” and stops with “heads”. With the first method - the probability to obtain “heads” at every toss being one-half - B’s expectation of gain is again infinite:

The debate thus turned around the possibility to change the definition of expecta­tion because the usual one gave results at odds with “good sense”: this was possible by changing the apprehension of the possible gains. As Cramer put it in 1728 - an approach which turned out to be the same as Buffon’s and Daniel’s solutions - the reason for the discrepancy between the mathematical calculation and common sense “results from the fact that, in theory, mathematicians evaluate money in proportion to its quantity while, in practice, people with common sense evaluate money in proportion to the use they can make of it” (Cramer, in Bernoulli 1738 [1954]: 33, translation modified). For the player, what matters is not the sum of money, but, in Daniel’s words, the “emolumentum” - the benefit, the advantage, usually translated as “utility” - an individual gains from it, which depends on the wealth already possessed. Daniel formalized and developed this idea in an outstanding way.