Countable Additivity and the de Finetti Lottery

Abstract

De Finetti would claim that we can make sense of a draw in which each positive integer has equal probability of winning. This requires a uniform probability distribution over the natural numbers, violating countable additivity. Countable additivity thus appears not to be a fundamental constraint on subjective probability. It does, however, seem mandated by Dutch Book arguments similar to those that support the other axioms of the probability calculus as compulsory for subjective interpretations. These two lines of reasoning can be reconciled through a slight generalization of the Dutch Book framework. Countable additivity may indeed be abandoned for de Finetti's lottery, but this poses no serious threat to its adoption in most applications of subjective probability. 1. Introduction 2. The de Finetti lottery 3. Two objections to equiprobability 3.1 The ‘No random mechanism’ argument 3.2 The Dutch Book argument 4. Equiprobability and relative betting quotients 5. The re-labelling paradox 5.1 The paradox 5.2 Resolution: from symmetry to relative probability 6. Beyond the de Finetti lottery