On the St. Petersburg paradox

Abstract

Set theory and measure theory were developed in the early part of this century, and subsequently A. N. Kolmogoroff [1] formulated an axiomatic foundation for the mathematical theory of probability. His work was further elaborated by others. Most mathematicians today are agreed that the foundations of probability theory are as well established as any other branch of mathematical analysis. While there are differences among practitioners concerning the range of applicability of probability models to real world phenomena and to the interpretations of conclusions obtained from them, and the differences are sometime wide, they are not dissimilar to those which arise when models from other mathematical disciplines are applied to the real world. This is only a recent stage in the history of mathematical probability. For much of the more than three centuries since its effective beginnings in an interchange of correspondence between Blaise Pascal and Pierre de Fermât, perhaps no mathematical subject has engendered greater controversy. Paradoxes and fallacious arguments abounded. (The term "paradox" as used here is akin to its original meaning of contrary to ordinary opinion or understanding.) Most of these have been satisfactorily resolved, but perhaps the most notorious of all, the "St. Petersburg paradox," is still discussed in the modern literature on probability and related fields, often evoking spirited arguments and counter arguments. While this is sometimes an embarrassment for practitioners, some comments of J. F. Steffensen, a Danish mathematician and actuary, in an analogous situation are relevant: