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History of Political Economy 33.4 (2001) 743-772

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Frequentist Probability and Choice under Uncertainty

Alberto Baccini

Decision theory in conditions of uncertainty is now a regular part of microeconomics, at least when it comes to models of expected-utility maximization. Expected utility is the instrument that makes it possible to choose between actions whose consequences are uncertain. Rational choice selects from among those actions the one that will guarantee to the decision maker the greatest possible expected utility, where utility is the sum of the products of utility and the probability of all the possible consequences of the action.

The origin of the notion of expected utility has been universally attributed to Daniel Bernoulli, who introduced it in order to find a solution to the so-called St. Petersburg paradox. In the eighteenth and nineteenth centuries, debate arose in regard to this paradox and the application of probability theory to problems of human behavior or decision making. The critical point in the debate is the question of the utility function.1 Bernoulli's solution was to introduce a logarithmic utility function, in place of the expected value—which gave origin to the paradox. Paul [End Page 743] Samuelson (1977), who was absolutely “fascinated” by the question, dedicated a long article to dissecting the paradox; he centered his attention exclusively on the utility function and showed that to generalize Bernoulli's result, one had to introduce a utility function that was strictly concave and, according to the lesson of Karl Menger (Bassett 1987), bounded. According to Samuelson's perspective, the historical development of the expected-utility model would therefore be centered mainly on the question of the utility function. The other term of the question—probability—appeared in substance not to be problematic.

In my opinion, this representation of the history of the model of expected utility—and of the St. Petersburg paradox—is not completely correct and probably would not be presented to an English probabilist of the end of the nineteenth century in the terms in which it is recounted today. Between 1843—just before the date of publication of Robert Leslie Ellis's papers (1844, 1854), in which the notion of frequentist probability is introduced—and 1888—the date of the third edition of John Venn's Logic of Chance—the English tradition of the theory of probability put the Laplacean vision in crisis (Hacking 1990). That tradition also rethought the usability of the probability theory for the theory of human action and, consequently, discovered in probability, and not in utility, the critical point of the function of expected utility—and of the cogency of the paradox of St. Petersburg. John Venn's systematization of a new and winning theory of probability sanctioned the impossibility of using probability to construct a theory of human behavior. This was the consequence of the transition—forgive the expositional simplification—from the classical semantics of probability to the frequentist theory, which radically modified the sphere of applicability of the theory of probability.

Despite the profuse commitment in his, some forty-years long, Treatise on Probabilities,2 Francis Y. Edgeworth was not able to place probability theory among the instruments useful for decision theory. However, he opened the way to the revision by John Maynard Keynes, who had to build a semantics of probability that was—at least partly—new, in [End Page 744] order to remove it from the impasse into which Venn had thrust it and to recover it as a “guide of life” (Keynes 1921, 323).

The aim of the following pages is to give a brief illustration of a small part of this history, by concentrating on the question of the impossibility of a theory of expected utility within the context of the frequentist theory of probability. The authors to be considered are Venn and Edgeworth. The illustration will be made with reference to the St. Petersburg paradox.

A Question of Method

The fundamental idea at the core of the reconstruction rather brutally outlined above is that, only after having rebuilt the logical foundations of probability is it possible to...


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pp. 743-772
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