St. Petersburg Paradoxes: Defanged, Dissected, and Historically Described

Abstract

THE St. Petersburg paradox constitutes a fascinating chapter in the history of ideas. What other subject can link together Edward Gibbon and the father-inlaw of Thomas Mann? Or can, in the conceit of Maynard Keynes, link together the Bernoullis and Darwin?' Substantively, the Petersburg paradox served as a dramatic paradigm, alerting people to the fact that their utility of gain exceeded their utility of equivalent money lost. As a byproduct, the discussion generated an interest of its own: the idiocies it evoked from various writers are scarcely less fascinating than the lasting insights. Yet when I came recently to review the matter, I found to my surprise that no one seems to have provided anything like a complete survey of the subject. The standard sources provide us with tantalizing glimpses: the useful 1954 English translation of the classic Latin work of Daniel Bernoulli [4, 1738]; the references and discussion in histories of probability, such as those of Isaac Todhunter [39, 1865] and Leonid Maistrov [17, 1966]; the selective but perspicacious accounts in the work of Keynes [15, 1921] and George Stigler [37, 1950]; the new lease on life given to the subject by Karl Menger [19, 1934] with the discussion there of SuperPetersburg paradoxes creatable when the utility function is unbounded; the recent observation by Kenneth Arrow [2, 1971] that not all stochastic processes can be ordered by the expected value of their utility outcomes when such Mengerian super-paradoxes are allowed to exist. The number of post-Mengerian writers is large, including L. S. Shapley [36, 1972], D. L. Brito [6, 1975], Samuelson [28, 1960], Robert J. Aumann [3, 1975], and many others. Given my own limited linguistic abilities and leisure time, I have not been able to provide anything like a definitive survey, one that checks back on and quotes copiously from the original sources. After all, the list of writers connected with the St. Petersburg paradox reads like a veritable who's who in probability and the social sciences. Here is but a sample: Nicholas Bernoulli (1687-1759), Montmort (1678-1719), Gabriel Cramer (1704IThe continuity and oneness of modern European thought may be illustrated, if such things amuse the reader, by the reflection that Condorcet derived from Bernoulli, that Godwin was inspired by Condorcet, that Malthus was stimulated by Godwin's folly into stating his famous doctrine, and that from the reading of Malthus on Population Darwin received his earliest impulse [15, Keynes, 1921, p. 83, n. 1].

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