The Petersburg Paradox at 300

In 1713 Nicolas Bernoulli sent to de Montmort several mathematical problems, the fifth of which was at odds with the then prevailing belief that the advantage of games of hazard follows from their expected value. In spite of the infinite expected value of this game, no gambler would venture a major stake in this game. In this year, de Montmort published this problem in his Essay d'analyse sur les jeux de hazard. By dint of this book the problem became known to the mathematics profession and elicited solution proposals by Gabriel Cramer, Daniel Bernoulli (after whom it became known as the Petersburg Paradox), and Georges de Buffon. Karl Menger was the first to discover that bounded utility is a necessary and sufficient condition to warrant a finite expected value of the Petersburg Paradox. It was, in particular, Menger's article which provided an important cue for the development of expected utility by von Neumann and Morgenstern. The present paper gives a concise account of the origin of the Petersburg Paradox and its solution proposals. In its third section, it provides a rigorous analysis of the Petersburg Paradox from the uniform methodological vantage point of d'Alembert's ratio text. Moreover, it is shown that appropriate mappings of the winnings or of the probabilities can solve or regain a Petersburg Paradox, where the use of probabilities seems to have been overlooked by the profession.