Carnap and de Finetti on Bets and the Probability of Singular Events: The Dutch Book Argument Reconsidered

Abstract

Recent contributions by Baillie and Ellis1 will have made readers of this Journal familiar with the so-called 'Dutch Book Argument'. This provides the rationality constraints for a widely accepted procedure of numerically measuring subjective degrees of belief in the occurrence of singular events by means of personal betting quotients in such a way that they can be taken as a concrete application of the abstract calculus of probability. The basic term here-probability-is defined by several axioms which state, roughly speaking, that the probabilities of mutually exclusive events are real numbers in the range from o to i, both bounds included, which sum up to I. A betting quotient q is the ratio of the sum hazarded (here called the 'stake') by the individual in question on the occurrence of the event E to the sum S (here called the 'total stake' or the 'gross gain') which will be won if E occurs. The more the individual believes in the occurrence of E, the more he will be prepared to risk for a particular sum S. Hence the betting quotient increases with degree of belief. Since the minimal degree of belief will result in risking nothing, and since the maximal degree will lead to a sum hazarded which does not exceed the expected gross gain, betting quotients satisfy by definition the first axiom of the calculus of probability. And it has been argued that the Dutch Book