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On the St. Petersburg paradox. (English) Zbl 0762.01007
The originator of the St. Petersburg paradox was Niklaus (1) Bernoulli (1687-1759). He sent several game problems to P. R. de Montmort (esp. no. 4 and 5 with infinite expectation but in which the experienced gambler would risk at most twenty ducats). He received an unsatisfactory solution by the Swiss mathematician Gabriel Cramer (1704-1752) and later in 1731 the solution by Daniel (1) Bernoulli (1700-1782), which has been published in Commentarii Akad. Sci. Petropolis 5, 175-192 (1738). Therefore d’Alembert coined the expression ‘St. Petersburg Paradox’. A historical exposition of its genesis is given and a sketch of some of the attempts at its solution is worthwhile since in the modern era it has attracted the attention not only of the probabilists, statisticians and game theorists, but also economists. The early development of probability theory is also sketched, e.g. G. Cardano (1539), Guido Grandi (1671-1742), Blaise Pascal (1623-1662) with his famous wager in the Pensées (a forerunner of decision theory), Ch. Huygens (1629-1695) with his five problems, Gottfried W. Leibniz (1646-1716) with his degree of certainty. Some earlier and new reviews of the problem are mentioned as E. Czuber (1882), Paul A. Samuelson (1977), and Gérard Jorland (1987), the latter one contains several errors. In 1985 A. Martin-Löf obtained a limit theorem for the total gain in a series of St. Petersburg games. A computer simulation relevant to the paradox is being added. 48 references.
Reviewer: H.Grimm

MSC:
01A50 History of mathematics in the 18th century
60B10 Convergence of probability measures
60-03 History of probability theory
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