The St. Petersburg Paradox as a Divergent Double Limit

Abstract

1. OFTEN, to analyze a paradox is to dispel it. The St. Petersburg Paradox, in which Paul refuses to pay Peter the extraordinarily high price that ordinary calculation of mathematical of moneygain computes for the privilege of Paul's receiving 21 ducats if heads first appears at the i-th toss of a fair coin, made an important contribution to the history of philosophy and probability. It dramatized the fact that men feel they value losses and gains at something different from their expected or arithmetic-mean values. Had all Pauls not already instinctively felt this, they would not have recoiled in shock from the prospect of paying Peter the infinite true mathematical odds, but it took the formulation of the problem by Cramer, Bernoulli, and others to rationalize what men already felt.' Now it may be cogently argued that a long series of--say 1 billioncoin tosses is really sufficient to drive home the point that rational men maximize their moral expectation (i.e., expected value of utility of wealth) rather than maximizing their simple money expectation. And yet, how much more dramatic is infinity than any large finite number! A present-day savant might grant the eye-opening service rendered by the St. Petersburg Paradox and still argue that it has by now done its work, and that-like most of the scaffoldings used to construct the edifice called the modern mind-it can now be dispensed with. Accepting the fact that each man acts as if he has an ultimately bounded utility function, or finite wealth and limited borrowing power, or only a finite life span in which to toss coins, such sophisticates argue that the St. Petersburg Paradox can never arise in real life. Paul will never be willing to give as much as Peter will demand for such a contract; and hence the indicated activity will take place at the equilibrivum level of zero intensity. Zero cancels out infinity, so to speak. 2. I do not disagree with this line of argument. But I propose to