**Abstract** |
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.
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